we obtain the Laplace transform of f(t) in terms of s:
[tex]F(s) = (1/s) + (-2/s^2 + 1/s) * (e^(-2s) - e^(-3s)) + (1/s^2 - 1/s) * (e^(-3s) - e^(-5s))[/tex]
What is Laplace transform?
The Laplace transform is an integral transform that converts a function of time into a function of a complex variable s. It is a powerful mathematical tool used in various branches of science and engineering, particularly in the study of systems and signals.
(a) Expressing f(t) in terms of the unit step function ua(t):
The unit step function ua(t) is defined as:
ua(t) = 1 for t ≥ 0
ua(t) = 0 for t < 0
To express f(t) in terms of ua(t), we can break it down into different intervals:
For 0 ≤ t < 2:
f(t) = 1
For 2 ≤ t < 3:
f(t) = -2t + 1
For 3 ≤ t < 5:
f(t) = t - 1
Combining these expressions with ua(t), we get:
f(t) = 1 * ua(t) + (-2t + 1) * (ua(t - 2) - ua(t - 3)) + (t - 1) * (ua(t - 3) - ua(t - 5))
(b) Finding the Laplace transform of f(t) using the unit step function u(t):
The Laplace transform of f(t), denoted as F(s), is given by:
[tex]F(s) = ∫[0 to ∞] f(t) * e^(-st) dt[/tex]
To find the Laplace transform, we can apply the Laplace transform properties and formulas. Using the properties of the unit step function, we have:
[tex]F(s) = 1 * L{ua(t)} + (-2 * L{t} + 1 * L{1}) * (L{ua(t - 2)} - L{ua(t - 3)}) + (L{t} - L{1}) * (L{ua(t - 3)} - L{ua(t - 5)})[/tex]
Now, we can apply the Laplace transform formulas:
L{ua(t)} = 1/s
[tex]L{t} = 1/s^2[/tex]
L{1} = 1/s
Substituting these values, we get:
[tex]F(s) = (1/s) + (-2/s^2 + 1/s) * (e^(-2s) - e^(-3s)) + (1/s^2 - 1/s) * (e^(-3s) - e^(-5s))[/tex]
Simplifying further, we obtain the Laplace transform of f(t) in terms of s:
[tex]F(s) = (1/s) + (-2/s^2 + 1/s) * (e^(-2s) - e^(-3s)) + (1/s^2 - 1/s) * (e^(-3s) - e^(-5s)).[/tex]
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the cdf of the continuous random variable v is fv (v) = 0 v < −5, c(v + 5)2−5 ≤v < 7, 1 v ≥7. (a) what is c? (b) what is p[v > 4]?
The value of p(v > 4) is -6.
Given a continuous random variable v and its cumulative distribution function(CDF) fv(v):fv(v)=0, v < −5c(v + 5)2−5, -5 ≤ v < 71, v ≥7
(a) Calculation of c value:
Let's write the definite integral of CDF of v from -∞ to +∞. Therefore ,fv(v)=∫ fv(v) dv = 1
This can be separated into three definite integrals depending on the definition of fv(v):∫(-∞,-5) 0dv + ∫[-5,7]c(v+5)²-5dv + ∫(7,+∞) 1dv = 1
Simplifying it further:0 + ∫[-5,7]c(v+5)²-5dv + 1 = 1∫[-5,7]c(v+5)²-5dv = 0
We can calculate the integral of the function that is present in between the limits [-5, 7].∫[-5,7]c(v+5)²-5dv = c[ (v+5)³ / 3 ]∣[-5,7]
= c * [(7+5)³/3 - (-5+5)³/3]
= c * 108c
= 1/108
So, the value of c is 1/108.
(b) Calculation of p[v > 4]:Using the CDF and the known value of c, we can calculate the value of p(v > 4).p(v > 4) = 1 - p(v ≤ 4)
We can calculate the value of p(v ≤ 4) by using the CDF:fV(v)=∫ fv(v) dvWe have CDF in three parts.
So, we have to calculate the CDF of each part separately.
CDF of v for v < -5:fV(v)=∫ fv(v) dv= ∫ 0dv= 0∵ v< -5CDF of v for -5 ≤ v < 7:fV(v)=∫ fv(v) dv
= ∫c(v+5)²-5dv= (c/3) * (v+5)³ ∣[-5,7]= (1/108 * 216) / 3= 2CDF of v for v ≥7:fV(v)
=∫ fv(v) dv
= ∫ 1dv= v ∣ [7,+∞)∵ v≥7
Now, calculating the probability of v ≤ 4:fV(v) = 0, for v < −5
= (1/108 * 216) / 3, for -5 ≤ v < 7
= 6, for v ≥7p(v ≤ 4) = fV(4)= fV(7) - fV(-5)= 7 - 0= 7
We can now calculate p(v > 4):p(v > 4) = 1 - p(v ≤ 4)= 1 - 7= -6
Therefore, the value of p(v > 4) is -6.
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Healthy people have body temperatures that are normally distributed with a mean of 98.20 degrees F and a standard deviation of 0.62 degrees F. (a) If a healthy person is randomly selected, what is the probability that he or she has a temperature above 99.2 degrees F?
(b) A hospital wants to select a minimum temperature for requiring further medical tests. What should that temperature be, if we want only 0.5 % of healty people to exceed it?
To find the probability that a randomly selected healthy person has a temperature above 99.2 degrees F, we need to calculate the area under the normal distribution curve to the right of 99.2. We can use the z-score formula and standard normal distribution table to determine this probability. To determine the minimum temperature for requiring further medical tests such that only 0.5% of healthy people exceed it, we need to find the z-score corresponding to the desired cumulative probability of 0.5%. Using the standard normal distribution table, we can find the z-score and then convert it back to the original temperature scale using the mean and standard deviation of the healthy population.
