Chapter 0, Prerequisites Video Solutions, Algebra and Trigonometry

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.

Amy Jiang

Numerade Educator

00:56

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 $C=$ _________.

Massimo Antonelli

Numerade Educator

01:00

Problem 3

$3-12=$ Using Models 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 $\$ 120 .$

Jennifer Stoner

Numerade Educator

01:04

Problem 4

Using Models 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.

Taylor Shimono

Numerade Educator

03:33

Problem 5

Using Models 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 milh for 3.5 $\mathrm{h}$ , how far has it traveled?

Kyle Icban

Numerade Educator

00:55

Problem 6

Using Models 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.

Taylor Shimono

Numerade Educator

07:20

Problem 7

Using Models Use the model given to answer the questions about the object or process being modeled.

The gas mileage $M($ in milgal) 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?

Alix Segil

Numerade Educator

02:13

Problem 8

Using Models 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 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?

Taylor Shimono

Numerade Educator

06:23

Problem 9

Using Models 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 sur-
face 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 show-
ing above the surface is 4 $\mathrm{km}^{3}$ .
(b) Find the volume showing above the surface for an ice-
berg with total volume 19 $\mathrm{km}^{3}$ .

Daniel Pezzi

Numerade Educator

02:33

Problem 10

Using Models 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?

Taylor Shimono

Numerade Educator

07:29

Problem 11

Using Models Use the model given to answer the questions about the object or process being modeled.

An ocean diver models the pressure $P$ ( in $\mathrm{lb} / \mathrm{in}^{2} )$ )
at depth $d$ (in ft) by

$$P=14.7+0.45 d$$

(a) Make a table that gives the pressure for each 10 -ft
change in depth, from a depth of 0 ft to 60 $\mathrm{ft}$ .
(b) If the pressure is $30 \mathrm{lb} / \mathrm{in}^{2},$ what is the depth?

James Choi

Numerade Educator

03:07

Problem 12

Using Models Use the model given to answer the questions about the object or process being modeled.

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) If the pressure is $30 \mathrm{lb} / \mathrm{in}^{2},$ what is the depth?
(b) What is the population of an Arizona town whose water
usage is $120,000$ gal per day?

Taylor Shimono

Numerade Educator

01:11

Problem 13

Making Models : Write an algebraic formula that models the given quantity.

13. The number $N$ of cents in $q$ quarters

Ruirui Liu

Numerade Educator

00:50

Problem 14

Making Models : Write an algebraic formula that models the given quantity.

14. The average $A$ of two numbers $a$ and $b$

Taylor Shimono

Numerade Educator

View

Problem 15

Making Models : Write an algebraic formula that models the given quantity.

15. The cost $C$ of purchasing $x$ gallons of gas at $\$ 3.50$ a gallon

Kelli Peritz

Numerade Educator

01:50

Problem 16

Making Models : Write an algebraic formula that models the given quantity.

16. The amount $T$ of a 15$\%$ tip on a restaurant bill of $x$ dollar

Taylor Shimono

Numerade Educator

01:47

Problem 17

Making Models : Write an algebraic formula that models the given quantity.

17. The distance $d$ in miles that a car travels in $t$ hours at 60 $\mathrm{milh}$

Reese Mikesell

Numerade Educator

01:21

Problem 18

Making Models : Write an algebraic formula that models the given quantity.

18. The speed $r$ of a boat that travels $d$ miles in 3 $\mathrm{h}$

Taylor Shimono

Numerade Educator

03:26

Problem 19

Cost of a Pizza 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
(c) If a pizza costs $\$ 16,$ how many toppings does it have?

$$n=1 \quad n=4$$

Karthik Balakrishnan

Numerade Educator

04:57

Problem 20

$\begin{array}{l}{\text { Renting a Car At a certain car rental agency a compact car }} \\ {\text { rents for } \$ 30 \text { a day and } 10 \phi \text { a mile. }} \\ {\text { (a) How much does it cost to rent a car for 3 days if the car }} \\ {\text { is driven } 280 \text { mi? }} \\ {\text { (b) Find a formula that models the cost } C \text { of renting this car }} \\ {\text { for } n \text { days if it is driven } m \text { miles. }}\end{array}$
(c) If the cost for a 3 -day rental was $\$ 140,$ how many miles
was the car driven?

Taylor Shimono

Numerade Educator

02:32

Problem 21

Energy cost for a Car 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.

Daniel Malawsky

Numerade Educator

02:44

Problem 22

Volume of Fruit Crate $A$ 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 $x .$

Taylor Shimono

Numerade Educator

06:21

Problem 23

Grade Point Average In many universities students are given
grade points for each credit unit according to the following
scale:
$$\begin{array}{ll}{A} & {4 \text { points }} \\ {B} & {3 \text { points }} \\ {C} & {2 \text { points }} \\ {D} & {1 \text { point }} \\ {F} & {0 \text { point }}\end{array}$$
For example, a grade of $\mathrm{A}$ 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, $B$ in $b$ units, $\operatorname{Cin} c$
units, $D$ in $d$ units, and $F$ in $f$ units.
(b) Find the GPA of a student who has earned a grade of A
in two 3 -unit courses, B in one 4 -unit course, and $C$ in
three 3 -unit courses.

Shubham Jain

Numerade Educator