consider an object moving along the parametrized curve with equations: x(t)=et + e–t, y(t)=e–t where t is in the time interval [0,7 ]
seconds.
(a) The maximum speed of the object on the time interval is at time .
(b) The minimum speed of the object on the time interval is at time .

As per the formula of surface area of cube, the length of the cube is 5.45 meters.

The general formula to calculate the surface area of the cube is calculated as,

=> SA = 6a²

here a represents the length of cube.

Here we know that  the side of a cube with a surface area of 180 square meters than a cube with the surface area of 120 square meters.

When we apply the value on the formula, then we get the expression like the following,

=> 180 = 6a²

where a refers the length of the cube.

=> a² = 30

=> a = 5.45

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To reduce your department by 15%, you would need to let go of 33 workers.

Reducing a department by 15% means that 15% of the current workforce in that department will be let go. To calculate this, you first need to determine the percentage of the total workforce that your department represents. In this case, your department has 220 workers out of 512 workers in the whole company, which represents 43% of the total workforce.

Then, you can calculate the number of workers that need to be let go by multiplying the total number of workers in your department by the percentage that needs to be cut. In this case, that would be

220 x 0.15 = 33 workers

This means that 33 out of the 220 workers in your department would need to be let go in order to achieve a 15% reduction.

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On solving the provided question we can say that here the probability P(X<3) = 0.7102 and P(X>3) = 0.5801.

Probability theory, a subfield of mathematics, gauges the likelihood of an occurrence or a claim being true. An event's probability is a number between 0 and 1, where approximately 0 indicates how unlikely the event is to occur and 1 indicates certainty. A probability is a numerical representation of the likelihood or likelihood that a particular event will occur. Alternative ways to express probabilities are as percentages from 0% to 100% or from 0 to 1. the percentage of occurrences in a complete set of equally likely possibilities that result in a certain occurrence compared to the total number of outcomes.

P(X = 3) = 0.2903

P(X>3) = 0.5801

P(X<3) = 0.7102

mean=2.8

standard deviation=1.2961

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The rate in centimeters per second, at which the radius is increasing at the instant when the radius is 5 centimeters is 0.16 cm/s.

Using a known value of a function at a certain point along with its rate of change at that moment, one application of derivatives is to estimate an unknown value of a function at a particular point.

Given:

Both radius r and area A is a function of time, so in fact:

A(t) = π [r(t)]²

Deriving this we get that the area changes and the radius changes as:

[tex]\frac{dA(t)}{dt} = 2\pi r(t)\frac{dr(t)}{dt}[/tex]

But,

[tex]\frac{dA(t)}{dt} = 5 \frac{cm^{2} }{s}[/tex]

Considering when r(t) = 5cm at the instant we get,

[tex]5 = 2\pi . 5\frac{dr(t)}{dt}[/tex]

Simplify 5 and take 2π to the other side and divide,

The rate of change of radius is given by:

[tex]\frac{dr(t)}{dt} =\frac{1}{2\pi } = 0.16 cm/s[/tex]

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The eight term of the geometric sequence 125/9, 25/3,5,3,.... is 243/625.

These steps to answer:

Question above is a geometric sequence consisting of 4 terms to find the 8th term, then we can use the geometric sequence formula below

Tn= a. r^n-1

is known

a= 125/9

r=T4/T2

r=3/5

then to find T4

Tn= a. r^n-1

T8=125/9. (3/5^8-1)

T8=125/9. (3/5)^7)

T8=125/9. (2187/78125)

T8=243/625

A geometric sequence is a sequence that satisfies the quotient of a term with the preceding terms which are of course consecutive. This thing has a constant value. Not only that, geometric sequences are also known as 'measurable sequences' which are still closely related to arithmetic sequences and series.

Examples of geometric sequences are a, b, and c. Then c/b = b/a = constant, this is where the quotient of adjacent terms will be obtained and then it is said to be the ratio of the geometric sequence which is given the symbol "r".

Another example, which is much easier to understand, is for example, if you have a sequence and a series: 2, 4, 8, 16, 32, ….. etc., then from the sequence and series it can be seen between the first and second terms and so on, have the same multiplier.

So, to find the nth term, you can easily find the ratio first. By knowing 'r', then you will easily find Tn.