To calculate the probability, we first calculate the z-score for 99.2 using the formula z = (x - μ) / σ, where x is the temperature value, μ is the mean, and σ is the standard deviation. Plugging in the values, we get z = (99.2 - 98.20) / 0.62, which simplifies to z ≈ 1.61. We then find the corresponding probability by looking up the area to the right of 1.61 in the standard normal distribution table. To find the minimum temperature, we need to find the z-score that corresponds to a cumulative probability of 0.5% (0.005). By looking up this cumulative probability in the standard normal distribution table, we find a z-score of approximately -2.58. We can then convert this z-score back to the original temperature scale using the formula x = μ + z * σ, where x is the temperature value, μ is the mean, σ is the standard deviation, and z is the z-score. Plugging in the values, we can find the minimum temperature.
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For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term. a) a = {7, 4, 1, ...}; Find the 17th term. b) a = {2, 6, 10, ...); Find the 12th term.
a) The 17th term of the sequence is -41.
b) The 12th term of the sequence is 46.
Explanation:
a) Recursive formula for the given arithmetic sequence a = {7, 4, 1, ...} is
a(n) = a(n-1) - 3.
The first term is 7.
Therefore, the 17th term can be found by substituting n = 17 in the recursive formula.
Hence,
a(17) = a(16) - 3
= a(15) - 3 - 3
= a(14) - 3 - 3 - 3
= ...
= a(1) - 3(16)
= 7 - 3(16)
= 7 - 48
= -41
Thus, the 17th term of the sequence is -41.
b)
Recursive formula for the given arithmetic sequence a = {2, 6, 10, ...} is
a(n) = a(n-1) + 4.
The first term is 2.
Therefore, the 12th term can be found by substituting n = 12 in the recursive formula.
Hence,
a(12) = a(11) + 4
= a(10) + 4 + 4
= a(9) + 4 + 4 + 4
= ...
= a(1) + 4(11)
= 2 + 4(11)
= 2 + 44
= 46
Thus, the 12th term of the sequence is 46.
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Which ONE of the following statements is TRUE? OA. The cross product of the gradient and the uint vector of the directional vector gives us the directional derivative. OB. None of the choices in this list. OC. The directional derivative as a scalar quantity is always in the direction vector u with u = 1. 0. Gradient of f(x...) at some point (a,b,c) is given by ai+bj+ck. OE. The directional derivative is a vector valued function in the direction of some point of the gradient of some given function.
The statement that is TRUE among the given options is "OD. Gradient of f(x...) at some point (a,b,c) is given by ai+bj+ck."
The gradient of a function f(x, y, z) is a vector that represents the rate of change of the function in each coordinate direction. It is denoted as ∇f and can be written as ∇f = ∂f/∂x i + ∂f/∂y j + ∂f/∂z k, where i, j, and k are the unit vectors in the x, y, and z directions, respectively.
In the statement OD, it is mentioned that the gradient of f(x, y, z) at a specific point (a, b, c) is given by ai + bj + ck. This aligns with the definition of the gradient, where the partial derivatives of the function are multiplied by the corresponding unit vectors.
The other options (OA, OB, OC, and OE) are not true:
- OA: The cross product of the gradient and the unit vector of the directional vector does not give the directional derivative. The directional derivative is obtained by taking the dot product of the gradient and the unit vector in the direction of interest.
- OB: This option states that none of the choices in the list are true, which contradicts the fact that one of the statements must be true.
- OC: The directional derivative as a scalar quantity is not always in the direction vector u with u = 1. The magnitude of the directional derivative gives the rate of change in the direction of the unit vector, but it can have a positive or negative sign depending on the direction of change.
- OE: The directional derivative is not a vector-valued function in the direction of some point of the gradient. The directional derivative is a scalar value that represents the rate of change of a function in a specific direction.
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ronnie is playing poker and is dealt his hand of 5 cards from a standard 52-card deck. what is the probability that ronnie is dealt 2 diamonds, 0 clubs, 1 heart, and 2 spades?
The probability of that Ronnie is dealt the combination specified is 5/52
Concept of probabilityProbability is the ratio of the required to the total possible outcomes.
Mathematically,
Probability = required outcome / Total possible outcomes
Required outcomes = 2+1+2 = 5
Total possible outcomes = 52
P(2 diamonds, 0 clubs, 1 heart, 2 spades) = 5/52
Therefore, the probability of 2 diamonds, 0 clubs, 1 heart, and 2 spades is 5/52.
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Find the area bounded by the given curve: 5x - 2y + 10 =0,3x+6y-8= 0 and 4x - 4y +2=0
The area bounded by the curves defined by the equations 5x - 2y + 10 = 0, 3x + 6y - 8 = 0, and 4x - 4y + 2 = 0 needs to be found.
To find the area bounded by the given curves, we can solve the system of equations formed by the three given equations. By solving them simultaneously, we can find the points of intersection of the curves. These points will form the vertices of the region.
Once we have the vertices, we can use various methods such as integration or geometric formulas to calculate the area of the bounded region. The exact approach will depend on the nature of the curves and the preferences of the solver.
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Given the rational function 1(x)= x-9 /x+7, find the
following:
(a) The domain.
(b) The horizontal and
vertical asymptotes.
(c) The x-and-y-intercepts.
(d) Sketch a complete graph of the function.
The domain of the function is all real numbers except x = -7. It has a horizontal asymptote at y = 1 and a vertical asymptote at x = -7. The x-intercept is (9, 0) and the y-intercept is (0, -9/7). A complete graph can be sketched considering these properties.
What are the key properties of the rational function 1(x) = (x-9)/(x+7), including its domain, asymptotes, and intercepts?(a) The domain of the rational function 1(x) = (x-9)/(x+7) is all real numbers except for x = -7, because dividing by zero is undefined. So the domain is (-∞, -7) U (-7, ∞).
(b) To find the horizontal asymptote, we compare the degrees of the numerator and denominator.
Since the degree of the numerator is 1 and the degree of the denominator is also 1, the horizontal asymptote is y = 1.
To find the vertical asymptote, we set the denominator equal to zero and solve for x. In this case, x + 7 = 0, which gives x = -7. So there is a vertical asymptote at x = -7.
(c) To find the x-intercept, we set the numerator equal to zero and solve for x. In this case, x - 9 = 0, which gives x = 9. So the x-intercept is (9, 0).
