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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.1.46c
Chapter 6, Problem 6.1.46c

Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions
r1(t) = 0.25t²+37.46t+722.47 (April) and
r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).
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c. The Spokane River flows out of Lake Coeur d’Alene, which contains approximately 0.67mi³ of water. Determine the percentage of Lake Coeur d’Alene’s volume that flows through Spokane in April and June.

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Identify the discharge functions for April and June: April is given by \(r_1(t) = 0.25t^{2} + 37.46t + 722.47\) and June by \(r_2(t) = 0.90t^{2} - 69.06t + 2053.12\), where \(t\) is in days and \(r(t)\) is in millions of cubic feet per day.
Calculate the total volume of water discharged in each month by integrating the discharge functions over the time interval from \(t=0\) to \(t=30\) (assuming 30 days in April and June). This means computing the definite integrals: \(V_1 = \int_0^{30} r_1(t) \, dt\) for April and \(V_2 = \int_0^{30} r_2(t) \, dt\) for June.
Convert the volume of water discharged from millions of cubic feet to cubic miles to be consistent with the volume of Lake Coeur d’Alene. Use the conversion factor: 1 cubic mile = 147,197,952,000 cubic feet. Since the discharge is in millions of cubic feet, multiply by \$10^6$ before converting.
Calculate the percentage of the lake's volume that flows through Spokane by dividing the volume discharged in each month by the total volume of the lake (0.67 cubic miles) and then multiplying by 100. That is, for April: \(\text{Percentage}_1 = \frac{V_1}{0.67} \times 100\) and for June: \(\text{Percentage}_2 = \frac{V_2}{0.67} \times 100\).
Interpret the results to understand how much of the lake's water volume passes through Spokane in each month, which provides insight into the river's flow relative to the lake's capacity.

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Key Concepts

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

Definite Integral as Accumulated Quantity

The definite integral of a rate function over a time interval represents the total accumulated quantity during that period. In this problem, integrating the discharge rate functions r1(t) and r2(t) over the days in April and June gives the total volume of water flowing through the river in those months.
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Unit Conversion and Volume Comparison

To compare the total volume of water discharged with the volume of Lake Coeur d’Alene, it is essential to convert units consistently. The river discharge is given in millions of cubic feet per day, while the lake volume is in cubic miles. Converting cubic miles to cubic feet allows for a meaningful percentage comparison.
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Quadratic Functions and Their Graphs

The discharge rates are modeled by quadratic functions, which describe how the flow changes over time. Understanding the shape and behavior of these parabolas helps interpret the flow trends and ensures correct integration limits and calculations for total discharge.
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Related Practice
Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.


d. Write an integral for the volume of the solid.

Textbook Question

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

c. Write an integral for the volume of the solid using the shell method.

Textbook Question

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

d. A piecewise function for s(t)

Textbook Question

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.


c. When will the tank be full?

Textbook Question

Let R be the region bounded by the curve y=√cos x and the x-axis on [0, π/2]. A solid of revolution is obtained by revolving R about the x-axis (see figures). 


c. Write an integral for the volume of the solid.

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Textbook Question

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function

v(t)={3t if 0≤t<2060 if 20≤t<45240−4t if t≥45v(t)= \(\begin{cases}\)3 t & \(\text\) { if } 0 \(\leq\) t<20 \\ 60 & \(\text\) { if } 20 \(\leq\) t<45 \\ 240-4 t & \(\text\) { if } t \(\geq\) 45\(\end{cases}\)

where is measured in seconds and v has units of m/s. 

d. What is the position of the automobile when t=75?