Part A
From the question, we know that
Rate at which pool is filled = 8g² + 3g - 4
Rate at which water leaves the pool = 9g² - 2g - 5
Hence, height of the pool would be = Rate at which pool is filled - Rate at which water leaves the pool
⇒ Height of the pool = 8g² + 3g - 4 - (9g² - 2g - 5)
⇒ Height of the pool = -g² + 5g + 1
Part B
Let water height be H
H (g) = -g² + 5g + 1
When g=1, H(1) = -(1)² + 5 (1) + 1 = 5 units
When g=2, H(2) = -(2)² + 5 (2) + 1 = -4+11 = 7 units
When g=3, H(3) = -(3)² + 5 (3) + 1 = -9+16 = 7 units
When g=4, H(4) = -(4)² + 5 (4) + 1 = -16+21 = 5 units
Part C
Now, we need to determine the value for g for which height would be maximum.
Looking at the expression that determines height, -g² + 5g + 1, we see that the coefficient of g² is negative. Hence the equation would represent a downward facing parabola, which means that the function H(g) will have a maxima point.
To find out the maxima point, differentiate H(g) with respect to g, and equate the resulting expression to zero.
= -2g + 5 = 0
⇒ 2g = 5
⇒ g = 2.5
So at g = 2.5, the height of the water in the pool is maximum. [Note: nearest tenth means rounding till the first decimal point, hence the answer is g=2.5]