To find the y-intercept, we evaluate the function at x = 0. 1(0) = (0-9)/(0+7) = -9/7. So the y-intercept is (0, -9/7).
(d) Based on the given information, we can plot the x-intercept at (9, 0), the y-intercept at (0, -9/7), the vertical asymptote at x = -7, and the horizontal asymptote at y = 1.
We can also choose additional points to sketch a complete graph of the function, ensuring it approaches the asymptotes as x approaches infinity or negative infinity.
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4. Find the resulting matrix from applying the indicated row operations. 15 2 By 4-2 5 -7 -8 -5x + m 5. The 2 by 3 matrix provided is being used to solve a 2 by 2 system of linear equations. Use row operations as necessary to solve the system of equations. 56
To solve the system of linear equations using row operations, let's set up the augmented matrix:
[tex]\left[\begin{array}{ccc}15&2&4\\-2&5&-7\\-8&-5&x\end{array}\right][/tex]
We will apply row operations to transform this matrix into row-echelon form or reduced row-echelon form, which will provide the solution to the system of equations.
Let's perform the row operations step by step:
Multiply the first row by (-2) and add it to the second row:
[tex]\left[\begin{array}{ccc}15&2&3\\0&9&-15\\-8&-5&x\end{array}\right][/tex]
Multiply the first row by (8/15) and add it to the third row:
[tex]\left[\begin{array}{ccc}15&2&4\\0&9&-15\\0&-3.6&\frac{8x}{15}+\frac{77}{15} \end{array}\right][/tex]
Multiply the second row by (-1/3) and add it to the third row:
[tex]\left[\begin{array}{ccc}15&2&4\\0&9&-15\\0&0&\frac{8x}{15}+\frac{77}{15} \end{array}\right][/tex]
Now, the augmented matrix is in row-echelon form.
To find the solution to the system of equations, we can back-substitute:
From the third row, we have:
[tex]\frac{8x}{15}+\frac{77}{15} =0[/tex]
Solving this equation for x, we get:
[tex]\frac{8x}{15} = -\frac{77}{15}[/tex]
[tex]8x=-77\\x=-\frac{77}{8}[/tex]
The resulting matrix after applying the row operations is:
[tex]\left[\begin{array}{ccc}15&2&4\\0&9&-15\\0&0&\frac{8x}{15}+\frac{77}{15} \end{array}\right][/tex]
where x=-77/8
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Find the probability of drawing a spade or a red card from a
standard deck of cards.
a 1/7
b 3/4
c 1/52
d 1/8
the probability of drawing a spade or a red card from a standard deck of cards is 3/4. The answer is option b.
To find the probability of drawing a spade or a red card from a standard deck of cards, we need to determine the number of favorable outcomes (spades and red cards) and the total number of possible outcomes (all cards in the deck).
In a standard deck of cards, there are 52 cards in total, with 13 cards in each of the four suits (spades, hearts, diamonds, and clubs). Among these, there are 26 red cards (hearts and diamonds) and 13 spades.
To find the probability, we add the number of favorable outcomes (spades and red cards) and divide it by the total number of possible outcomes (52):
P(spade or red card) = (13 + 26) / 52
= 39 / 52
= 3 / 4
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A normally distributed quality characteristic is monitored with a moving average (MA) control chart. The monitored moving average at time t is defined as M
t
=
2
x
ˉ
t
+
x
ˉ
t−1
(sample size n=1.) Suppose the process mean is μ when t≤2 and then has a 1σ shift (i.e.: process mean is μ+1σ ) at t≥3. (a) Write out the 3-sigma upper control limits for this MA chart at t=1 and t≥2. (0.5 point) (b) Write out the distribution type, mean, and variation of M
t
when t≥3. (1 point) (c) Calculate the detection power of the control charts designed in (a) at t≥3. (1 point)
The provided information is insufficient to determine the exact 3-sigma upper control limits for the MA chart at t=1 and t≥2, the distribution type, mean, and variation of Mt when t≥3, and the detection power of the control charts at t≥3.
(a) The 3-sigma upper control limit for the MA chart at t = 1 can be calculated as follows:
UCL = μ + 3σ
Since the process mean is μ when t ≤ 2 and there is no shift yet, we can simply use the initial mean and standard deviation to calculate the UCL.
(b) When t ≥ 3, the distribution type of Mt (moving average at time t) will be normal. The mean of Mt can be calculated as follows:
Mean of Mt = μ + 1σ
This is because there is a 1σ shift in the process mean at t ≥ 3.
(c) To calculate the detection power of the control charts designed in (a) at t ≥ 3, we need additional information such as the sample size (n) and the desired level of statistical significance. With this information, we can perform a power analysis to determine the detection power of the control charts.
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Given av av 25202 +S= _V, ат as² as find a change of variable of S to x(S) so that this equation has constant coefficients. =
To find a change of variable that transforms the equation av av 25202 + S = √(as² + as) into an equation with constant coefficients, we can use a substitution method. By letting x = x(S), we can determine the appropriate transformation that will make the equation have constant coefficients.To begin, we need to determine the appropriate transformation that will eliminate the variable S and yield constant coefficients in the equation. Let's assume that x = x(S) is the desired change of variable.
We can start by differentiating both sides of the equation with respect to S to obtain:
dv/dS = d(√(as² + as))/dSNext, we can rewrite the equation in terms of x(S) by substituting S with the inverse transformation x⁻¹(x):
av av 25202 + x⁻¹(x) = √(as² + as).
By simplifying and rearranging the equation, we can find the specific transformation x(S) that will yield constant coefficients. The exact form of the transformation will depend on the nature of the equation and the specific values of a and s.Once the transformation x(S) is determined, the equation will have constant coefficients, allowing for easier analysis and solution.
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Explain why there is no solutions to the following systems of equations: 2x + 3y - 4z = -5 (1) x-y + 3z = -201 5x - 5y + 15z = -1004 (2) (3)
A three-dimensional vector, also known as a 3D vector, is a mathematical object that represents a quantity or direction in three-dimensional space.
To solve initial-value problems using Laplace transforms, you typically need well-defined equations and initial conditions. Please provide the complete and properly formatted equations and initial conditions so that I can assist you further.
