Question

11. Rebecca cut shapes out of construction paper to create a face. The nose and mouth were created using an equilateral triangle and a semicircle, as shown. What is the ratio of the area of the nose to the area of the mouth if the perimeters of the nose and the mouth are equal? Express your answer as a decimal to the nearest hundredth. 0​

Answer 1

The ratio of the area of the nose to the area of the mouth, when the perimeters are equal, is approximately 1.5.

To find the ratio of the area of the nose to the area of the mouth, let's denote the side length of the equilateral triangle as "s" (since it has equal sides) and the radius of the semicircle as "r". Given that the perimeters are equal, we can set up an equation:

3s = πr + s + πr

Simplify the equation:

3s = 2πr + s

Subtract s from both sides:

2s = 2πr

Divide both sides by 2:

s = πr

Now, the area of the nose (equilateral triangle) is given by (3/4)πr^2, and the area of the mouth (semicircle) is (1/2)πr^2.

Substitute s = πr into the equation for the area of the nose:

Area of the nose = (3/4)πr^2

Now, find the ratio:

Ratio = (Area of the nose) / (Area of the mouth) = ((3/4)πr^2) / ((1/2)πr^2) = (3/4) / (1/2) = 3/2, which is approximately 1.5.

In summary, the ratio is 1.5.