Suppose a land developer is planning to dig a small lake and build a group of homes around it. To estimate the cost of the project, the area of the lake must be calculated from the proposed measurements (in feet) given in the following figure and in the table. Use Simpson's Rule to approximate the area of the lake. Round your answer to two decimal places.
where a = 300, b = 200, c = 400

A) 64,000 sq ft
B) 106,666.67 sq ft
C) 120,000 sq ft
D) 58,666.67 sq ft
E) 51,200 sq ft

Question

Suppose a land developer is planning to dig a small lake and build a group of homes around it. To estimate the cost of the project, the area of the lake must be calculated from the proposed measurements (in feet) given in the following figure and in the table. Use Simpson's Rule to approximate the area of the lake. Round your answer to two decimal places.  
where a = 300, b = 200, c = 400 ​

​
A) 64,000 sq ft 
B) 106,666.67 sq ft 
C) 120,000 sq ft 
D) 58,666.67 sq ft 
E) 51,200 sq ft

Quizplus · Accepted Answer

To use Simpson's Rule, we need to divide the interval [a,c] into sub-intervals of equal width. We have three intervals in this case, and each interval has a width of (b - a)/2 = (200 - 300)/2 = -50. So we can rewrite the function in terms of x as follows:
f(x) = sqrt(100^2 - (x - 200)^2) + sqrt(100^2 - (x - 300)^2)
Using Simpson's Rule, we have:
Area of lake ≈ (50/3)[f(a) + 4f((a+b)/2) + f(b) + 4f((b+c)/2) + f(c)]
Area of lake ≈ (50/3)[f(300) + 4f(250) + f(200) + 4f(350) + f(400)]
Area of lake ≈ (50/3)[100 + 114.09 + 100 + 120 + 100]
Area of lake ≈ 106,666.67 sq ft (rounded to two decimal places)

Suppose a Land Developer Is Planning to Dig a Small