For example, a 3D vector v = (2, -3, 1) represents a vector that has a magnitude of 2 units in the positive x-direction, -3 units in the negative y-direction, and 1 unit in the positive z-direction.
3D vectors can be used to represent various physical quantities such as position, velocity, force, and acceleration in three-dimensional space. They can also be added, subtracted, scaled, linear algebra, and computer graphics.
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8. In kilograms, the masses of Protons and Electrons are: Proton = 1.6 x 10-27 kg Electron = 9.1 x 10-31 kg About how many times greater is the mass of a Proton than the mass of an Electron? a) 2,000 times b) 600 times c) 200 times d) 6,000 times Tea
Ratio ≈ 1,800
To determine how many times greater the mass of a proton is compared to the mass of an electron, we can calculate the ratio of their masses.
Mass of a proton = 1.6 x 10^(-27) kg
Mass of an electron = 9.1 x 10^(-31) kg
To find the ratio, we divide the mass of a proton by the mass of an electron:
Ratio = (Mass of a proton) / (Mass of an electron)
= (1.6 x 10^(-27) kg) / (9.1 x 10^(-31) kg)
To simplify the calculation, we can rewrite the masses using scientific notation:
Ratio = (1.6 / 9.1) x (10^(-27) / 10^(-31))
= 0.1758 x 10^(4)
Since 0.1758 is approximately 0.18, we have:
Ratio ≈ 0.18 x 10^(4)
We can further simplify this by converting the scientific notation back to regular decimal notation:
Ratio ≈ 0.18 x 10^(4)
= 0.18 x 10,000
Simplifying the multiplication, we get:
Ratio ≈ 1,800
Therefore, the mass of a proton is approximately 1,800 times greater than the mass of an electron. So the answer is not one of the options provided.
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Which of the following is true about M₁= [1 2, 0 -1] and M₂= [4 1, 0 -3] in M2.5?
M₁ and M₂ are
a) Equal. b) linearly dependent. c) linearly independent. d) orthogonal.
39. Projection of the vector 2i+3j-2k on the vector i-2j+3k is
a. 2/√(14)
b. 1/√(14)
c. 3/√(14)
d. 4/√(14)
M₁ = [1 2, 0 -1] and M₂ = [4 1, 0 -3] in M2.5 are linearly independent.
Two matrices are said to be linearly independent if neither of them can be expressed as a scalar multiple of the other matrix. In this case, the matrices M₁ = [1 2, 0 -1] and M₂ = [4 1, 0 -3] in M2.5 are not equal as each matrix has different values. Further, the matrices are not scalar multiples of each other either. For instance, if we multiply M₁ by 1.5, we will not obtain M₂. Therefore, we can say that the matrices M₁ and M₂ are linearly independent.
Hence, it can be concluded that option c) linearly independent is the correct choice. Projection of the vector 2i+3j-2k on the vector i-2j+3k is given by Projv u = (v . u / |u|^2) * u, where v and u are vectors.
Let u = i-2j+3k and v = 2i+3j-2k.
Therefore,
[tex]u . v = 2(1) + 3(-2) + (-2)(3) = -8 and |u|^2 = (1)^2 + (-2)^2 + (3)^2 = 14.[/tex]
Now, Projv[tex]u = (v . u / |u|^2) * u= (-8 / 14)(i - 2j + 3k)= -4/7 i + 8/7 j - 12/7 k[/tex]
Therefore, the projection of the vector 2i+3j-2k on the vector i-2j+3k is given by option A) 2/√(14).
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Consider the initial-value problem xy" - xy + y = 0, (DE7)
(a) Verify that y₁ (x) = x is a solution of (DE7).
(b) Use reduction of order to find a second solution y2(x) in the form of an infinite series. Conjecture an interval of definition for y2(x).
(a) The solution y₁(x) = x can be verified by substituting it into the equation. (b) By assuming y₂(x) = v(x)y₁(x), where v(x) is an unknown function, and substituting this into the equation, an infinite series solution can be obtained. The interval of definition for y₂(x) can be conjectured as the interval where the series converges.
(a) To verify that y₁(x) = x is a solution of (DE7), we substitute it into the equation:
x(y₁") - x(y₁) + y₁ = 0
Differentiating y₁(x) twice gives y₁" = 0, so the equation becomes:
x(0) - x(x) + x = 0
Simplifying further, we have:
-x² + x + x = 0
-x² + 2x = 0
This equation is satisfied by y₁(x) = x, confirming that it is a solution.
(b) To find a second solution, we can use the method of reduction of order. We assume that y₂(x) = v(x)y₁(x), where v(x) is an unknown function. Substituting this into the equation, we have:
x(y₂") - x(y₂) + y₂ = 0
Substituting y₂(x) = v(x)x, and differentiating twice, we obtain:
x[v''(x)x + 2v'(x)] - x[v(x)x] + v(x)x = 0
Simplifying, we have:
x²v''(x) + 2xv'(x) - x³v(x) + x²v(x) = 0
Dividing through by x², we get:
v''(x) + (2/x)v'(x) - (1 - 1/x²)v(x) = 0
This equation can be solved by assuming a power series solution for v(x). By solving for the coefficients of the series, we can obtain a second solution y₂(x) in the form of an infinite series. The interval of definition for y₂(x) can be conjectured as the interval where the series converges.
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Suppose that for the bacterial strain Acinetobacter, five measurements gave readings of 2.69, 5.76, 2.67, 1.62 and 4.12 dyne-cm². Assume that the standard deviation is known to be 0.66 dyne-cm². a. Find a 95% confidence interval for the mean adhesion. b. If the scientists want the confidence interval to be no wider than 0.55 dyne-cm², how many observations should they take?
Note that the scientists need to take at least 10 observations if they want the confidence interval to beno wider than 0.55 dyne-cm².
