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College Algebra 7th

James Stewart, Lothar Redlin, Saleem Watson

Chapter 0

Prerequisites

Educators

AG

Problem 1

The model $L=4 S$ gives the total number of legs that $S$ sheep have. Using this model, we find that 12 sheep have $L=$____ legs.

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Problem 2

Suppose gas costs $\$ 3.50$ a gallon. We make a model for the cost $C$ of buying $x$ gallons of gas by writing the formula $C=$____ .

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Problem 3

Use the model given to answer the questions about the object or process being modeled.
The sales tax $T$ in a certain county is modeled by the formula $T=0.06 x$. Find the sales tax on an item whose price is $\$ 120 .$

AG
Ankit G.
Numerade Educator

Problem 4

Use the model given to answer the questions about the object or process being modeled.
Mintonville School District residents pay a wage tax $T$ that is modeled by the formula $T=0.005 x$ Find the wage tax paid by a resident who earns $\$ 62,000$ per year.

AG
Ankit G.
Numerade Educator

Problem 5

Use the model given to answer the questions about the object or process being modeled.
The distance $d$ (in mi) driven by a car traveling at a speed of $v$ miles per hour for $t$ hours is given
by
$$
d=v t
$$
If the car is driven at $70 \mathrm{mi} / \mathrm{h}$ for $3.5 \mathrm{h}$, how far has it traveled?

AG
Ankit G.
Numerade Educator

Problem 6

Use the model given to answer the questions about the object or process being modeled.
The volume $V$ of a cylindrical can is modeled by the formula
$$
V=\pi r^{2} h
$$
where $r$ is the radius and $h$ is the height of the can. Find the volume of a can with radius 3 in. and height 5 in.

AG
Ankit G.
Numerade Educator

Problem 7

Use the model given to answer the questions about the object or process being modeled.
The gas mileage $M$ (in mi/gal) of a car is modeled by $M=N / G$, where $N$ is the number of miles driven and $G$ is the number of gallons of gas used.
a. Find the gas mileage $M$ for a car that drove 240 mi on 8 gal of gas.
b. A car with a gas mileage $M=25$ mi/gal is driven 175 mi. How many gallons of gas are used?

AG
Ankit G.
Numerade Educator

Problem 8

Use the model given to answer the questions about the object or process being modeled.
A mountain climber models the temperature $T$ (in $^{\circ} \mathrm{F}$ ) at elevation $h$ (in $\mathrm{ft}$ ) by
$$
T=70-0.003 h
$$
a. Find the temperature $T$ at an elevation of $1500 \mathrm{ft}$.
b. If the temperature is $64^{\circ} \mathrm{F}$, what is the elevation?

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Problem 9

Use the model given to answer the questions about the object or process being modeled.
The portion of a floating iceberg that is below the water surface is much larger than the portion above the surface. The total volume $V$ of an iceberg is modeled by
$$
V=9.5 S
$$
where $S$ is the volume showing above the surface.
a. Find the total volume of an iceberg if the volume showing above the surface is $4 \mathrm{km}^{3}$.
b. Find the volume showing above the surface for an iceberg with total volume $19 \mathrm{km}^{3}$.

AG
Ankit G.
Numerade Educator

Problem 10

Use the model given to answer the questions about the object or process being modeled.
The power $P$ measured in horsepower (hp) needed to drive a certain ship at a speed of $s$ knots is modeled by
$$
P=0.06 s^{3}
$$
a. Find the power needed to drive the ship at 12 knots.
b. At what speed will a 7.5 -hp engine drive the ship?
a. Find the power needed to drive the ship at 12 knots.
b. At what speed will a 7.5 -hp engine drive the ship?

AG
Ankit G.
Numerade Educator

Problem 11

An ocean diver models the pressure $P (\text { in } 1 b / \text { in }^{2} \text { ) at depth } d$ (in ft) by
$$P=14.7+0.45 d$$a. Make a table that gives the pressure for each $10-\mathrm{ft}$ change in depth, from a depth of $0 \mathrm{ft}$ to 60 ft..
b. If the pressure is $30 \mathrm{lb} / \mathrm{in}^{2},$ what is the depth?

