Question

A park walkway surrounds a fountain as shown. Find the area of the walkway. Round to the nearest foot.

Answer 1

The fountain is depicted by the white circle in the picture. The surrounding walkway is depicted by the grey areas.

From the sketch shown above, the semi-circle inscribed in the rectangle is one half of the fountain. We shall calculate the area of the semi-circle and subtract this from the area of the rectangle.

The area of the rectangle is;

`\begin{gathered} \text{Area}=l* w \\ \text{Area}=30*42.5 \\ \text{Area}=1275ft^2 \\ \text{The area of the semicircle is,} \\ \text{Area=}(1)/(2)(\pi* r^2) \\ \text{The diameter is 18 ft, and therefore the radius is 9 ft} \\ \text{Area}=(1)/(2)(3.14*9^2) \\ \text{Area}=(1)/(2)(3.14*81) \\ \text{Area}=(1)/(2)(254.34) \\ \text{Area}=127.17ft^2 \end{gathered}`

Therefore, the area of the shaded region would be,

Area = 1275 - 127.17

Area = 1147.83

Next step is to calculate the other half of the figure (the right side), as follows;

Observe that the outer semi-circle is the shaded region while the inner one is the white portion.

The area is

`\begin{gathered} \text{Shaded region;} \\ \text{Area}=(1)/(2)(\pi* r^2) \\ \text{Area}=(1)/(2)(3.14*15^2) \\ \text{Area}=(1)/(2)(3.14*225) \\ \text{Area}=(1)/(2)(706.5) \\ \text{Area}=353.25ft^2 \\ \text{White region;} \\ \text{Area}=(1)/(2)(\pi*9^2) \\ \text{Area}=(1)/(2)(3.14*81) \\ \text{Area}=127.17ft^2 \end{gathered}`

The area of the shaded region is;

Area = 353.25 - 127.17

Area = 226.38

Therefore the total area of the walkway surrounding the fountain is;

Area = 1147.83 + 226.38

Area = 1374.21

Area = 1,374 feet squared (rounded to the nearest foot)