find square miles of increased alarm system

Question

An alarm system can be heard for 40 miles in all directions (its effective reach).  If the alarm is upgraded to be heard for 60 miles in all directions, by how many square miles has the effective reach of the alarm increased (approx.)?

Answer 1

A₄₀ = π(40)²

A₄₀ = 1600π

 

A₆₀ = π(60)²

A₆₀ = 3600π

 

Increase = 3600π - 1600π

Increase = 2000π

Increase ≈ 6283.18531

 

The increase is approximately 6,283.19 mi².

Answer 2

This is a difference of areas of two circles.  Since the alarm can be heard in all directions, you are working with circles.  Keep in mind the distance in one direction (40 miles to start) is the radius.

 

The area of a circle = Pi*R²

 

So, the original circle has an area of Pi*40² = 5026 square miles

The second circle has an area of Pi*60² = 11309 square miles

The difference is 11309 - 5026 =  6283 square miles.

 

I hope this helps.