A rectangular swimming pool 10 ft wide by 20 ft long and of un...

Question

A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water.

c. At what rate is the water level rising if the pool is filled at a rate of 10ft³/min?

Accepted Answer

Welcome back, everyone. A rectangular garden pond measuring 8 ft in width and 12 ft in length is being filled with water at a steady rate. If the filling rate is 6 cubic feet per minute, what is the rate at which the water depth is increasing? Were given 4 answer choices A says 1 divided by 20 ft per minute. B 1 divided by 16 ft per minute, C 1 divided by 13 ft per minute, and D 1 divided by 10 ft per minute. So first of all, let's define the volume of a rectangular box because this is what we have. The volume of a rectangular box can be calculated by multiplying length of the base by the width of the base, and finally multiplying that by height. So the area of the base multiplied by height. In the context of this problem, we know that this rectangular garden pond is measuring 8 ft in width, so our width is 8 ft. And length L is 12 ft. We also know that the filling rate is 6 cubic. Feet per minute, right? So this is the rate of change of a volume with respect to time, which is DV divided by DT. It is 6 cubic feet per minute. Now, what we're going to do is simply evaluate the rate of change of depth. And in this context our depth is h. We're going to take the derivative of both sides, but before we do that, knowing that L and W are constants, we're going to multiply them, right? So we're going to get E equals 8 multiplied by 12, which is 96, right? H. So that's our function, and now we want to differentiate it with respect to time. So we're going to get the DB divided by DT on the left side. On the right side, we have a constant 96 multiplied by the age divided by DT because depth is changing relative to time. And now if we want to solve for the H divided by DT, this is the target variable, the rate of change of depth. This is going to be one. Divided by 96, multiplied by D V divided by DT, right? So we are essentially dividing both sides by 96. And what we're going to get Is 1 divided by 96 multiplied by 6 ft. Cubed per minute. Now what about the units? Well, we have to recall that 96 was a product of feet multiplied by feet, so that's feet squared. And now we can see that our units cancel out and we get 6 divided by 96, which is 1 divided by 16 ft per minute. Which is our final answer to this problem, and it corresponds to the answer choice B. Thank you for watching.

A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water.
c. At what rate is the water level rising if the pool is filled at a rate of 10ft³/min?

Key Concepts

Volume of a Rectangular Prism

Related Rates

Differentiation

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