Question

Express the statement as an equation. Use the given information to find the constant of proportionality.

H is jointly proportional to the square of l and w. If $l=2$and $w=\frac{1}{3}$, then $H=36$.

Short answer

Equation that represents the statement is$H=k{\left(lw\right)}^2$.

Constant of proportionality is 81.

Step-by-step solution

Step 1. Understanding Direct variation.

Direct Variation is said to be the relationship between two variables in which one is a constant multiple of the other.

For example:

$y=kx$

Now, if x and z are jointly proportional to y, then,

$y=k\left(xz\right)$

Here $k\ne 0$, y is directly proportional to x and vice versa.

Step 2. Write the equation.

As H is jointly proportional to squares of l and w, hence

$H=k{\left(lw\right)}^2$

Where k is constant of proportionality.

Step 3. Plug the values given in the equation to determine the value of the constant of proportionality.

Given values:

$\begin{array}{l}H=36\\ l=2\\ w=\frac{1}{3}\end{array}$

Putting values we get,

$\begin{array}{l}36=k{\left(2\times \frac{1}{3}\right)}^2\\ 36=k\left(\frac{4}{9}\right)\\ k=\frac{36\times 9}{4}\\ k=81\end{array}$