3. Create your own piecewise function with at least two functions. Explain, using complete
sentences, the steps for graphing the function. Graph the function by hand or using a
graphing software of your choice (remember to submit the graph).

The value of c for same sign of roots are,

⇒ c > 0

Given that;

Quadratic equation is,

⇒ x² - 2x + c = 0

Since, We know that;

For the roots of ax²+bx+c=0 to have same signs,

a(x2+b/ax+c/a), the last term, i.e. c/a>0, because if you factorize the quadratic, to arrive at positive constant you either have to have two negative numbers multiplied or two positive multiplied by each other.

Here,

Quadratic equation is,

⇒ x² - 2x + c = 0

Hence, The condition for same sign of roots are,

⇒ c > 0

Thus, The value of c for same sign of roots are,

⇒ c > 0

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Answer:

2. (i)  p = 4

   (ii)  p < -4

   (iii)  0 < c ≤ 1

Step-by-step explanation:

To find the value(s) of p for which x² - 2x - 3 = p will have the equal or real roots, we can use the discriminant formula.

[tex]\boxed{\begin{minipage}{9 cm}\underline{Discriminant}\\\\$b^2-4ac$ \quad when $ax^2+bx+c=0$\\\\when $b^2-4ac > 0 \implies$ two real roots.\\when $b^2-4ac=0 \implies$ one real root (equal roots).\\when $b^2-4ac < 0 \implies$ no real roots (two complex roots).\\\end{minipage}}[/tex]

Rearrange the given equation so that it is in the form ax² + bx + c = 0:

[tex]x^2-2x-3-p=0[/tex]

Therefore, the values of a, b and c are:

To find the value of p for which the given quadratic has equal roots, substitute the values of a, b and c into the discriminant formula, set it equal to zero, and solve for p:

[tex]\begin{aligned}b^2-4ac&=0\\(-2)^2-4(1)(-3-p)&=0\\4-4(-3-p)&=0\\4+12+4p&=0\\16+4p&=0\\4p&=-16\\p&=-4\end{aligned}[/tex]

Therefore, the value of p for which the given quadratic will have equal roots is p = -4.

Rearrange the given equation so that it is in the form ax² + bx + c = 0:

[tex]x^2-2x-3-p=0[/tex]

Therefore, the values of a, b and c are:

To find the values of p for which the given quadratic has no real roots, substitute the values of a, b and c into the discriminant formula, set it to less than zero, and solve for p:

[tex]\begin{aligned}b^2-4ac& < 0\\(-2)^2-4(1)(-3-p)& < 0\\4-4(-3-p)& < 0\\4+12+4p& < 0\\16+4p& < 0\\4p& < - 16\\p& < -4\end{aligned}[/tex]

Therefore, the value of p for which the given quadratic will no real roots is p < -4.

For the roots of x² - 2x + c = 0 to have the same sign, both roots either have to be less that 0 or more than 0.

From part (i), we know that x² - 2x - 3 - p = 0 has equal roots when p = -4.

Substitute p = -4 into the equation:

[tex]\begin{aligned}x^2 - 2x - 3 - p &= 0\\x^2-2x-3-(-4)&=0\\x^2-2x-3+4&=0\\x^2-2x+1&=0\end{aligned}[/tex]

Comparing this to x² - 2x + c = 0, we can see that x² - 2x + c = 0 has equal roots when c = 1.

The leading coefficient of x² - 2x + c = 0 is positive, so the parabola opens upwards. We know it has equal roots when c = 1, so the vertex touches the x-axis when c = 1. Therefore, it has no real roots when c > 1 (since the vertex will be above the x-axis in this interval).

Therefore, we can determine that x² - 2x + c = 0 has two real roots when c ≤ 1.

When c = 0, the equation of the parabola is y = x² - 2x.

The roots are the points at which the curve crosses the x-axis (when y = 0). As y = 0 when x = 0, one of the roots is (0, 0) when c = 0.

Therefore, when c < 0, one root will be negative and the other will be positive.

Note there is no value of c where both roots are negative, as the x-value of the vertex is positive.

So the values of c for which the roots of x² - 2x + c = 0 will have the same sign (positive) are 0 < c ≤ 1.