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Answer: C

Step-by-step explanation:

Yes, the conditions for constructing a t confidence interval are met. The sample size is 10, which is greater than 30, and the sample is a random sample of 10 deliveries from a large number of deliveries made.

1. Constructing a t confidence interval requires that the sample size is greater than 30, so the first condition that must be met is that the sample size is greater than 30.

2. The second condition that must be met is that the sample is a random sample of the population of interest. In this case, the sample is a random sample of 10 deliveries from a large number of deliveries made.

Since both conditions are met, the conditions for constructing a t confidence interval are met.

Yes, the conditions for constructing a t confidence interval are met. The sample size is 10, which is greater than 30, and the sample is a random sample of 10 deliveries from a large number of deliveries made.

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The value of r in the given equation is 0.04 by the process of solving the equation for the unknown.

The given equation: 816 = 600(1 + r)

Where it represents the amount of money earned on a simple interest savings account.

We need to find out the value of r in the equation.

To find the value of unknown r, both the LHS and RHS of the equation must be divided by 600.

Therefore, 816/600 = [600(1 + r)]/600

816/600 = 1(1 + r)

816/600 = 1 + r

r = 816/600 - 1

r = 0.04

Therefore, the amount of money earned on a simple interest savings account using the given equation is 0.04

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There are 22 different types of functions based on range, element, equation, and domain.

Based on the range it includes Modulus Function, rational function, signum function, even and odd functions, periodic functions, greatest integer function, inverse function, and composite functions. On the basis of elements, it includes One One function, many one functions, onto function, one one and onto function, into function, and constant function. On the basis of the equation, it includes the identity function, linear function, quadratic function, cubic function, and polynomial functions. On the basis of the domain, it has algebraic functions, trigonometric functions, and logarithmic functions.

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The vertices of the polygon before and after dilation are shown below

preimage         Image

J (-5, 3)            J' (-5/2, 3/2)

K (2, 3)            K' (1, 3/2)

L (2, -3)           L' (1, -3/2)

M (-5, -3)        M' (-5/2, -3/2)

Dilation is a method of transformation that magnify or shrink the preimage depending on the scale factor

The rule used in  transformation is as follows

(x, y) for a scale factor of r → (rx, ry)

Applying the rule to the vertices  of the polygon we have

preimage                                  Image

J (-5, 3)      (1/2 * -5, 1/2 * 3)      J' (-5/2, 3/2)

K (2, 3)       (1/2 * 2, 1/2 * 3)       K' (1, 3/2)

L (2, -3)      (1/2 * 2, 1/2 * -3)       L' (1, -3/2)

M (-5, -3)    (1/2 * -5, 1/2 * -3)     M' (-5/2, -3/2)

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Answer:

Perpendicular Bisector

The algebraic representation of this dilation is: (x', y') = (-0.9x, -1.5y)

A dilation is a transformation that changes the size of a geometric figure, but not its shape. The center of dilation is a fixed point, and all other points in the figure are expanded or contracted in relation to it. In this case, the center of dilation is the origin, and the original figure is m'n'o'p' and the image figure is mnop.

The algebraic representation of a dilation is given by the equation:

(x', y') = (kx, ky)

Where (x', y') are the coordinates of the image point, (x, y) are the coordinates of the original point and k is the scale factor of the dilation.

We can find the scale factor k by comparing the coordinates of any two corresponding vertices of the figures. For example, if we take the vertex m' and m, we have:

m'(-5, -4) and m(4.5, 6)

Here we can see that the x-coordinate of m' is -5 and the x-coordinate of m is 4.5 and the y-coordinate of m' is -4 and the y-coordinate of m is 6.

So, k = mx/mx' = 4.5/-5 = -0.9 and k = my/my' = 6/-4 = -1.5

We can use this scale factor to find the coordinates of any other point in the image figure.

Therefore, the algebraic representation of this dilation is: (x', y') = (-0.9x, -1.5y)

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Therefore , the solution of the given problem of integer comes out to be 1/100 = 0.01

Zero, a positive integer, or a negative integer denoted by a minus sign are all examples of integers. A negative number is the additive reciprocal of a positive number that it corresponds to. A bold Z or a bold "mathbb Z" is frequently used in mathematical notation to denote a group of integers. A positive, negative, or zero integer—not a fraction—is referred to as an integer (pronounced IN-tuh-jer). The numbers -5, 1, 5, 8, 97, and 3,043 are examples of integers. 1.43, 1 3/4, 3.14, and other numbers are non-integer examples. Integers are a collection of integers and their antipodes. Decimals and fractions are not part of the set of integers.

