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Suppose that $z$ varies jointly with the cube of $x$ and the square of $y$ . Find the constant of proportionality $k$ if $z = 3600$ when $y = 3$ and $x =5$ . Using the $k$ from above write the variation equation in terms of $x$ and $y$ . Using the $k$ from above find $z$ given that $x =$ $12$ and $x=$ $13$ .

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Write and solve direct variation equations

Full solution

Q. Suppose that

z

varies jointly with the cube of

x

and the square of

y

. Find the constant of proportionality

k

if

z = 3600

when

y = 3

and

x =5

. Using the

k

from above write the variation equation in terms of

x

and

y

. Using the

k

from above find

z

given that

x =

12

and

x=

13

.

Identify general form: Identify the general form of joint variation. $\newline$ The general form of joint variation is $z = k \times x^n \times y^m$ , where $k$ is the constant of proportionality, $n$ and $m$ are the powers to which $x$ and $y$ are raised, respectively. $\newline$ In this case, since $z$ varies jointly with the cube of $x$ and the square of $y$ , the equation becomes $z = k \times x^3 \times y^2$ .
Substitute values into equation: Substitute $z = 3600$ , $y = 3$ , and $x = 5$ into the equation $z = k \times x^3 \times y^2$ . $3600 = k \times 5^3 \times 3^2$
Calculate $x^3$ and $y^2$ : Calculate the values of $x^3$ and $y^2$ . $\newline$ $5^3 = 125$ $\newline$ $3^2 = 9$
Substitute calculated values: Substitute the calculated values back into the equation. $3600 = k \times 125 \times 9$
Solve for constant: Solve the equation for the constant of proportionality, $k$ . $3600 = k \times 1125$ $k = \frac{3600}{1125}$ $k = 3.2$
Write joint variation equation: Write the joint variation equation using the value of $k$ . $\newline$ Substitute $k = 3.2$ into $z = k \times x^3 \times y^2$ . $\newline$ The joint variation equation is $z = 3.2 \times x^3 \times y^2$ .
Find $z$ with new values: Find $z$ given that $y = 12$ and $x = 13$ using the joint variation equation. $\newline$ Substitute $y = 12$ and $x = 13$ into $z = 3.2 \times x^3 \times y^2$ . $\newline$ $z = 3.2 \times 13^3 \times 12^2$
Calculate $13^3$ and $12^2$ : Calculate the values of $13^3$ and $12^2$ . $\newline$ $13^3 = 2197$ $\newline$ $12^2 = 144$
Substitute values for z: Substitute the calculated values back into the equation to find z. $\newline$ $z = 3.2 \times 2197 \times 144$
Perform multiplication: Perform the multiplication to find the value of $z$ . $z = 3.2 \times 316,608$ $z = 1,013,145.6$

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