The perimeter of the figure is 24.50 feet feet, and the area of the figure is 45.68 square feet

From the question, we have the following parameters that can be used in our computation:

Radius, r = 4 feet (see attachment)

The perimeter of the figure is

Perimeter = Circumference of 3/4 circle + Perimeter of triangle - 2r

So, we have

Perimeter  = 3/4 * 2 * 3.14 * 4 + 4 + 4 + 4√2 - 2 * 4

Evalate

Perimeter  = 24.50 feet

The area of the figure is:

Area = Area of 3/4 circle + Area of triangle

So, we have

Area = 3/4 * 3.14 * 4² + 1/2 * 4 * 4

Evaluate

Area = 45.68

Hence, the area is 45.68 square feet

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The requried probability that the mean of a sample of 70 computers would differ from the population means by less than 1.9 months is 0.8671, rounded to four decimal places.

The mean of the sampling distribution of the sample means is:

u = 105

and the variance of the sampling distribution of the sample means is:

σ² = 81/70

We can standardize the distribution of the sample means by using the z-score formula:

z = (x - u) / (σ/ √(n))

where x is the sample mean, mu is the population mean, sigma is the population standard deviation, and n is the sample size.

We want to find the probability that the sample mean differs from the population mean by less than 1.9 months, or |x - u| < 1.9. This is equivalent to finding the probability that the standardized sample mean falls within the range:

-1.9 / (σ/ √(n)) < z < 1.9 / (σ/ √(n))

Substituting the values we have, we get:

-1.9 / (9 / √(70)) < z < 1.9 / (9 / √(70))

Simplifying, we get:

-1.1158 < z < 1.1158

Using a standard normal distribution table, we can find the probability that the z-score falls within this range:

P(-1.1158 < z < 1.1158) = 0.8671

Therefore, the probability that the mean of a sample of 70 computers would differ from the population means by less than 1.9 months is 0.8671, rounded to four decimal places.

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The solution to the system of equations, using the Gauss-Jordan method, is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&14.28\\0&1&0&-21.4\\0&0&1&1.2\end{array}\right][/tex]

The matrix representing the system of equations is given as follows:

[tex]\left[\begin{array}{cccc}-5&-3&-4&-12\\0&-2&-7&38\\0&1&4&-22\end{array}\right][/tex]

First we want a value of 1 at line 1, column 1, hence we multiply the first line by -1/5, that is:

R1 -> -1/5R1

Hence:

[tex]\left[\begin{array}{cccc}1&0.6&0.8&2.4\\0&-2&-7&38\\0&1&4&-22\end{array}\right][/tex]

First we want a value of 1 at line 2, column 2, hence we multiply the second line by -1/2, that is:

R2 -> -1/2R2

Hence:

[tex]\left[\begin{array}{cccc}1&0.6&0.8&2.4\\0&1&1.5&-19\\0&1&4&-22\end{array}\right][/tex]

We want an element of zero at line 3, column 2, hence:

R3 -> R3 - R2

Hence:

[tex]\left[\begin{array}{cccc}1&0.6&0.8&2.4\\0&1&1.5&-19\\0&0&2.5&-3\end{array}\right][/tex]

First we want a value of 1 at line 3, column 3, hence we multiply the third line by 2.5, that is:

R3 -> 1/2.5R3

Hence:

[tex]\left[\begin{array}{cccc}1&0.6&0.8&2.4\\0&1&1.5&-19\\0&0&1&1.2\end{array}\right][/tex]

Then the solution to the system of equations is given as follows:

Hence the row-echelon form is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&14.28\\0&1&0&-21.4\\0&0&1&1.2\end{array}\right][/tex]

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Answer:

0.17

Step-by-step explanation:

Probability for Event A: 1/3

Probability for Event B: 1/2

Total Probability: 1/6

Answer:

B)

Step-by-step explanation:

 Geometric mean:

     [tex]\sf 5 , a_2,a_3, a_4, a_5,1215[/tex]

       [tex]\bf a_1 = first \ term = 5[/tex]

n = number of terms = 6

r = common ratio

            [tex]\boxed{\bf a_n = a_1 * r^{n-1}}[/tex]

             [tex]a^{6} = 5 *r^{6-1}\\\\1215 = 5*r^5\\\\ \dfrac{1215}{5}=r^5\\\\243=r^5\\\\3^5=r^5[/tex]

As powers are same, compare the bases,

        r = 3

Each term is obtained by multiplying the previous term by the common ratio. r = 3

    [tex]\sf a_2 = 5*3 = 15\\\\a_3= 15*3 = 45\\\\a_4 = 45 *3 =135\\\\ a_5 = 135*3 = 405[/tex]

The area of the triangle given at the end of the answer is of:

B. 24 units².

The area of a rectangle of base b and height h is given by half the multiplication of dimensions, as follows:

A = 0.5bh.