Here,

=> 1/4 =0.25

=> 1/20 = 0.05

=> 1/25 = 0.04

=> 1/50 = 0.02

=> 1/100 = 0.01

Therefore , the solution of the given problem of integer comes out to be 1/100 = 0.01

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In order for the number to form a terminating decimal, 2 and 5 must be its only prime factors. Obviously any number less than [tex]0.01[/tex] or any value of [tex]n > 100[/tex] will not have a nonzero hundredths digit.

We now count the possibilities for [tex]n[/tex]:

[tex]5^0 : 2^0, 2^1,..., 2^6 \rightarrow 7 \text{ values}\\5^1 : 2^0, 2^1,..., 2^4 \rightarrow 5 \text{ values}\\5^2 : 2^0, 2^1, 2^2 \rightarrow 3 \text{ values}[/tex]

for a subtotal of [tex]15[/tex] values.

However, some of these have a nonzero tenths digit and a zero hundredths digit. In other words, we need to remove all of the single-digit terminating decimals from this list. These are [tex]\dfrac{1}{1} = 1.00,\dfrac{1}{2} = 0.50,\dfrac{1}{5}=0.20, \text{ and }\dfrac{1}{10} = 0.10 \implies 4 \text{ values}[/tex]

which means the final answer is [tex]15-4=11[/tex].

Answer:

The total number of questions in the test is 40. This can be calculated by dividing 80% by 32, which equals 0.25. Multiplying 0.25 by 32 gives 8, and then multiplying 8 by 5 gives 40.

Step-by-step explanation:

Answer:

40

Step-by-step explanation:

Based on the given conditions, formulate: 32/80%

Multiply both the numerator and denominator with the same integer:320/8

Cross out the common factor:40

get the result:40

Answer: 40

Answer:

8.1

Step-by-step explanation:

12/x = tan56

12/x =1.48

so x = 12/1.48 =8.1

Answer:

a) In order to write the ratio h:k in its simplest form, we need to find the greatest common divisor (GCD) of h and k and divide both h and k by that value.

The GCD of 2 and 6 is 2.

So the simplified ratio h:k is 2/2:6/2 = 1:3

b) In order to write the ratio k:l in its simplest form, we need to find the greatest common divisor (GCD) of k and l and divide both k and l by that value.

The GCD of 6 and 9 is 3.

So the simplified ratio k:l is 6/3:9/3 = 2:3

Answer:

the ratio of h and k in simplest form is 1:3

the ratio of k and l is 2:3

Answer:

Step-by-step explanation:

substitute p for 8, because p=8

question 1 :    8+4=12    ----Yes

question 2: 8x8 =64-------Yes

question 3  5x8=35, not 8----No

question 4 2+8=10, not 12----No

Answer:

47

Step-by-step explanation:

The number of pages that the student will read in 7 hours, given the number of pages read in 4 hours, is

The constant variation is 14 pages.

This constant variation is direct.

The quantity that connects two variables that are directly or inversely proportional to one another is known as the constant of variation.

Variation in mathematics demonstrates how one variable fluctuates in respect to another. Typically, a ratio is used to illustrate this relationship. When we remark that a variation is continuous, we are referring to how consistently the ratio changes.

The constant of variation in this instance therefore, is:

= Number of pages read / Number of hours

= 56 / 4

= 14 pages

This constant of variation is direct.