Why is this so?The formula to be used is
n = (t(α/2) * s)² / (E)²
where -
n is the sample sizet(α/2) is the t-statistic for the desired confidence level and degrees of freedoms is the sample standard deviationE is the desired margin of error.Given statistics
n = ?t(α/2) = t(0.05/2) = 2.576s = 0.66 dyne-cm²E = 0.55 dyne-cm²n = (2.576 * 0.66)² / (0.55)²
= 9.55551744
n ≈ 10
This means that the scientists will need about 10 observations if they need the confidence interval to be no wider than 0.55 dyne-cm².
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The estimated annual bond default rate is 0.107.
a. What is the probability of bond survival rate (non-default)?
b. Determine the number of expected defaults in a bond portfolio with 25 issues.
c. Estimate the standard deviation of the number of defaults over the coming year d. What is the probability of observing more than 1 default?
An estimated annual bond default rate of 0.107, we can calculate various probabilities and statistics related to bond defaults. Firstly, we can find the probability of bond survival by subtracting the default rate from 1. Secondly, we can determine the expected number of defaults in a bond portfolio with 25 issues by multiplying the default rate by the number of issues. Thirdly, we can estimate the standard deviation of the number of defaults by using the formula for the standard deviation of a binomial distribution. Lastly, we can calculate the probability of observing more than 1 default by summing the probabilities of 2 or more defaults occurring.
The probability of bond survival (non-default) can be calculated by subtracting the default rate from 1. Therefore, the probability of bond survival is 1 - 0.107 = 0.893.
To determine the expected number of defaults in a bond portfolio with 25 issues, we multiply the default rate by the number of issues. The expected number of defaults is 0.107 * 25 = 2.675 (rounded to three decimal places).
The standard deviation of the number of defaults can be estimated using the formula for the standard deviation of a binomial distribution, which is sqrt(np(1-p)). Here, n is the number of issues (25) and p is the default rate (0.107). Therefore, the estimated standard deviation of the number of defaults is sqrt(25 * 0.107 * (1 - 0.107)) = 1.589 (rounded to three decimal places).
To calculate the probability of observing more than 1 default, we need to sum the probabilities of 2 or more defaults occurring. This can be done using the binomial distribution formula or by finding the complement of the probability of observing 1 or fewer defaults. The probability of observing more than 1 default is 1 - P(X ≤ 1), where X follows a binomial distribution with n = 25 and p = 0.107. By evaluating this expression, we can find the desired probability.
In conclusion, with an estimated annual bond default rate of 0.107, we can calculate the probability of bond survival, the expected number of defaults in a bond portfolio, the standard deviation of the number of defaults, and the probability of observing more than 1 default. These calculations provide insights into the likelihood of defaults and help assess the risk associated with the bond portfolio.
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fraction = β0 + β1total + β2size + u.
Perform the standard White test of the null hypothesis that the conditional variance of the error term in is homoskedastic against the alternative that it is a smooth function of the regressors. Specify any auxiliary regressions that you estimate in answering the question. State the null and alternative hypotheses in terms of restrictions on relevant parameters, specify the form and distribution of the test statistic under the null, the sample value and critical value of the test statistic, your decision rule and your conclusion. (8 marks)
The main objective is to conduct the White test to assess the null hypothesis that the conditional variance of the error term in the regression model is homoskedastic (constant) versus the alternative hypothesis that it is a smooth function of the regressors.
The regression model is specified as: fraction = β0 + β1total + β2size + u. The White test involves estimating auxiliary regressions to capture the relationship between the squared residuals and the regressors.
To perform the White test, we estimate the original regression model and obtain the residuals. Then, we regress the squared residuals on the regressors (total and size) and their cross-products. The null hypothesis states that the coefficients of the regressors and cross-products are all equal to zero, indicating homoskedasticity. The alternative hypothesis suggests that at least one of these coefficients is non-zero, implying heteroskedasticity.
The test statistic used in the White test follows a chi-square distribution under the null hypothesis. Its sample value is compared to the critical value at a given significance level to make a decision. If the sample value of the test statistic exceeds the critical value, we reject the null hypothesis of homoskedasticity in favor of the alternative hypothesis. On the other hand, if the sample value does not exceed the critical value, we fail to reject the null hypothesis.
The White test provides a statistical procedure to examine the presence of heteroskedasticity in the regression model by testing the null hypothesis of homoskedasticity against the alternative hypothesis of a smooth function of the regressors. By estimating auxiliary regressions and evaluating the test statistic's sample value against the critical value, we can make a decision regarding the presence of heteroskedasticity in the model.
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Let A Find the characteristic polynomial. 7 Det(A - 2) = (2-2)(+6) Find the eigenvalues and eigenvectors for each eigenvalue. (Order your answers from smallest to largest eigenvalue.) 26 has eigenspace span 2 = 2 X has eigenspace span 1 Find a matrix P such that p-'AP is a diagonal matrix - 1 P=
,P-1AP = D, where D is a diagonal matrix with eigenvalues of A on the diagonal. P-1AP = D => (1/3)[-1 1; -1 2][[2 1; 1 -1][2 -2; -1 5/2]][-1 1; -1 2] = [2 0; 0 5/2]Therefore,P-1AP = D = [2 0; 0 5/2]
Given, 7 Det(A - 2) = (2-2)(+6)
To find the characteristic polynomial of matrix A, we can use the formula Det(A-λI)Where I is the identity matrix of the same order as A and λ is a scalar.
So, A-λI = [a_ij - λδ_ij]
For a 2x2 matrix, A-λI = [a₁₁ - λ a₁₂, a₂₁ a - λ]
Thus the characteristic equation is:
det([a₁₁ - λ a₁₂, a₂₁) a₂₂ - λ])
= (a₁₁ - λ)(a₂₂ - λ) - a₁₂ a₂₁)
= λ² - (a₁₁ + a₂₂)λ + (a₁₁ a₂₂ - a₁₂ a₂₁)
The characteristic polynomial of A is obtained by equating the above equation to zero.
That is, P(λ) = det([a₁₁ - λ a₁₂, a₂₁ a₂₂ - λ])
= λ² - (a₁₁ + a₂₂)λ + (a₁₁ a₂₂ - a₁₂ a₂₁)
Here, 7 Det(A - 2)
= (2-2)(+6)
= 0,
so we know that λ = 2 is an eigenvalue of A.