Considering the image presented for this problem, the dimensions for the triangle are given as follows:

b = 12, h = 4.

Hence the area is given as follows:

A = 0.5 x 4 x 12

A = 24 units².

This means that option B is the correct option in the context of this problem.

The triangle is given by the image presented at the end of the answer.

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Answer:

8

Step-by-step explanation:

I will just calculate the population of every detail for ease.
800 total students.
800 * 60% or 0.6 = 480
480 girls.
480 * 50% or 0.5 = 240
240 music class girls.
240 * 10% or 0.1 = 24
24 music class girls.
so 24 music class girls that are into volleyball and 32 team girls.
32-24 = 8
8 of the team girls are not into music.

Answer:

train is the mode of transportation that can be use

Answer:

r(x) = x(x - 7)

Step-by-step explanation:

given the roots of a parabola x = a , x = b , then

the corresponding factors are (x - a) and (x - b)

from the graph the roots are where the graph crosses the x- axis

the graph crosses the x- axis at x = 0 and x = 7 , then

the corresponding factors are (x - 0) and (x - 7) , that is x and (x - 7)

the equation is then the product of the factors , that is

r(x) = x(x - 7)

Austin's rate of motion is (1/70)π Radians per second.

To determine Austin's rate of motion in radians per second, we need to use the formula for angular velocity:

ω = Δθ / Δt

Where:

ω = angular velocity (in radians per second)

Δθ = change in angular displacement (in radians)

Δt = change in time (in seconds)

We know that Austin runs three-quarters of a circular track, which means he covers an arc length that is equal to three-quarters of the circumference of the circle. Let's call the radius of the circle "r". Then, the arc length covered by Austin is given by:

s = (3/4) * 2πr

s = (3/2)πr

We also know that it takes Austin 1 minute and 45 seconds to cover this distance. This is the same as 105 seconds (since 1 minute = 60 seconds).

So, Δt = 105 seconds

Now, we can calculate the change in angular displacement (Δθ). The total angle around a circle is 2π radians, so the angle covered by Austin is given by:

Δθ = (3/4) * 2π

Δθ = (3/2)π

Therefore, Austin's rate of motion (ω) in radians per second is:

ω = Δθ / Δt

ω = [(3/2)π] / 105

ω = (3/210)π

ω = (1/70)π radians per second

So, Austin's rate of motion is (1/70)π radians per second.

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The height of the cylinder is determined as 36.2 inches.

The height of the cylinder is calculated by applying the formula for surface area of a cylinder as shown below;

S.A = 2πr(r + h)

where;

The height of the cylinder is calculated as follows;

2,285.92 = 2π x 8.2 (8.2 + h)

2,285.92 = 51.52(8.2 + h)

8.2 + h = 2,285.92/51.52

8.2 + h = 44.37

h = 36.2 inches

Thus, the height of the cylinder for the given radius and surface area is determined by using the equation for surface area of a cylinder.

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The complete question is below:

The surface area of a cylinder is 2,285.92 square inches. What is the height? if the radius of the cylinder is 8.2 inches.

30.8 pounds of the individual's weight is made up of fat, rounded to the nearest tenth.

To calculate the pounds of body fat, we need to multiply the body weight by the body fat percentage in decimal form:
173 pounds x 0.178 = 30.794 pounds
Therefore, approximately 30.8 pounds of his weight is made up of fat.
To find out how many pounds of an individual's weight is made up of fat, you can use the following formula:
Total weight x Body fat percentage = Fat weight
In this case:
173 pounds x 0.178 (17.8%) ≈ 30.8 pounds
So, approximately 30.8 pounds of the individual's weight is made up of fat, rounded to the nearest tenth.

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Answer:

Step-by-step explanation:

Let's set David's score to D, and Aaron's score to A. If so, we get the equation that satisfies the constraints:

D = 3*A-5, D + A = 215

Set D in terms of A:

D = 215 - A

Input it into the equation:

215 - A = 3*A - 5 -> add 5 and A to both sides to set numbers to the left and variables to the right

215 + 5 = 3* A + A ->

220 = 4A -> divide by 4

220/4 = A ->

55 = A -> substitute A into D + A = 215

D + 55 = 215 -> subtract 55 from both sides

D = 215 - 55 = 160

So, out final answer is:

D = 160, A = 55

Answer:

D.