The number of pages read in 7 hours is:

= Constant of variation x number of hours

= 14 x 7

= 98 pages

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Answer: Dilation by 1/4 and translation

Step-by-step explanation:

The general solution of the system x' (t) is[tex]X(t)=c_1\left(\left[\begin{array}{c}-5 \\2\end{array}\right] \cos 3 t-\left[\begin{array}{l}1 \\0\end{array}\right] \sin 3 t\right)[/tex][tex]+c_2\left(\left[\begin{array}{c}1 \\0\end{array}\right] \cos 3 t+\left[\begin{array}{c}-5 \\2\end{array}\right] \sin 3 t\right)\end{gathered}[/tex]

The given system of equation is X'A=X

where

[tex]$$A=\left[\begin{array}{cc}-15 & -39 \\6 & 15\end{array}\right]$$[/tex]

Eigenvector of a square matrix is defined as a non-vector in which when a given matrix is multiplied,  it is equal to a scalar multiple of that vector. Let us suppose that A is an n x n square matrix, and if v be a non-zero vector, then the product of matrix A, and vector v is defined as the product of a scalar quantity λ and the given vector, such that:

Where

v = Eigenvector and λ be the scalar quantity that is termed as eigenvalue associated with given matrix A

The values of A are given by

[tex]$|A-\lambda I|=0 \\\left|\begin{array}{cc}-15-\lambda & -39 \\6 & 15-\lambda\end{array}\right|=0 \\[/tex]

[tex](-15-\lambda)(15-\lambda)-(6)(-39)=0 \\[/tex]

[tex]\Rightarrow-(15+\lambda)(15-\lambda)+(6)(39)=0 \\[/tex]

[tex]\\\Rightarrow(15+\lambda)(15-\lambda)-(6)(39)=0 \\\\\Rightarrow 225-\lambda^2-234=0 \\[/tex]

[tex]\Rightarrow-\lambda^2-9=0 \\[/tex]

[tex]\Rightarrow \lambda^2=-9 \\[/tex]

[tex]\Rightarrow \lambda=\pm 3 i[/tex]

Now, eigen vector u corresponding to [tex]$\lambda=3 i$[/tex] is given by

[tex]$$\begin{gathered}{[A-3 i I] u=O} \\{\left[\begin{array}{cc}-15-3 i & -39 \\6 & 15-3 i\end{array}\right]\left[\begin{array}{l}u_1 \\u_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\\\\end{gathered}$$[/tex]

Applying [tex]R_2 \rightarrow R_2-6 R_1[/tex]

[tex]& {\left[\begin{array}{cc}1 & \frac{-39}{-15-3 i} \\0 & 0\end{array}\right]\left[\begin{array}{l}u_1 \\u_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\[/tex]

[tex]& \Rightarrow u_1+\frac{39}{15+3 i} u_2=0 \\[/tex]

[tex]& \Rightarrow u_1=-\frac{13}{5+i} u_2=-\frac{13}{5+i} * \frac{5-i}{5-i} u_2=-\frac{13(5-i)}{26} u_2=-\frac{5-i}{2} u_2 \\[/tex]

Thus, by choosing [tex]$u_2[/tex]=1 eigenvector corresponding to [tex]$\lambda=3 i$[/tex] is [tex]$$u=\left[\begin{array}{c}-\frac{5-i}{2} \\1\end{array}\right]$$[/tex]

[tex]R_1 \rightarrow \frac{1}{-15+3 i} R_1 \\[/tex]

[tex]{\left[\begin{array}{cc}1 & \frac{-39}{-15+3 i} \\6 & 15+3 i\end{array}\right]\left[\begin{array}{l}v_1 \\v_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\[/tex]

Applying [tex]R_2 \rightarrow R_2-6 R_1 \\[/tex]

[tex]{\left[\begin{array}{cc}1 & \frac{-39}{-15+3 i} \\0 & 0\end{array}\right]\left[\begin{array}{l}v_1 \\v_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\[/tex]

[tex]\Rightarrow v_1+\frac{39}{15-3 i} v_2=0 \\\Rightarrow v_1=-\frac{39}{15-3 i}[/tex]

[tex]\begin{gathered}\Rightarrow v_1+\frac{39}{15-3 i} v_2=0 \\\v_2=-\frac{13(5+i)}{26} \\\Rightarrow v_1==-\frac{5+i}{2}\end{gathered}$$[/tex]

Thus, by choosing [tex]$v_2=1$[/tex] eigenvector corresponding to [tex]$\lambda=-3 i$[/tex] is

[tex]$$v=\left[\begin{array}{c}-\frac{5+i}{2} \\1\end{array}\right]$$[/tex]

