University Calculus: Early Transcendentals

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 16

First-Order Differential Equations - all with Video Answers

Educators

Section 3

Applications

Problem 1

Coasting bicycle $\quad$ A $66-\mathrm{kg}$ cyclist on a 7 -kg bicycle starts coasting on level ground at $9 \mathrm{m} / \mathrm{sec} .$ The $k$ in Equation (1) is about 3.9 kg/sec.
a. About how far will the cyclist coast before reaching a complete stop?
b. How long will it take the cyclist's speed to drop to $1 \mathrm{m} / \mathrm{sec} ?$

Matthew A.

Matthew A.

Numerade Educator

Problem 2

Coasting battleship Suppose that an Iowa class battleship has mass around 51,000 metric tons $(51,000,000 \mathrm{kg})$ and a $k$ value in Equation (1) of about 59,000 kg/sec. Assume that the ship loses power when it is moving at a speed of $9 \mathrm{m} / \mathrm{sec}$
a. About how far will the ship coast before it is dead in the water?
b. About how long will it take the ship's speed to drop to $1 \mathrm{m} / \mathrm{sec} ?$

Matthew A.

Numerade Educator

Problem 3

The data in Table 16.4 were collected with a motion detector and a CBL Tit by Valerie Sharritts, then a mathematics teacher at St. Francis DeSales High School in Columbus, Ohio. The table shows the distance $s$ (meters) coasted on inline skates in $t$ sec by her daughter Ashley when she was 10 years old. Find a model for Ashley's position given by the data in Table 16.4 in the form of Equation ( $2) .$ Her initial velocity was $v_{0}=2.75 \mathrm{m} / \mathrm{sec},$ her mass $m=39.92 \mathrm{kg}$ (she weighed 88 ib), and her total coasting distance was $4.91 \mathrm{m}$

Matthew A.

Numerade Educator

Problem 4

Coasting to a stop Table 16.5 shows the distance $s$ (meters) coasted on inline skates in terms of time $t$ (seconds) by Kelly Schmitzer. Find a model for her position in the form of Equation ( 2 ). Her initial velocity was $v_{0}=0.80 \mathrm{m} / \mathrm{sec},$ her mass $m=49.90 \mathrm{kg}$ $(110 \mathrm{lb}),$ and her total coasting distance was $1.32 \mathrm{m}$

Matthew A.

Numerade Educator

Problem 5

Find the orthogonal trajectories of the family of curves. Sketch several members of each family.
$$y=m x$$

Matthew A.

Numerade Educator

Problem 6

Find the orthogonal trajectories of the family of curves. Sketch several members of each family.
$$y=c x^{2}$$

Matthew A.

Numerade Educator

Problem 7

Find the orthogonal trajectories of the family of curves. Sketch several members of each family.
$$k x^{2}+y^{2}=1$$

Matthew A.

Numerade Educator

Problem 8

Find the orthogonal trajectories of the family of curves. Sketch several members of each family.
$$2 x^{2}+y^{2}=c^{2}$$

Matthew A.

Numerade Educator

Problem 9

Find the orthogonal trajectories of the family of curves. Sketch several members of each family.
$$y=c e^{-x}$$

Matthew A.

Numerade Educator

Problem 10

Find the orthogonal trajectories of the family of curves. Sketch several members of each family.
$$y=e^{d x}$$

Matthew A.

Numerade Educator

Problem 11

Show that the curves $2 x^{2}+3 y^{2}=5$ and $y^{2}=x^{3}$ are orthogonal.

Matthew A.

Numerade Educator

Problem 12

Find the family of solutions of the given differential equation and the family of orthogonal trajectories. Sketch both families.
a. $x d x+y d y=0$
b. $x d y-2 y d x=0$

Matthew A.

Numerade Educator

Problem 13

A tank initially contains 100 gal of brine in which
50 lb of salt is dissolved. A brine containing 2 Ib/gal of salt runs into the tank at the rate of 5 gal/min. The mixture is kept uniform by stirring and flows out of the tank at the rate of 4 gal/min.
a. At what rate (pounds per minute) does salt enter the tank at time $t ?$
b. What is the volume of brine in the tank at time $t ?$
c. At what rate (pounds per minute) does salt leave the tank at time $t ?$
d. Write down and solve the initial value problem describing the mixing process.
e. Find the concentration of salt in the tank 25 min after the process starts.

Sajin S.

Numerade Educator

Problem 14

A 200 -gal tank is half full of distilled water. At time $t=0,$ a solution containing $0.5 \mathrm{lb} / \mathrm{gal}$ of concentrate enters the tank at the rate of 5 gal/min, and the well-stirred mixture is withdrawn at the rate of 3 gal/min.
a. At what time will the tank be full?
b. At the time the tank is full, how many pounds of concentrate will it contain?

Sajin S.

Numerade Educator

Problem 15

A tank contains 100 gal of fresh water. A solution containing $11 b /$ gal of soluble lawn fertilizer runs into the tank at the rate of 1 gal/min, and the mixture is pumped out of the tank at the rate of 3 gal/min. Find the maximum amount of fertilizer in the tank and the time required to reach the maximum.

Sajin S.

Numerade Educator

Problem 16

An executive conference room of a corporation contains $4500 \mathrm{ft}^{3}$ of air initially free of carbon monoxide. Starting at time $t=0,$ cigarette smoke containing $4 \%$ carbon monoxide is blown into the room at the rate of $0.3 \mathrm{ft}^{3} / \mathrm{min} .$ A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of $0.3 \mathrm{ft}^{3} / \mathrm{min}$. Find the time when the concentration of carbon monoxide in the room reaches $0.01 \%$

Sajin S.

Numerade Educator