AG
Ankit G.
Numerade Educator

Problem 12

Arizonans use an average of 40 gal of water per person each day. The number of gallons $W$ of water used by $x$ Arizonans each day is modeled by $W=40 x$
a. Make a table that gives the number of gallons of water used for each 1000 -person change in population, from 0 to 5000.
b. What is the population of an Arizona town whose water usage is 120,000 gal per day?

AG
Ankit G.
Numerade Educator

Problem 13

Write an algebraic formula that models the given quantity.
The number $N$ of cents in $q$ quarters

AG
Ankit G.
Numerade Educator

Problem 14

Write an algebraic formula that models the given quantity.
The average $A$ of two numbers $a$ and $b$

AG
Ankit G.
Numerade Educator

Problem 15

Write an algebraic formula that models the given quantity.
The cost $C$ of purchasing $x$ gallons of gas at $\$ 3.50$ a gallon

AG
Ankit G.
Numerade Educator

Problem 16

Write an algebraic formula that models the given quantity.
The amount $T$ of a $15 \%$ tip on a restaurant bill of $x$ dollars

AG
Ankit G.
Numerade Educator

Problem 17

Write an algebraic formula that models the given quantity.
The distance $d$ in miles that a car travels in $t$ hours at $60 \mathrm{mi} / \mathrm{h}$

AG
Ankit G.
Numerade Educator

Problem 18

Write an algebraic formula that models the given quantity.
The speed $r$ of a boat that travels $d$ miles in $3 \mathrm{h}$

AG
Ankit G.
Numerade Educator

Problem 19

A pizza parlor charges $\$ 12$ for a cheese pizza and $\$ 1$ for each topping.
a. How much does a 3 -topping pizza cost?
b. Find a formula that models the cost $C$ of a pizza with $n$ toppings.
c. If a pizza costs $\$ 16,$ how many toppings does it have?

AG
Ankit G.
Numerade Educator

Problem 20

a. How much does it cost to rent a car for 3 days if the car is driven $280 \mathrm{mi} ?$
b. Find a formula that models the cost $C$ of renting this car for $n$ days if it is driven $m$ miles.
c. If the cost for a 3 -day rental was $\$ 140$, how many miles was the car driven?

AG
Ankit G.
Numerade Educator

Problem 21

The cost of the electricity needed to drive an all-electric car is about 4 cents per mile. The cost of the gasoline needed to drive the average gasoline-powered car is about 12 cents per mile.
a. Find a formula that models the energy cost $C$ of driving $x$ miles for
i. the all-electric car and
ii. the average gasoline-powered car.
b. Find the cost of driving 10,000 mi with each type of car.

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Problem 22

A fruit crate has square ends and is twice as long as it is wide.
a. Find the volume of the crate if its width is 20 in.
b. Find a formula for the volume $V$ of the crate in terms of its width $x$

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Problem 23

In many universities students are given grade points for each credit unit according to the following scale:
A 4
points
B 3
points
C 2
points
$D \quad 1$ point
$F \quad 0$ point
For example, a grade of A in a 3 -unit course earns $4 \times 3=12$ grade points and a grade of $\mathrm{B}$ in a 5-unit course earns 3 $\times 5=15$ grade points. A student's grade point average (GPA) for these two courses is the total number of grade points earned divided by the number of units; in this case the GPA is $(12+15) / 8=3.375$
a. Find a formula for the GPA of a student who earns a grade of A in $a$ units of course work, $\mathrm{B}$ in
$b$ units, $\mathrm{C}$ in $c$ units, $\mathrm{D}$ in $d$ units, and $\mathrm{F}$ in $f$ units.
b. Find the GPA of a student who has earned a grade of A in two 3 -unit courses, $\mathrm{B}$ in one 4 -unit course, and C in three 3-unit courses.

AG
Ankit G.
Numerade Educator