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Chapter 2: Q. 15 (page 120)

A semicircle of radius r is inscribed in a rectangle so that the diameter of the semicircle is the length of the rectangle. See the figure.

(a) Express the area A of the rectangle as a function of the radius r of the semicircle.

(b) Express the perimeter p of the rectangle as a function of r.

Short Answer

Expert verified

Part (a) Area A of rectangle as a function of the radius, r of the semicircle is 2r2.

Part (b) The perimeter pof the rectangle as a function of r is6r.

Step by step solution

01

Part (a) Step 1. Given

The length of the rectangle equals the diameter of the semicircle=2r

The breadth of the rectangle equals the radius of the semicircle=r

02

Part (a) Step 2. Substitute given values in the formula of the area of the rectangle.

Area A of the rectangle =lb

A=lb=2r(r)=2r2

Area of the rectangle as a function of the radius of the semicircle is2r2.

03

Part (b) Step 1. Substitute the give values of length and breadth of the rectangle in the formula for the perimeter of the rectangle.

Length of the rectangle=2r

Breadth of the rectangle=r

Perimeter p of the rectangle=2(l+b)

p=2(l+b)p=2(2r+r)p=2(3r)p=6r

The perimeter of the rectangle as a function of the radius of the semicircle is 6r.

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