Question

The monthly demand for a company’s sports caps varies directly as the amount spent on advertising and inversely as the square of the price per cap. At $15 per cap, when $2500 is spent each week on ads, the demand is 300 caps. When advertising is increased to $3000, what price yields a demand of 300 caps?

Answer 1

Answer: $\ 16.432$

Step-by-step explanation:

Let be "y" the monthly demand for the company’s sports caps, "x" the amount in dollar spent on advertising and "z"the price ind dollars per cap.

The model for this situation is:

$y=\frac{kx}{z^2}$

Where "k" is the constant of proportionality.

Since $y=300$ when $z=15$ and $x=2,500$, we can substitute values into $y=\frac{kx}{z^2}$ and solve for "k" to finds its value:

$300=\frac{k(2,500)}{15^2}\\\\\frac{(300)(15^2)}{2,500}=k\\\\k=27$

Then, in order to calculate what price yields a demand of 300 caps when advertising is increased to $3,000, we must substitute the following values into $y=\frac{kx}{z^2}$:

$x=3,000\\\\y=300\\\\k=27$

Then:

$300=\frac{(27)(3,000)}{z^2}$

Solving for "z" we get:

$z^2=\frac{(27)(3,000)}{300}\\\\z=\sqrt{\frac{(27)(3,000)}{300}}\\\\z=16.432$