Question

Use the composite figure below to answer questions 4 and 5.

Four congruent circles create rounded corners on a rectangular carpet. A quarter of the circle overlaps each corner.
Find the perimeter of the figure to the nearest
A 1,414.5 inches
B. 2,640.5 inches
C.2,532.0 inches
D. 2,230.1 inches

Answer 1

B

Explanation:

Let's find perimeter of the 4 circles first.

Each circle is 3/4th of the whole circle. Perimeter of a circle is given by
`2\pi r` and since we have 3/4th of whole, so the perimeter of one circle would be
`(3)/(4)*2\pi r=(3)/(2)\pi r`

Now, what is the radius?

Since the whole bottom is 915 and the top part is 895 (it excludes 2 radii of 2 circles). So we have 915 - 895 = 20 (20 is equally distributed in 2 radii, so the radius of 1 circle is 20/2 = 10).

Hence, perimeter of 1 circle is
`(3)/(2)\pi r=(3)/(2)\pi (10)=15\pi`

And, 4 of these will have perimeter of
`4*15\pi = <strong>60\pi`

Now, let's find perimeter of the rectangle.

Top and bottom is 895 each, so that's 895 * 2 = 1790

Now, to find the right and left side, we use pythagorean theorem which is

leg^2 + leg^2 = hypotenuse^2

Where two legs are 915 and other one let's call it x, and hypotenuse is 1000-(10+10)=980. Thus,

915^2 + x^2 = 980^2

x^2 = 980^2 - 915^2

x^2 = 123,175

x = Sqrt(123,175)

x = 350.96

x includes the two radius so we need to subtract 10 + 10 = 20, thus we have 350.96 - 20 = 330.96 (left side). So left PLUS right side = 330.96 * 2 = 661.92

Thus, perimeter = 60π + 1790 + 661.92 = 2640.5

B is right.