Question

The diagram shows a circular pond, of radius r metres, surrounded by a circular path.The circular path has a constant width of 1.5 metres.The area of the path is the area of the pond.(a) Show that 2r2 − 60r − 45 = 0

Answer 1

The formula for calculating the area of a circle is expressed as;

A = πr²

• Area of the pond = πr²
• Area of the path = Area of the path + pond - area of pond

• Area of path = π(r+1.5)² - πr²

Since the  area of the path is equal to the area of the pond, hence;

π(r+1.5)² - πr² = πr²

Add πr² to both sides

π(r+1.5)² - πr² + πr² =  πr² + πr²

π(r+1.5)² = 2πr²

(r+1.5)² =   2r²

Expand the parenthesis

r²+3r + 2.25 = 2r²

r²+3r + 2.25 - 2r² = 0

-r² + 3r + 2.25 = 0

r² - 3r - 2.25 = 0

Multiply through by 2

2r² - 6.0r - 4.5 = 0

This proves the required equation

Learn more on area of composite figures here: brainly.com/question/15981553