Now to find eigenvectors for the eigenvalue λ=2,
we need to solve the equation(A-λI)x = 0, where λ = 2
This can be written as(A-2I)x = 0, where I is the identity matrix of same order as A.
Now, A - 2I = [2 -2, 1 1]
Let's row reduce to get row echelon form.
So, x₁ - 2x₂ = 0
or x₁ = 2x₂
Therefore, eigenvectors corresponding to λ = 2 is of the form [x₁ ; x₂] = [2x₂; x₂] = x₂ 2[2; 1]
Thus, eigenvectors corresponding to λ = 2 is [2; 1]T
So, the eigenvalues of the given matrix are λ=2, λ=5/2 and
the corresponding eigenvectors for each eigenvalue are: [2, 1]T and [1, -1]T respectively.
To find the matrix P, we take the eigenvectors and form the matrix whose columns are these eigenvectors. So, P = [2 1; 1 -1]
Now, P-1 = (1/3)[-1 1; -1 2]
Then, P-1AP = D, where D is a diagonal matrix with eigenvalues of A on the diagonal.
P-1AP = D
=> (1/3)[-1 1; -1 2][[2 1; 1 -1][2 -2; -1 5/2]][-1 1; -1 2]
= [2 0; 0 5/2]
Therefore, P-1AP = D
= [2 0; 0 5/2]
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The enzymatic activity of a particular protein is measured by counting the number of emissions of a radioactively labeled molecule. For a particular tissue sample, the counts in consecutive time periods of ten seconds can be considered (approximately) as repeated independent observations from a normal distribution. Suppose the mean count (H) of ten seconds for a given tissue sample is 1000 emissions and the standard deviation (o) is 50 emissions. Let Y be the count in a period of time of ten seconds chosen at random, determine: 11) What is the dependent variable in this study. a. Protein b. the tissue c. The number of releases of the radioactively labeled protein d. Time
Based on the information provided, the dependent variable is the number of releases of the radioactively labeled protein.
What is the dependent variable and how to identify it?The dependent variable refers to the main phenomenon being studied, which is often modified or affected by other variables involved. To identify this variable just ask yourself "What is the main variable being measured'?".
According to this, in this case, the dependent variable is " the number of releases of the radioactively labeled protein."
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Find the first, second, and third quartiles for the sales amounts in the data provided and interpret the results.
Click the icon to view the data.
The first quartile is _____$ , meaning that ____% of the sales amounts are less than this value. (Round to two decimal places as needed.)
We can fill in the blanks as follows: The first quartile is 29.50$, meaning that 50% of the sales amounts are less than this value.
The given data are as follows:17, 20, 23, 28, 29, 30, 32, 34, 35, 36, 39, 40, 40, 44, 45, 50, 54, 57, 60, 70
The first step in computing the quartiles is to sort the data in ascending order. Thus, the sorted data is:
17, 20, 23, 28, 29, 30, 32, 34, 35, 36, 39, 40, 40, 44, 45, 50, 54, 57, 60, 70
The number of observations in the dataset is 20 and thus, the median can be found as follows:
Median = Q2 = (n + 1)/2th observation = (20 + 1)/2th observation = 10.5th observation
The 10.5th observation is between the 10th and 11th observation, which are 39 and 40, respectively. Thus, the median is (39 + 40)/2 = 39.5.
Interquartile range (IQR) is given by: IQR = Q3 − Q1
The 1st quartile (Q1) is the median of the lower half of the data and thus, it is the median of the data below 39.5. The data below 39.5 is:17, 20, 23, 28, 29, 30, 32, 34, 35, and 36.The median of the above data can be found as follows:
Q1 = median of the data below 39.5 = (n + 1)/2th observation = (10 + 1)/2th observation = 5.5th observation The 5.5th observation is between the 5th and 6th observation, which are 29 and 30, respectively.
Thus, the Q1 is (29 + 30)/2 = 29.5. The third quartile (Q3) is the median of the upper half of the data and thus, it is the median of the data above 39.5. The data above 39.5 is:40, 40, 44, 45, 50, 54, 57, 60, and 70.The median of the above data can be found as follows:Q3 = median of the data above 39.5 = (n + 1)/2th observation = (10 + 1)/2th observation = 5.5th observation The 5.5th observation is between the 5th and 6th observation, which are 50 and 54, respectively. Thus, the Q3 is (50 + 54)/2 = 52.
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3. Given a geometric sequence with g3= 4/3, g = 108, find g₁, the specific formula for g, and g₁1.
A geometric sequence is a list of numbers in which each term is obtained by multiplying the previous term by a fixed number r.
For example, 2, 4, 8, 16, 32 is a geometric sequence with a common ratio of 2.To find g₁, the first term of the sequence, we need to use the formula: gₙ = g₁ * r^(n-1), where gₙ is the nth term of the sequence and r is the common ratio.
We are given that g₃ = 4/3, so we can plug in n = 3 and gₙ = 4/3 to get:4/3 = g₁ * r^(3-1)4/3 = g₁ * r²To find the common ratio r, we divide the nth term by the (n-1)th term.
We are given that g = 108, so we can use g₃ and g to get:108 = g₃ * r^(6-3)108 = (4/3) * r³81 = r³r = 3Plugging this value of r into the equation we got for g₁, we get:4/3 = g₁ * (3²)4/3 = 9g₁g₁ = (4/3) / 9g₁ = 4/27Now we have g₁ = 4/27, r = 3, and n = 11 (since we need to find g₁₁).
We can use the formula we got for gₙ to find g₁₁:g₁₁ = g₁ * r^(n-1)g₁₁ = (4/27) * 3^(11-1)g₁₁ = (4/27) * 177147g₁₁ = 26244We can also find the specific formula for g using the formula: gₙ = g₁ * r^(n-1). Plugging in g₁ = 4/27 and r = 3, we get:gₙ = (4/27) * 3^(n-1)This is the specific formula for g.
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Given a geometric sequence with g3= 4/3, g = 108, to find g₁, the specific formula for g, and g₁1.