Step-by-step explanation:

6a² = surface area for cube

6(5)² = 150

2[tex]\pi[/tex](r)(h)+2[tex]\pi[/tex]r²  = surface area cylinder

2[tex]\pi[/tex](2)(3)+2[tex]\pi[/tex](4) = 20pi

150 + 20pi = 212.8318531

The %RE value for the Improved Euler (Heun) method is zero, which means that the method is exact. The %RE value for the Explicit Euler's method is 0.14%, which means that the method is a good approximation.

First rewrite the differential equation as follows:

[tex]\frac{dy}{dt } = (\frac{4}{t } - 6t^2)y[/tex]

Now integrate both sides of the equation:

[tex]\int\limits{\frac{dy}{dt}} \, = \int\limits\, (\frac{4}{t } - 6t^2)y dt[/tex]

This gives us:

[tex]y = 4ln(t) - 2t^3 + C[/tex]

where C = arbitrary constant.

Now, use the initial condition y(1) = 2 to find the value of C:

[tex]2 = 4ln(1) - 2(1)^3 + C[/tex]

This gives us C = 6.

Therefore, the solution of the initial-value problem is:

[tex]y = 4ln(t) - 2t^3 + 6[/tex]

Now, use the Explicit Euler's and Improved Euler (Heun) methods to compute y(1.3) numerically:

Explicit Euler's method:

First find the step size h:

h = (1.3 - 1) = 0.3

Now, use the Explicit Euler's method to compute y(1.3) as follows:

[tex]y(1.3) = y(1) + \frac{h}{2} * \frac{dy}{dt}[/tex]

[tex]y(1.3) = 2 + 0.3 * (4ln(1.3) - 2(1.3)^3 + 6)[/tex]

y(1.3) = 2.119

Improved Euler (Heun) method:

First find the step size h:

h = (1.3 - 1) = 0.3

Now, use the Improved Euler (Heun) method to compute y(1.3) as follows:

[tex]y(1.3) = y(1) + \frac{h}{2} * (\frac{dy}{dt} + \frac{dy}{dt'})[/tex]

where dy/dt' is the value of dy/dt evaluated at t = 1.3:

[tex]\frac{dy}{dt'} = 4ln(1.3) - 2(1.3)^3 + 6[/tex]

[tex]y(1.3) = 2 + \frac{0.3}{2} * (4ln(1.3) - 2(1.3)^3 + 6 + 4ln(1.3) - 2(1.3)^3 + 6)[/tex]

y(1.3) = 2.122

Now, calculate the %RE values for both methods:

Explicit Euler's method:

%RE = [tex]\frac{(y(1.3) - y_{true})}{y_{true}} * 100[/tex]

where y_true is the exact value of y(1.3) = 2.122:

%RE = (2.119 - 2.122)/2.122 × 100 = 0.14%

Improved Euler (Heun) method:

%RE = [tex]\frac{(y(1.3) - y_{true})}{y_{true}} * 100[/tex]

%RE = (2.122 - 2.122)/2.122 × 100 = 0%

The %RE value for the Improved Euler (Heun) method is zero, which means that the method is exact. The %RE value for the Explicit Euler's method is 0.14%, which means that the method is a good approximation.

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The circumference of the circle 48.4 inches.

The circumference of a circle is the perimeter of the circle. Therefore, let's find the circumference of the circle as follows:

length of arc = ∅ / 360 × 2πr

where

Therefore, let's find the radius

length of arc = 35 / 360 × 2 × 3.14 × r

4.6 = 219.8 / 360 r

4.6 = 0.61055555555 r

divide both sides by 0.61

r = 4.6 / 0.6

r = 7.7 inches

Therefore,

circumference = 2 × 3.14 × 7.7

circumference = 48.356

circumference = 48.4 inches

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The value of permutation expression ₅₉P₂ is 3422.

A permutation is an arrangement of objects in a definite order.

To find the value of ₅₉P₂, we use the formula below

Formula:

Where:

Substitute these values into equation 1

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The first derivative of the inverse trigonometric function is equal to f'(x) = - 2 / [1 + (2 · x + 1)².

Herein we find the definition of an inverse trigonometric function, whose first derivative must be found by using derivative rules and especially derivative formulas for inverse trigonometric functions and chain rule. First, write the original function:

f(x) = cot⁻¹ (2 · x + 1)

Second, use derivative rules to determine the derivative:

f'(x) = [- 1 / [1 + (2 · x + 1)²]] · 2

f'(x) = - 2 / [1 + (2 · x + 1)²

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Answer:

length = 7 width = 2

Step-by-step explanation:

for the perimeter 7+7+2+2 is 18 and the area is 7×2 is 14 so the length is 7 and width is 2

Answer:37.5 average

Step-by-step explanation:I think

Answer: A ≈ 127.31 units²

Step-by-step explanation:

     We can use the given formula for the area of a regular hexagon, where a is equal to a given side (here, that is 7).

