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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
Textbook Question

{Use of Tech} Spring runoff The flow of a small stream is monitored for 90 days between May 1 and August 1. The total water that flows past a gauging station is given by v(t) = <matrix 2x2> where V is measured in cubic feet and t is measured in days, with t=0 corresponding to May 1.

c. Describe the flow of the stream over the 3-month period. Specifically, when is the flow rate a maximum?

Textbook Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 64 ft/s from a height of 32 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+64t+32.

c. What is the height of the stone at the highest point?

Textbook Question

Derivatives using tables Let h(x)=f(g(x))h(x)=f(g(x)) and p(x)=g(f(x))p(x)=g(f(x)). Use the table to compute the following derivatives.

<IMAGE>

c. p′(4)p^{\(\prime\)}\(\left\)(4\(\right\))

Textbook Question

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x) = (x−1) sin^−1 x on [−1,1]

Textbook Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 19.6 m/s from a height of 24.5 m above the ground. The height (in meters) of the stone above the ground t seconds after it is thrown is s(t) = -4.9t²+19.6t+24.5.

c. What is the height of the stone at the highest point?

Textbook Question

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

c. Solve the equation y(x²+4)=8 for y to find an explicit expression for y and then calculate dy/dx.