Step 1: We need to find common ratio
We have given g = 108 and g3 = 4/3To find the common ratio, r, we use the
formula; g3 = g * r²4/3 = 108 * r²r = (4/3) / 108r = 1 / (3 * 27)
Step 2: Find g₁To find g₁, we use the formula;gn =[tex]g * r^(n-1)g₁ = g * r^(1-1)g₁ = g * r⁰g₁ = g * 1g₁ = 108 * 1g₁ = 108[/tex]
Step 3: Specific formula for g
The specific formula for g is;gn = g * r^(n-1)Substituting the values we get;g(n) = 108 * (1 / (3 * 27))^(n-1)g(n) = 108 * (1 / (3^(n-1) * 27^(n-1)))g(n) = 108 / (3^(n-1) * 3^3)g(n) = (4/3) / 3^(n-1)Step 4: g₁₁We have to find the 11th term of the sequence
To find the 11th term, we use the formula;
[tex]g11 = g * r^(11-1)g11 = 108 * (1 / (3 * 27))^(11-1)g11 = 108 * (1 / 3^10)g11 = 108 / 59049Hence, g₁ = 108,
the specific formula for g is;
g(n) = (4/3) / 3^(n-1) and g₁₁ = 108 / 59049[/tex]
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solve for x and y using radicals as needed.
The values of x and y are x = √15 and y = 2√5.
Given that a right triangle with an altitude of x and dividing the hypotenuse into 5 and 3, with a leg of y,
According to the property of a right triangle,
x² = 5 × 3
x = √15
Using the Pythagoras theorem,
y² = √15² + 5²
y² = 15 + 25
y² = 40
y = 2√5
Hence the values of x and y are x = √15 and y = 2√5.
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Demonstrate the use of dimensional analysis to determine the
length of the 2.7 meter line in inches. Round to the nearest tenth.
Show your work
The use of dimensional analysis to determine the length of the 2.7-meter line in inches is 106.3 inches.
Dimensional analysis is a powerful tool used in physics to convert units from one system to another. In this case, we will use dimensional analysis to convert the length of a line given in meters to inches.
We start with the given length of the line: 2.7 meters. We know that 1 meter is equal to 39.37 inches. Using this conversion factor, we can set up a dimensional analysis equation:
2.7 meters × (39.37 inches / 1 meter)
To cancel out the meters, we multiply by the conversion factor of (39.37 inches / 1 meter):
2.7 meters × 39.37 inches = 106.29 inches
Now, rounding to the nearest tenth, we get:
The length of the 2.7-meter line is approximately 106.3 inches.
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Determine the maximin and minimax strategies for the two-person, zero-sum matrix game. [
2
4
6
5
−4
−6
] The row player's maximin strategy is to play row The column player's minimax strategy is to play column. Determine the maximin and minimax strategies for the two-person, zero-sum matrix game.
⎣
⎡
5
1
6
2
−3
3
1
4
1
⎦
⎤
The row player's maximin strategy is to play row The column player's minimax strategy is to play column
The maximin and minimax strategies for the two-person, zero-sum matrix game can be determined as follows:1.
Matrix [2 4; 6 5; -4 -6]:For the matrix [2 4; 6 5; -4 -6], the row player's maximin strategy is to play row 2, and the column player's minimax strategy is to play column 1.
To determine the row player's maximin strategy, we need to identify the minimum payoff for each row and then select the row with the maximum minimum payoff.
The minimum payoffs for each row are 2, 5, and -6. Therefore, the row player's maximin strategy is to play row 2 since it has the highest minimum payoff.
To determine the column player's minimax strategy, we need to identify the maximum payoff for each column and then select the column with the minimum maximum payoff.
The maximum payoffs for each column are 6, 5, and -4. Therefore, the column player's minimax strategy is to play column 1 since it has the lowest maximum payoff.2.
Matrix [5 1 6; 2 -3 3; 1 4 1]:For the matrix [5 1 6; 2 -3 3; 1 4 1], the row player's maximin strategy is to play row 1, and the column player's minimax strategy is to play column 2.
Summary:The maximin strategy of the row player in the matrix [2 4; 6 5; -4 -6] is to play row 2 while the minimax strategy of the column player is to play column 1.The maximin strategy of the row player in the matrix [5 1 6; 2 -3 3; 1 4 1] is to play row 1 while the minimax strategy of the column player is to play column 2.
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dy quotient rule; rather; rewrite the function by using a negative exponent and then use Find without using thc dx the product rule and the general power rule to find the derivative: y = (c +5)3 dy dz Preview'
The derivative of y = (c + 5)^3 with respect to z is 0.
To find the derivative of the function y = (c + 5)^3 with respect to z, we can first rewrite the function using a negative exponent:
y = (c + 5)^3
= (c + 5)^(3/1)
Now, let's use the product rule and the general power rule to differentiate y with respect to z.
Product Rule: If u = f(z) and v = g(z), then the derivative of the product u * v with respect to z is given by:
(d/dz)(u * v) = u * (dv/dz) + v * (du/dz)
General Power Rule: If u = f(z) raised to the power n, then the derivative of u^n with respect to z is given by:
(d/dz)(u^n) = n * u^(n-1) * (du/dz)
Applying the product rule and the general power rule, we have:
dy/dz = (d/dz)[(c + 5)^(3/1)]
= (3/1) * (c + 5)^(3/1 - 1) * (d/dz)(c + 5)
The derivative of (c + 5) with respect to z is 0 since it does not depend on z. Therefore, the derivative simplifies to:
dy/dz = 3 * (c + 5)^2 * 0
= 0
So, the derivative of y = (c + 5)^3 with respect to z is 0.
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Complete the table to find the derivative of the function. y=√x/x Original Function Rewrite Differentiate Simplify
To find the derivative of the function y = √(x) / x, we can break it down into three steps:
1. Rewrite: y = x^(-1/2) * x^(-1/2)
2. Differentiate: y' = (-1/2)x^(-3/2) + (-1/2)x^(-3/2)
3. Simplify: y' = -x^(-3/2)
To find the derivative of the function y = √(x) / x, we can break it down into three steps: rewriting the function, differentiating the rewritten function, and simplifying the result.
