Answer:
(a) Let h represents the height of water and w represents the width of the water,
Since, the depth of the water is increasing at a rate of 2 inches per hour,
So, after t hours,
The height of water, h(t) = 2t inches = t/6 ft,
( ∵ 1 foot = 12 inches ⇒ 1 inch = 1/12 ft )
Thus, the distance distance from the centre to the top of the water, d = 9 - h(t) ( see in the diagram )
,
By the Pythagoras theorem,
Since, diameter of the semicircular cross section is 18 ft,
So, 0 ≤ w ≤ 18,
i.e Range = [0, 18]
Also, w will be defined if 108t - t² ≥ 0
⇒ (108 - t)t ≥ 0,
⇒ 0 ≤ t ≤ 108
i.e Domain = [0, 108]
(b) If w = 6,
By using quadratic formula,
Hence, After 3.1 hours or 104.9 hours will the surface of the water have width of 6 feet.
(c)
For 0 ≤ w ≤ 18,
0 ≤ t ≤ 108,
So, Domain = [0, 18]
Range = [0, 108]