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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.11.5c
Chapter 3, Problem 3.11.5c

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?

Verified step by step guidance
1
First, identify the variables involved in the problem. Let h be the height of the water level in the pool, which is changing over time.
The volume V of the water in the pool can be expressed as V = length × width × height, or V = 20 × 10 × h. This simplifies to V = 200h.
Since the pool is being filled at a rate of 10 ft³/min, this is the rate of change of the volume with respect to time, denoted as dV/dt = 10 ft³/min.
To find the rate at which the water level is rising, we need to find dh/dt. Use the relationship between the rates: dV/dt = 200 × dh/dt.
Solve for dh/dt by dividing both sides of the equation by 200: dh/dt = (dV/dt) / 200. Substitute dV/dt = 10 ft³/min into the equation to find dh/dt.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Volume of a Rectangular Prism

The volume of a rectangular prism, such as a swimming pool, is calculated using the formula V = length × width × height. In this case, the pool's dimensions are given, and understanding this formula is essential to relate the volume of water being added to the change in water level.
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Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to determine how fast the water level (height) is rising as the volume of water (10 ft³/min) is added, which requires applying the concept of derivatives to relate the rates of change.
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Differentiation

Differentiation is a fundamental concept in calculus that deals with finding the rate of change of a function. In this context, we will differentiate the volume formula with respect to time to find the rate at which the height of the water is increasing as the pool is filled.
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Related Practice
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