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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 $x =5$ .

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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

.

Identify general form: Identify the general form of joint variation. $\newline$ Joint variation is expressed as $z = k \times x^n \times y^m$ , where $k$ is the constant of proportionality, and $n$ and $m$ are the powers to which $x$ and $y$ are raised, respectively. 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 joint variation equation $z = k \times x^3 \times y^2$ . $\newline$ $3600 = k \times 5^3 \times 3^2$
Calculate $x^3$ and $y^2$ : Calculate the values of $x^3$ and $y^2$ . $\newline$ $5^3 = 5 \times 5 \times 5 = 125$ $\newline$ $3^2 = 3 \times 3 = 9$
Substitute calculated values: Substitute the calculated values into the equation. $\newline$ $3600 = k \times 125 \times 9$
Solve for constant of proportionality: Solve the equation for the constant of proportionality, $k$ . $3600 = k \times (125 \times 9)$ $3600 = k \times 1125$ $k = \frac{3600}{1125}$
Perform division: Perform the division to find the value of $k$ . $k = \frac{3600}{1125}$ $k = 3.2$

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