Question

Very important 50 pts

Answer 1

134.9 cm² (1 d.p.)

Explanation:

The given compound shape is composed of two elements:

• A rectangle measuring 8 cm in width and 14 cm in length.
• A sector of a circle with a central angle of 41° and a radius equivalent to the width of the rectangle, which is 8 cm.

To calculate the total area of the compound shape, simply add together the areas of both the rectangle and the circular sector.

Area of the rectangle

The area of a rectangle can be calculated using the formula A = w · l, where 'w' represents the width, and 'l' represents the length. Therefore, the area of the rectangle is:

`\begin{aligned}\textsf{Area of the rectangle}&=\sf 8\;cm \cdot 14\;cm\\\\&=\sf 112\; cm^2\end{aligned}`

Area of the circular sector

The area of the sector of a circle can be calculated using the following formula:

`\boxed{\begin{array}{l}\underline{\textsf{Area of a sector}}\\\\A=\left((\theta)/(360^(\circ))\right) \pi r^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the radius.}\\\phantom{ww}\bullet\;\;\textsf{$\theta$ is the angle measured in degrees.}\end{array}}`

In this case:

• θ = 41°
• r = 8 cm

Therefore, the area of the sector of the circle is:

`\begin{aligned}\textsf{Area of the sector}&=\left((41^(\circ))/(360^(\circ))\right) \pi \cdot (8)^2\\\\&=(41)/(360)\pi \cdot 64\\\\&=(2624)/(360)\pi\\\\&=22.8987197...\; \sf cm^2\\\\&=22.9\; \sf cm^2\;(1\;d.p.)\end{aligned}`

Area of the compound shape

Therefore, we can calculate the area of the compound shape by summing the two areas:

`\begin{aligned}\textsf{Area of compound shape}&=\textsf{Area of rectangle}+\textsf{Area of sector}\\\\&=\sf 112\; cm^2+22.9\;cm^2\\\\&=\sf 134.9\; cm^2\end{aligned}`

So, the area of the compound shape is 134.9 cm² (rounded to one decimal place).