Question

the surface area of this cone is 505.54 square centimeters. what is the slant height of this cone?
round your answer to the nearest hundredth.
7 cm
$s \approx$ centimeters

Explanation:

Step1: Recall total surface area formula

The total surface area \( SA \) of a cone is \( SA = \pi r^2 + \pi r s \), where \( r = 7 \, \text{cm} \), \( SA = 505.54 \, \text{cm}^2 \), and \( s \) is slant height.

Step2: Calculate base area first

Base area \( \pi r^2 = \pi (7)^2 = 49\pi \approx 153.94 \, \text{cm}^2 \)

Step3: Isolate lateral surface area

Subtract base area from total area:
\( \pi r s = SA - \pi r^2 = 505.54 - 153.94 = 351.6 \, \text{cm}^2 \)

Step4: Solve for slant height \( s \)

Rearrange to solve for \( s \):
\( s = \frac{351.6}{\pi r} = \frac{351.6}{7\pi} \approx \frac{351.6}{21.9911} \)

Answer:

15.99 centimeters