Q23. - Algebra 1

Question

Find the area of each figure. Round to the nearest tenth.

Step-by-Step Solution

Answer

The area of the given figure is 4.0 square centimeters.

Step 1. State the formula.

The formula of the area of a rectangle is $A=lw$, where A is the area, l is the length and w is the width of the rectangle.

 

The formula of the area of a semicircle is $A=\frac{\pi r^2}{2}$, where A is the area and r is the radius of the semicircle.

 

The formula of the radius of a semicircle is $r=\frac{d}{2}$, where d is the diameter of the semicircle.

Step 2. List the given data.

The given figure is a combination of a rectangle and a semicircle.

 

The length and width of the rectangle are 2.9 cm and 1.2 cm respectively, while the diameter of the semicircle is equal to the width of the rectangle, that is, 1.2 cm.

 

Then, for the rectangle, $l=2.9\mathrm{cm}$, $w=1.2\mathrm{cm}$ and for the semicircle, $d=1.2\mathrm{cm}$.

Step 3. Calculate the radius of the semicircle.

Put $d=1.2$ in $r=\frac{d}{2}$ to get,

$$\begin{gathered} r=\frac{1.2}{2} \\ =0.6 \end{gathered}$$

So, $r=0.6$.

Step 3. Calculate the area.

Put $l=2.9$ and $w=1.2$ in $A=lw$ to get the area of the rectangle as,

$$\begin{gathered} A=\left(2.9\right)\left(1.2\right) \\ =3.48 \end{gathered}$$ 

Put $r=0.6$ and $\pi =\frac{22}{7}$ in $A=\frac{\pi r^2}{2}$ to get the area of the semicircle as,

$$\begin{gathered} A=\frac{\frac{22}{7}{\left(0.6\right)}^2}{2} \\ =\frac{\frac{22}{7}\left(0.36\right)}{2} \\ =\frac{11}{7}\left(0.36\right) \\ \approx 0.5657\left(\mathrm{roundedto}4\mathrm{decimalplaces}\right) \end{gathered}$$ 

Then, the total area of the figure is $3.48+0.5657=4.0457$, that is, approximately 4.0457.

Step 4. Determine the unit.

Since the units of length and width of the rectangle and the diameter of the semicircle are all given as centimeters, the unit of their area and thus the unit of the total area of the figure must be square centimeters.

 

So, the area of the given figure is approximately 4.0457 square centimeters. Rounding to the nearest tenth, the area is 4.0 square centimeters.