Question

A rose garden is formed by joining a rectangle and a semicircle, as shown below. The rectangle Is 33 ft long and 23 ft wide. If the gardener wants to build afence around the garden, how many feet of fence are required? (Use the value 3.14 for 11, and do not round your answer. Be sure to include the correct unit inyour answer.)

Answer 1

So we basically need to find the perimeter of the garden formed by a rectangle and a semicircle. This perimeter is composed of three of the four sides of the recatangle and the arc of the semicircle.

We have the lengths of the sides of the rectangle but we need to find that of the arc of the semicircle. A semicircle is composed of a 180° arc and a straight segment. Since a semi circle is the half of a circle then the length of the straight segment is equal to the circle's diameter. Then that length divided by 2 gives us the radius of the circle and the semicircle.

We know that the straight line of this semicircle is also one of the sides of the rectangle and its length is equal to 23 ft which means that the radius of the semicircle is:

`r=(23ft)/(2)=11.5ft`

And this radius is important because the length of the arc is given by:

`L=r\cdot\theta`

Where theta is the arc's measure in radians. Since this is a semicircle the measure of the arc is 180°. In order to convert it to radians we have to multiply it by 2π and divide it by 360:

`\theta=(180\cdot2\pi)/(360)=(180)/(360)\cdot2\cdot3.14=(1)/(2)\cdot2\cdot3.14=3.14`

Then the length of the semicircle's arc is:

`L=11.5ft\cdot3.14=36.11ft`

Then the sum of this length and the three sides of the rectangle mentioned before give us the perimeter of the garden:

`36.11ft+33ft+23ft+33ft=125.11ft`

Then the answer is 125.11ft.