Question

the formula for the slant height of a cone is $l = \frac{s - \pi r^{2}}{\pi r}$, where $s$ is surface area of the cone. use the formula to find the slant height, $l$, of a cone with a surface area of $1500\pi$ $ft^{2}$ and a radius of 15 ft.
$l = \square$ ft

Explanation:

Step1: Substitute given values

Substitute $S=1500\pi$ and $r=15$ into the formula:
$l = \frac{1500\pi - \pi(15)^2}{\pi(15)}$

Step2: Calculate $\pi r^2$ term

Compute $\pi(15)^2$:
$\pi(15)^2 = 225\pi$

Step3: Simplify numerator

Subtract terms in the numerator:
$1500\pi - 225\pi = 1275\pi$

Step4: Simplify denominator

Compute $\pi(15)$:
$\pi(15) = 15\pi$

Step5: Solve for $l$

Divide numerator by denominator, cancel $\pi$:
$l = \frac{1275\pi}{15\pi} = 85$

Answer:

85 ft