A circular grass plot 12 feet in diameter is cut by a staright gravel path 3 feet wide, on e of which passes through the center of the plot . the number of square feet in the remaining grassy area is

The number of square feet in the remaining grassy area is 162 square feet.

Step-by-step explanation:

The number of square feet in the remaining grassy area = Area of the circular grass plot - area of the straight gravel path

Area of the circular grass plot =

Since its diameter is 12 feet, the its radius =

                                       =

                                       = 6 feet

So that,

            Area of the circular grass plot = ×

                                     = × 36

                                     = 198 squared feet

Area of the straight gravel path = length  × width

                                                    = 12  × 3

                                                    = 36 square feet

Thus,

The number of square feet in the remaining grassy area = 198 - 36

                                                     162 square feet.

The correct answer is E. 30π - 9√3

Correct statement, question and options:

A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is :

A. 36π - 34

B. 30π - 15

C. 36π - 33

D. 35π - 9√3

E. 30π - 9√3

Source:

Previous question that can be found at chegg

Step-by-step explanation:

1. Let's review the information given to us to answer the question correctly:

Diameter of the circular grass = 12 feet ⇒ Radius = 6 feet

Width of the straight gravel path = 3 feet

For answering the question, first we have to calculate the area of the semicircle, this way:

A = π * r²/2

A = π * 6²/2

A = 18π ft²

Now, we have to calculate the area of the circular segment, this way:

A = Sector AOB – △AOB.

We see that △AOC is a 30°-60° right △ since OC = 3 (width of the straight gravel path) and  OA = 6 (radius of the circular grass).

Thus, ∠AOC = 60° and ∠AOB = 120°.

A= 120/360 * π r²

A= 120/360 * π * 6²

A = 36π/3 = 12π ft²

Finally, △AOB has base AB = 6√3 and height OC = 3, therefore:

A △AOB = (6√3 * 3)/2 = 9√3 ft²

Now, we calculate the total area of the grassy area this way:

18π ft² + 12π ft² - 9√3 ft²

30π - 9√3 ft²

The correct answer is E. 30π - 9√3