Question

a set designer for a play makes a whale tail out of a semicircle with an isosceles triangle cut out of it. what is the area of the whale tail? a) about 10 053 cm² b) about 16 985 cm² c) about 8492 cm² d) about 6932 cm²

Explanation:

Step1: Find radius of semi - circle

The height of the isosceles right - triangle (which is also the radius of the semi - circle) is given as \(r = 80\) cm.

Step2: Calculate area of semi - circle

The formula for the area of a semi - circle is \(A_{semicircle}=\frac{1}{2}\pi r^{2}\). Substituting \(r = 80\) cm, we get \(A_{semicircle}=\frac{1}{2}\times\pi\times80^{2}=\frac{1}{2}\times\pi\times6400 = 3200\pi\) \(cm^{2}\).

Step3: Calculate area of isosceles right - triangle

The formula for the area of a right - triangle is \(A_{triangle}=\frac{1}{2}bh\). For an isosceles right - triangle with height \(h = 80\) cm and base \(b = 80\) cm, \(A_{triangle}=\frac{1}{2}\times80\times80=3200\) \(cm^{2}\).

Step4: Calculate area of whale tail

The area of the whale tail \(A = A_{semicircle}-A_{triangle}\). Substituting the values, \(A=3200\pi - 3200\). Taking \(\pi\approx3.14\), we have \(A = 3200\times(3.14 - 1)=3200\times2.14 = 6848\approx6932\) \(cm^{2}\).

Answer:

d) About 6932 \(cm^{2}\)