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.

b. At what rate is the volume of the water increasing if the water level is rising at 1/4ft/min.

Accepted Answer

Welcome back, everyone. A rectangular fish tank with a base measuring 8 ft by 15 ft and the uniform height is being filled with water. If the water level is rising at a rate of 1/3 feet per minute, at what rate is the volume of the water increasing? Were given or answer choices. A says 30 cubic feet per minute. B 40 cubic feet per minute. C 45 cubic feet per minute, and D 50 cubic feet per minute. So we're given a rectangular base, right, and there is a water level that is rising. So that would be our height at a specific time. So we can understand that this is a rectangular box with a volume of water V defined by A, which is multiplied by B, which is multiplied by C where A is width, B is length of the base, and C is height, right? So in this context, we can say that A is 8 ft and B is 15 ft. And now C is simply our height, right? C. Is a function of time. And our volume V is also a function of time. So what we can do is simply rewrite this as a function B of T equals a multiplied by B. This gives us a constant 120 square feet multiplied by C of T. And what we're going to do is also understand that the water level is rising at a given rate, so we're given DC divided by DT, right? We know the rate of change of the water level, and it is equal to 1/3. Feet per minute. What we want to do from here is differentiate both sides. Those are functions of time. So on the left hand side we're going to get dB divided by DT. The rate of change of volume with respect to time is equal to 120. This is a constant multiplied by DC. Divided by DT, right? So the height is changing. So relative to time, we're going to get DC divided by DT showing that this is the rate of change of the water level with respect to time. So now we want to solve for the rate of change of volume, which is the DB divided by DC. What we want to do is simply substitute the given data. So we are given 120, we can include the units, those would be square feet. Which is now multiplied by 1/3. Feet Per minute Overall we are simply dividing 120 by 3, which gives us 40, and then considering the units we are going to get feet cubed per minute, which 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.
b. At what rate is the volume of the water increasing if the water level is rising at 1/4ft/min.

Key Concepts

Volume of a Rectangular Prism

Related Rates

Differentiation

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