Problem of the Week - Algebra and Up - April 11, 2021 to April 17, 2021

Algebra and Up: 

Question: A circular beach umbrella with an area of 16π square feet casts a circular shadow with a circumference of  16π feet. What is the circumference of an umbrella at the same height that casts a shadow whose area is 256π square feet? 

(For the purposes of this exercise, pretend umbrellas are flat circles instead of domes.)

Answer:  16π feet

Solution:  If the small umbrella has an area of 16π square feet, its radius must be 4 feet. If its shadow has a circumference of 16π feet, then its radius must be 8 feet. The radius of the shadow is double the radius of the umbrella. Since we know the area of the larger shadow is 256π square feet, its radius must be 16 feet. That means that the radius of the larger umbrella must be half of that, 8 feet, making its circumference 16π feet.