Answer:
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
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