Hence, the general solution is given by

[tex]X(t)=c_1 e^{3 i t} u+c_2 e^{-3 i t} v \text { [using } e^{i t}=\cos t+i \sin t \\[/tex]

[tex]X(t)=c_1\left[\begin{array}{c}-\frac{5-i}{2}(\cos 3 t+i \sin 3 t) \\(\cos 3 t+i \sin 3 t)\end{array}\right]+c_2\left[\begin{array}{c}-\frac{5+i}{2}(\cos 3 t-i \sin 3 t) \\(\cos 3 t-i \sin 3 t)\end{array}\right] \\[/tex]

[tex]X(t)=c_1\left[\begin{array}{c}-\frac{5}{2}(\cos 3 t+i \sin 3 t)+\frac{i}{2}(\cos 3 t+i \sin 3 t) \\(\cos 3 t+i \sin 3 t)\end{array}\right][/tex][tex]+c_2\left[\begin{array}{c}-\frac{5}{2}(\cos 3 t-i \sin 3 t)-\frac{i}{2}(\cos 3 t-i \sin 3 t) \\(\cos 3 t-i \sin 3 t)\end{array}\right] \\[/tex]

[tex]X(t)=c_1\left[\begin{array}{c}-\frac{5}{2} \cos 3 t-\frac{5 i}{2} \sin 3 t+\frac{i}{2} \cos 3 t-\frac{1}{2} \sin 3 t \\\cos 3 t+i \sin 3 t\end{array}\right][/tex][tex]+c_2\left[\begin{array}{c}-\frac{5}{2} \cos 3 t+\frac{5}{2} i \sin 3 t-\frac{i}{2} \cos 3 t-\frac{1}{2} \sin 3 t \\\cos 3 t-i \sin 3 t\end{array}\right ][/tex]

[tex]X(t)=c_1\left(\left[\begin{array}{c}-5 \\2\end{array}\right] \cos 3 t-\left[\begin{array}{l}1 \\0\end{array}\right] \sin 3 t\right)[/tex][tex]+c_2\left(\left[\begin{array}{c}1 \\0\end{array}\right] \cos 3 t+\left[\begin{array}{c}-5 \\2\end{array}\right] \sin 3 t\right)\end{gathered}[/tex]

Therefore, the general solution of the system X(t) is [tex]X(t)=c_1\left(\left[\begin{array}{c}-5 \\2\end{array}\right] \cos 3 t-\left[\begin{array}{l}1 \\0\end{array}\right] \sin 3 t\right)[/tex][tex]+c_2\left(\left[\begin{array}{c}1 \\0\end{array}\right] \cos 3 t+\left[\begin{array}{c}-5 \\2\end{array}\right] \sin 3 t\right)\end{gathered}[/tex].

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Answer:

16

Step-by-step explanation:

The side lengths of the original square was 2.  2 x 2 = 4

.25 x 2 = .5

The squares are now 1/2 x 1/2.  One of the 4 original squares will now need 4 new smaller size squares to cover the one square.

4 x 4 = 16

Answer:

|1|

Step-by-step explanation:

Because absolute value measures the positive distance from the numeral to 1, the answer could be negative or positive one, which is what the graph shows. So, |1| is the answer.

The first partial derivatives of the function z = x sin(xy) is x²cos(xy)

The term partial derivatives is defined as the rate of change of a function with respect to a variable and the derivatives are fundamental to the solution of problems in calculus and differential equations.

Here we have given that the function z = x sin(xy).

And as per the definition of partial derivative the value is calculated as,

Here we have given that

=> f(x, y) = x sin(xy)

And then here we need to find fx​ we treat y as constant and differentiate with respect to x, then we get

=> fx = sin(x y) + xy cos(xy)

Similarly now we have to find fy​ we treat x as constant and differentiate with respect to y

=> fy​ = x²cos(xy)

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The hypotenuse side of the right triangle is 7 units.

The value of x in the triangle is 5.92 units.

A right triangle is a triangle with one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.

The side of a right triangle can be found using Pythagoras's theorem as follows:

using Pythagoras's theorem,

c² = a² + b²

where

Therefore,

hypotenuse = 7

Let's find x.