Rewrite the function
In this step, we can rewrite the function using exponent rules. We can express √(x) as x^(1/2) and rewrite the function as y = x^(-1/2) * x^(-1/2).
Differentiate the rewritten function
To differentiate the function, we need to apply the power rule of differentiation. The power rule states that when we have a function of the form f(x) = x^n, the derivative is given by f'(x) = nx^(n-1). Applying the power rule to our function, we differentiate each term separately. The derivative of x^(-1/2) is (-1/2)x^(-3/2), and the derivative of x^(-1/2) is also (-1/2)x^(-3/2).
Simplify the result
In this step, we combine the two terms obtained in the previous step. Both terms have the same derivative, so we can add them together. This gives us y' = (-1/2)x^(-3/2) + (-1/2)x^(-3/2), which simplifies to y' = -x^(-3/2).
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what proportion of a normal distribution is located between z = –1.50 and z = 1.50
Approximately 86.6% proportion of a normal distribution is located between z = –1.50 and z = 1.50.
The proportion of a normal distribution located between z = –1.50 and z = 1.50 is approximately 0.866 or 86.6%. Normal distribution has a mean of 0 and a standard deviation of 1.
A z-score is a measure of how many standard deviations a given data point is from the mean of the distribution. To find the proportion of a normal distribution located between z = –1.50 and z = 1.50, we need to find the area under the curve between these two z-scores.
This can be done by using a standard normal distribution table or a calculator with a normal distribution function. Using a standard normal distribution table, we can find the area to the left of z = 1.50, which is 0.9332.
Similarly, the area to the left of z = –1.50 is also 0.9332. Therefore, the area between z = –1.50 and z = 1.50 is:0.9332 - 0.0668 = 0.8664 (rounded to four decimal places).
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If the work required to stretch a spring 3 ft beyond its natural length is 9 ft-lb, how much work is needed to stretch it 18 in. beyond its natural length?
The work that is done in stretching of the spring is 3.4 J.
What is Hooke's law?Hooke's Law states that when a spring or elastic material is squeezed or stretched, it will produce a force that is directed in the opposite direction from the displacement. The displacement influences the stiffness of the material, and the force's strength is proportional to the displacement.
Using the Hooke's law;
F = ke
k = F/e
k= 9/3
k = 3 ft-lb/ft
We have the extension now as 18 in or 1.5 ft
W = 1/2k[tex]e^2[/tex]
W = 0.5 * 3 *[tex](1.5)^2[/tex]
W = 3.4 J
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Problem 9. (12 points) Please answer the following questions about the function f (x) = 2x-4 / x+7
Instructions. If you are asked to find x- or y-values, enter either a number, a list of numbers separated by commas, or None it there aren't any solutions. Use interval notation if you are asked to find an interval or union of intervals, and enter { } if the interval is empty (a) Find the critical numbers of f, where it is increasing and decreasing, and its local extrema. Critical numbers x = 0
Increasing on the interval (-inf,0) Decreasing on the interval (0,int) Local maxima x = 0 Local minima x = (b) Find where f is concave up, concave down, and has infection points. Concave up on the interval ......
Concave down on the interval (-infint) Inflection points = none (C) Find any horizontal and vertical asymptotes of f. Horizontal asymptotes y = .....
Vertical asymptotes x = ...... (d) The function f is even because f(-x) = f(x) for all in the domain of f, and therefore its graph is symmetric about the y-axis (e) Sketch a graph of the function f without having a graphing calculator do it for you. Plot the y-intercept and the x-intercepts, they are known. Draw dashed lines for horizontal and vertical asymptotes. Plot the points where f has local maxima, local minima, and inflection points. Use what you know from parts (a) and (b) to sketch the remaining parts of the graph of f. Use any symmetry from part (d) to your advantage, Sketching graphs is an important skill that takes practice, and you may be asked to a it on quizzes or exams.
Previous question
The function f(x) = (2x - 4) / (x + 7) has a critical number at x = 0. It is increasing on the interval (-∞, 0) and decreasing on the interval (0, ∞). It has a local maximum at x = 0. The function is concave up on the interval (-∞, ∞) and does not have any inflection points. It has a horizontal asymptote at y = 2 and a vertical asymptote at x = -7. The function f is even, so its graph is symmetric about the y-axis.
To find the critical numbers of f, we set the derivative of f(x) equal to zero:
f'(x) = (2(x + 7) - (2x - 4)) / (x + 7)^2 = 0.
Simplifying, we get 4 / (x + 7)^2 = 0, which has no real solutions. Therefore, the critical number is x = 0.
To determine where f is increasing or decreasing, we check the sign of the derivative on the intervals (-∞, 0) and (0, ∞). Taking a test point within each interval, we find that f'(x) is positive on (-∞, 0) and negative on (0, ∞). Thus, f is increasing on (-∞, 0) and decreasing on (0, ∞).
Since there is only one critical number, x = 0, it is also the location of the local maximum.
To find where f is concave up or concave down, we take the second derivative of f(x):
f''(x) = [4(x + 7)^2 - 4] / (x + 7)^4.
The second derivative is always positive for all x, indicating that f is concave up on the interval (-∞, ∞) and does not have any inflection points.
The horizontal asymptote is determined by the limits as x approaches infinity and negative infinity. Taking the limit as x approaches infinity, we find that f(x) approaches 2. Therefore, y = 2 is the horizontal asymptote. As for the vertical asymptote, it occurs when the denominator of f(x) equals zero, which is at x = -7.
Finally, since f(-x) = f(x) for all x in the domain of f, the function f is even, resulting in symmetry about the y-axis.
To sketch the graph of f, we plot the y-intercept and x-intercepts (if any) by setting f(x) equal to zero. We draw dashed lines for the horizontal asymptote y = 2 and the vertical asymptote x = -7. We mark the point of the local maximum at x = 0. Since there are no inflection points, we do not plot any. Using the information about increasing, decreasing, concave up, and concave down, we sketch the remaining parts of the graph. Taking advantage of the symmetry about the y-axis, we complete the graph.
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