Question

1. find the area of the shaded region. express your answer in terms of π.

Explanation:

Step1: Calculate the area of the triangle

The formula for the area of a triangle is \(A_{\triangle}=\frac{1}{2}\times base\times height\).
The base of the triangle \(b = 5 + 4+3 + 2=14\mathrm{cm}\), and the height \(h = 14\mathrm{cm}\).
So, \(A_{\triangle}=\frac{1}{2}\times14\times14 = 98\mathrm{cm}^{2}\).

Step2: Calculate the areas of the circles

The formula for the area of a circle is \(A_{circle}=\pi r^{2}\).

• For the first circle with diameter \(d_1 = 5\mathrm{cm}\), radius \(r_1=\frac{5}{2}\mathrm{cm}\), \(A_1=\pi\times(\frac{5}{2})^{2}=\frac{25\pi}{4}\mathrm{cm}^{2}\).
• For the second circle with diameter \(d_2 = 4\mathrm{cm}\), radius \(r_2 = 2\mathrm{cm}\), \(A_2=\pi\times2^{2}=4\pi\mathrm{cm}^{2}\).
• For the third circle with diameter \(d_3 = 3\mathrm{cm}\), radius \(r_3=\frac{3}{2}\mathrm{cm}\), \(A_3=\pi\times(\frac{3}{2})^{2}=\frac{9\pi}{4}\mathrm{cm}^{2}\).

The sum of the areas of the three circles \(A_{circles}=A_1 + A_2+A_3=\frac{25\pi}{4}+4\pi+\frac{9\pi}{4}\).

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Step3: Calculate the area of the shaded region

The area of the shaded region \(A = A_{\triangle}-A_{circles}\).
Substitute \(A_{\triangle}=98\) and \(A_{circles}=12.5\pi\) into the formula, \(A=(98 - 12.5\pi)\mathrm{cm}^{2}\).

Answer:

\((98 - 12.5\pi)\mathrm{cm}^{2}\)