Answer: To find the volume of water in the pool, we can divide it into smaller sections and calculate the volume of each section separately. Let's divide the pool into horizontal slices.
The pool is circular with a 20-ft diameter, which means the radius (r) is half of the diameter, so r = 20/2 = 10 ft.
The depth increases linearly from 5 ft at the south end to 10 ft at the north end. The average depth of each slice can be calculated by taking the average of the depths at the two ends of the slice.
Let's consider a small slice of the pool between two consecutive east-west lines, with a width of Δx. The depth at the south end of the slice is 5 ft, and the depth at the north end is 10 ft. The average depth of this slice is (5 + 10) / 2 = 7.5 ft.
The volume of this slice can be calculated by multiplying its average depth by the width (Δx) and the circumference of the pool (2πr):
Volume of slice = average depth * Δx * circumference
= 7.5 ft * Δx * (2π * 10 ft)
To find the total volume of the pool, we need to sum up the volumes of all the slices. Since the depth is constant along east-west lines, the width Δx cancels out in the summation.
Total volume = ∑(Volume of each slice)
= ∑(7.5 ft * Δx * (2π * 10 ft))
= 7.5 ft * (2π * 10 ft) * ∑(Δx)
As the width Δx approaches zero, the summation ∑(Δx) becomes the integral of 1 with respect to x from the south end to the north end.
Total volume = 7.5 ft * (2π * 10 ft) * ∫(1 dx)
= 7.5 ft * (2π * 10 ft) * [x] between south end and north end
= 7.5 ft * (2π * 10 ft) * (north end - south end)
The north end of the pool is at a depth of 10 ft, and the south end is at a depth of 5 ft. So:
Total volume = 7.5 ft * (2π * 10 ft) * (10 ft - 5 ft)
= 7.5 ft * (2π * 10 ft) * 5 ft
= 7.5 * 2 * 10 * 5 * π * ft³
= 750 * π ft³
Now we can calculate the approximate value of the volume:
Total volume ≈ 750 * 3.14159 ft³
≈ 2356.194 ft³
Rounding to the nearest whole number, the volume of water in the pool is approximately 2356 ft³.