7² - (√14)² = x²

49 - 14 = x²

x² = 35

square root both sides

x = √35

x = 5.9160797831

x = 5.92 units

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Answer:

b) [tex]y = -3x - 2[/tex] and [tex]y = -3x + 5[/tex] are parallel because the slope of each line is -3

Therefore , the solution of the given problem of function comes out to be the function f(x) = x³ + 2x² + 7x - 11  = -1 and g(x) = 3f(x) = -3

The subject of mathematics includes quantities and their variations, equations and related structures, shapes and their locations, and places where they can be found. The term "function" refers to the relationship between a set of inputs, each of which has an associated output. A connection between inputs and outputs in which each input leads to a single, distinct result is known as a function. Each function is given a domain and a codomain, or scope.

Here,

Given:

we have been provided with -

f(x) = x³ + 2x² + 7x - 11  

so, x = 1

f(x) = 1 +2 +7 -11= -1

g(x) = 3f(x) = -3

Therefore , the solution of the given problem of function comes out to be the function f(x) = x³ + 2x² + 7x - 11  = -1 and g(x) = 3f(x) = -3

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An expression using subtraction to find the distance between the two divers is |B - A| or |A - B| or B - A as B > A and the distance using additive inverses is 4 unit.

The number line displays the separation in meters between two divers A and B and the shipwreck at position X:

A = -2.5

X = 0

B = 1.5

We have to write an expression using subtraction to find the distance between the two divers and the distance using additive inverses.

Distance between the two divers = |B - A| or |A - B| or B - A as B > A

Distance between the two divers = |1.5 - (-2.5)| or |-2.5 - 1.5| or  1.5 - (-2.5)

Additive inverse of a number "x" is -x. So Additive inverse of -2.5 = 2.5

Distance between the two divers = 1.5 + 2.5

Distance between the two divers = 4

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Answer:

C. y = 30x + 130    D. 5 people  Q3. The 1st number is 25

     y = 20x +180                                 The 2cnd number is 7

Step-by-step explanation:

C. The x represents the amount of people attending.

     30 and 20 are the price per person.

D. 30 x 5 + 130 = 280

    20 x 5 = 180 = 280

Q3. 32 + 18 = 50

50/2 = 25

25 + y = 32

y = 32 - 25

y = 7

hope this helps =)

y = 30x + 130 and y = 20x + 180 is the  system of equations that represents the model and The two numbers are 25 and 7 in second question.

Two or more expressions with an Equal sign is called as Equation.

Let y represents the total cost of the party, and x represents the number of the people attending the party.

The hockey suite costs $130 to rent the room and $30 per person.

y=130+30x

The basketball suite costs $180 to rent the room and $20 per person.

y=180+20x

So option c is correct and

If there was going to be a total of 4 people attending the party,

y=130+30(4)=250

y=180+20(4)=260

basket ball is most cost efficient.

If there was an amount of people that could attend the party so that both games would cost the same

130+30x=180+20x

10x=50

x=5

So 5 people could attend the party.

The sum of two numbers is 32

The difference is 18

x+y=32

x-y=18

x+y+x-y=32+18

2x=50

Divide both sides by 2

x=25

y=7

The two numbers are 25 and 7.

Hence, y = 30x + 130 and y = 20x + 180 is the  system of equations that represents the model and The two numbers are 25 and 7 in second question.

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Given the polynomial Q(x)=4x^3-3x+1,

P(x) = 8x^3 - 6x + 2

Generally, To find P(x), we can start by substituting

P(x) = 2*(Q(x) + P(x)) into the equation:

P(x) = 2*(Q(x) + P(x))

This simplifies to:

P(x) = 2*(4x^3 - 3x + 1 + P(x))

Expanding and rearranging the equation gives:

P(x) = 8x^3 - 6x + 2 + 2P(x)

Subtracting 2P(x) from both sides of the equation gives:

P(x) - 2P(x) = 8x^3 - 6x + 2

This simplifies to:

-P(x) = 8x^3 - 6x + 2

Adding P(x) to both sides of the equation gives:

0 = 8x^3 - 6x + 2 + P(x)

Therefore, the polynomial P(x) that satisfies the equation

P(x) = 2*(Q(x) + P(x)) is:

P(x) = 8x^3 - 6x + 2

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Answer:

Probably  c:  f(x) = y + 3

Step-by-step explanation:

See attached worksheet