        [tex]\displaystyle A=\frac{3\sqrt{3} }{2} a^2[/tex]

     We will substitute known values and solve by computing and simplifying.

        [tex]\displaystyle A=\frac{3\sqrt{3} }{2} a^2[/tex]

        [tex]\displaystyle A=\frac{3\sqrt{3} }{2} (7)^2[/tex]

        [tex]\displaystyle A=\frac{5.1961524}{2} 49[/tex]

        [tex]\displaystyle A=(2.59807621)(49)[/tex]

        [tex]\displaystyle A=127.3057[/tex]

        [tex]\displaystyle A\approx 127.31[/tex]

Answer:

All you have to do is to find the area of a circle to know the area it will contain.

Step-by-step explanation:

3. d=150cm= 150/100m=1.5m

r= 1.5/2= 0.75m

since a drum is circular under it will take a circular area

Area of a circle=πr^2

=π×(0.75)^2

=0.5625π = 1.77m^2

4. r=48dm=48/10=4.8m

since the clock is circular it will contact a circular area

Area of a circle= πr^2

=π(4.8)^2

=23.04π= 72.38m^2

5. since it's circular

C=75cm=75/100m=0.75m

Circumference of a circle=2πr

0.75=2πr

r=0.75/2π

r=0.119m

Area of a circle= πr^2

=π×(0.119)^2

=0.045m^2

I hope it helps.

The 10th percentile is 164 milliseconds, and the 75th percentile is 261 milliseconds.

To find the percentiles for the given data, we first need to arrange the data in ascending order:

152, 164, 175, 193, 212, 217, 222, 235, 250, 257, 261, 273, 296, 311

There are 15 data points, so the 10th percentile can be calculated as follows:

10th percentile = (10/100) * (15 - 1) + 1 = 1.4

The 10th percentile corresponds to the 2nd data point in the ordered list:

10th percentile = 164

Similarly, the 75th percentile can be calculated as follows:

75th percentile = (75/100) * (15 - 1) + 1 = 11.4

The 75th percentile corresponds to the 11th data point in the ordered list:

75th percentile = 261

Therefore, the 10th percentile is 164 milliseconds, and the 75th percentile is 261 milliseconds.

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After considering all the given data we conclude that the range is 45, the variance is 422.21, the standard deviation is 20.55

To evaluate the range of the given sample data, we subtract the smallest value from the largest value. For the given case, the largest value is 79 and the smallest value is 34. Therefore, the range is 79 - 34 = 45.
To evaluate the variance of the given sample data, we first need to find the mean. The mean is evaluated by adding up all of the values and dividing by the number of values. Including the given case, we have 10 values, so we add them up and divide by 10:
(51 + 63 + 36 + 43 + 34 + 62 + 73 + 39 + 53 + 79) / 10
= 51.3

Finally , we have to apply subtraction the mean from each value and square the result. Then we add up all of these squared differences and apply division by n-1 (here n is the number of values). This gives us the variance:
((51 - 51.3)² + (63 - 51.3)² + (36 - 51.3)² + (43 - 51.3)² + (34 - 51.3)² + (62 - 51.3)² + (73 - 51.3)² + (39 - 51.3)² + (53 - 51.3)² + (79 - 51.3)²) / (10-1) = 422.21
Lastly,  to evaluate the standard deviation, we take the square root of the variance:
√(422.21) = 20.55
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The mean of the distribution of the sample means (sampling distribution) for all possible samples of size 81 that could be drawn from the parent population of GPAs is equal to the mean of the parent population of GPAs.

The mean of the distribution of the sample means (sampling distribution) for all possible samples of size 81 that could be drawn from the parent population of GPAs is equal to the mean of the parent population. This is due to the central limit theorem, which states that as the sample size increases, the distribution of sample means approaches a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
Therefore, if we were to take all possible samples of size 81 from the population of GPAs and calculate the mean of each sample, the distribution of these means would be approximately normal with a mean equal to the population mean of GPAs. This means that if we take a large number of samples of size 81 and calculate the mean of each sample, the average of these means will be very close to the population mean of GPAs.
In conclusion, the mean of the distribution of the sample means (sampling distribution) for all possible samples of size 81 that could be drawn from the parent population of GPAs is equal to the mean of the parent population of GPAs.

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