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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 $y$ .

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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

.

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: Substitute $z = 3600$ , $y = 3$ , and $x = 5$ into the joint variation equation $z = k \times x^3 \times y^2$ . $3600 = k \times 5^3 \times 3^2$
Calculate cube and square: Calculate the cube of $x$ and the square of $y$ . $5^3 = 125$ and $3^2 = 9$
Substitute values in equation: Substitute the values of $x^3$ and $y^2$ into the equation. $\newline$ $3600 = k \times 125 \times 9$
Solve for constant: Solve the equation for the constant of proportionality, $k$ . $3600 = k \times 1125$ $k = \frac{3600}{1125}$
Perform division: Perform the division to find the value of $k$ . $k = 3.2$
Write joint variation equation: Write the joint variation equation using the value of $k$ . Substitute $k = 3.2$ into $z = k \times x^3 \times y^2$ . $z = 3.2 \times x^3 \times y^2$

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