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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. $\newline$ Joint variation is when one variable varies directly as the product of two or more other variables. The general form of joint variation with the cube of $x$ and the square of $y$ is $z = k \times x^3 \times y^2$ , where $k$ is the constant of proportionality.
Substitute Values into Equation: Substitute $z = 3600$ , $y = 3$ , and $x = 5$ into the equation $z = k \times x^3 \times y^2$ . $\newline$ $3600 = k \times 5^3 \times 3^2$
Solve for Constant: Solve the equation for the constant of proportionality, $k$ . $3600 = k \times 125 \times 9$ $3600 = k \times 1125$ $k = \frac{3600}{1125}$ $k = 3.2$
Write Joint Variation Equation: Write the joint variation equation using the value of $k$ found in Step $3$ . $\newline$ Substitute $k = 3.2$ into $z = k \times x^3 \times y^2$ . $\newline$ $z = 3.2 \times x^3 \times y^2$
Find z with New Values: Use the joint variation equation to find $z$ when $y = 12$ and $x = 13$ . Substitute $y = 12$ and $x = 13$ into $z = 3.2 \times x^3 \times y^2$ . $z = 3.2 \times 13^3 \times 12^2$
Calculate Final Value: Calculate the value of $z$ . $z = 3.2 \times 2197 \times 144$ $z = 3.2 \times 316368$ $z = 1011577.6$

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