At 1:00 p.m., Reservoir A contains more water than Reservoir B. Starting at 1:00 p.m., water is transferred from Reserv…

To find r in terms of D and M, set up linear models for the volume of each reservoir as a function of time t, in hours after 1:00 p.m.

Let A_0 and B_0 represent the initial volumes of Reservoir A and Reservoir B at 1:00 p.m., respectively, with A_0 > B_0.

Water leaves Reservoir A at rate r (transferred to B) and rate d (drained), so the volume of Reservoir A at time t is:

A(t) = A_0 - (r + d)t

Water enters Reservoir B at rate r, so the volume of Reservoir B at time t is:

B(t) = B_0 + rt

Now consider the combined volume of water in both reservoirs:

A(t) + B(t) = [A_0 - (r + d)t] + [B_0 + rt] = (A_0 + B_0) - dt

The combined volume decreases at a constant rate of d gallons per hour because the transferred water remains within the two-reservoir system. From 1:00 p.m. to 4:00 p.m. (t = 3 hours), the combined volume decreases by M gallons:

3d = M, which gives d = M / 3.

Next, consider the difference between the volume of water in Reservoir A and Reservoir B:

A(t) - B(t) = [A_0 - (r + d)t] - [B_0 + rt] = (A_0 - B_0) - (2r + d)t

The difference decreases at a constant rate of (2r + d) gallons per hour because transferring water at rate r reduces Reservoir A by r and simultaneously increases Reservoir B by r (reducing the gap by 2r), while the drainage at rate d further reduces Reservoir A (reducing the gap by d). From 1:00 p.m. to 6:00 p.m. (t = 5 hours), the difference decreases by D gallons:

5(2r + d) = D, which gives 2r + d = D / 5.

Substitute d = M / 3 into this equation:

2r + M / 3 = D / 5

2r = D / 5 - M / 3

2r = (3D - 5M) / 15

r = (3D - 5M) / 30

Choice A is incorrect because it assumes the difference decreases at a rate of (r + d) rather than (2r + d), neglecting the fact that transferring water affects both reservoirs and thus reduces the difference at twice the transfer rate.

Choice C is incorrect because it uses 2r - d instead of 2r + d, erroneously treating the drainage from Reservoir A as widening rather than narrowing the gap between the two reservoirs.

Choice D is incorrect because it reverses the two time intervals, assigning the 5-hour duration to the combined volume decrease and the 3-hour duration to the difference decrease.