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Coffee is draining from a conical filter (vertex down) into a cylindrical coffeepot at the rate of 8 cubic inches per minute. The filter and the coffeepot both have a diameter of 6 inches, and the height of the filter is also 6 inches. How fast is the level in the pot rising when the coffee in the cone is 5 inches deep?

A. 1.6 inches per minute
B. 2 inches per minute
C. 2.4 inches per minute
D. 3 inches per minute

1 Answer

4 votes

The answer is C. 2.4 inches per minute.

How can you solve how fast the level in the pot rises when the coffee in the cone is 5 inches deep?

V c(t) = (1/3)πr²h c(t) = (1/3)π(3²)(5) = 25π cubic inches

The rate of change of the coffee volume in the pot is equal to the draining rate:

dV p(t)/dt = Rate = 8 cubic inches per minute

V p(t) = πr²h p(t)

The total volume of coffee (V c(t) + V p(t)) is constant:

V c(t) + V p(t) = V c(0) + V p(0) (initial volume)

V c(t) + πr²h p(t) = V c(0) + V p(0)

dV c(t)/dt + 2πr²h p(t)dh p(t)/dt = 0

8 + 2πr²h p(t)dh p(t)/dt = 0

dh p(t)/dt = -8 / (2πr²h p(t))

dh p(t)/dt = -8 / (2π(3²)(5)) ≈ -2.4 inches per minute

The negative sign indicates that the height of the coffee in the pot is decreasing, not rising. However, we need the absolute value of the rate of change. Therefore, the level in the pot is rising at a rate of:

|dh p(t)/dt| ≈ 2.4 inches per minute

Therefore, the answer is C. 2.4 inches per minute.

User Mathieu Castets
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