Some Algebra Word Problems
Source: Blitzer, Robert Introductory
& Intermediate Algebra for College Students 3rd Edition
Pearson, Prentice Hall: Upper Saddle
River, New Jersey 2009 ISBN-13:
978-0-13-602895-6
Problems and
diagrams are rewritten. No claims of
profit are made.
A Simple Equation for Heart Beat
Problem:
Write
a formula that models that the heart rate, H, in beats per minute, for a person
who is a years old and likes to
achieve 9/10 of maximum (90% of maximum) heart rate during exercise. Find the heart during exercise for a 40 year
old.
Information: The book gives the maximum heart rate in
beats per minute as 220 – a. [pg. 29,
number 127 - Blitzer]
Discussion:
The maximum
heart rate is given by 220 – a. Since we
want to achieve 9/10 of the maximum, multiply the percentage by the heart
rate. An appropriate function is:
H = 9/10 * (220
– a)
At 40 years
old, let a = 40 and H = 9/10 * (220 – 40) = 9/10 * 180 = 162
The maximum
rate would be 162 beats per minute.
Doubling the Sphere
Problem:
What
happens to the volume of a sphere if its radius is doubled? [pg. 171 – number
89 – Blitzer]
Discussion:
The volume of a
sphere is V = 4/3 * π * r^3, where r is the radius.
If we double
the radius (sub 2*r for r), then the volume would be:
V = 4/3 * π *
(2*r)^3
V = 4/3 * π * 8 * r^3 (I)
Simplifying:
V = 32/3 * π *
r^3
Effectively,
doubling the radius increases the sphere’s volume eight times. (see (I))
What Angle Are Talking About?
Problem: (diagram page 171 number 91)
Find
the measure of angle x as indicated by the figure below. [pg. 171, number 91 - Blitzer]
Discussion:
Assume all
angles are measured in degrees. From the
diagram above, label angle A = 2 * x and angle B = 2 * x + 40.
Observe that A
+ B = 180°. We will use this to solve
for x.
A + B = 180
(2*x) + (2*x +
40) = 180
4*x + 40 = 180
4*x + 40 – 40 =
180 – 40
4*x = 140
4*x/4 = 140/4
x = 35
The measure of
x is 35°.
The Sum and Difference of Two Numbers
Problem:
The
sum of two numbers, x and y, is 28. The
difference between the numbers is 6.
What are x and y? [pg. 287,
number 85 – Blitzer]
Discussion:
The goal is to
find the values of the two unknowns, x and y.
The sum of x
and y is 28. Sum means addition. Hence, x + y = 28.
The difference
is 6, and difference means subtraction.
Since no variable is specified, let’s assume x is greater than y. The equation would be x – y = 6.
Given both
statements are true, we have a pair of simultaneous equations:
(I) x + y = 28
(II) x – y = 6
Now it is a
matter of solving for both x and y.
Note how the equations are marked above as (I) and (II).
First add
equations (I) and (II) to get:
2 * x = 34
2 * x / 2 = 34
/ 2
x = 17
I am going to
substitute x = 17 in equation (I) and solve for y (doing this in (II) would
work too).
17 + y = 28
17 + y – 17 =
28 – 17
y = 11
Check: 17 + 11 = 28 and 17 – 11 = 6
Yes, it checks
out. The solution is x = 17 and y = 11.
The Cost of Competing Telephone Plans
Problem:
You
are choosing between two-long distance telephone plans. Plan A has a monthly fee of $15 with a charge
of $0.08 per minute for all-long distance calls. Plan B has a monthly fee of $5 with a charge
of $0.10 per minute for long-distance calls.
Determine the number of minutes when the cost of the two plans will be
the same. What will be the cost? [pg. 301, number 17a – Blitzer]
Discussion:
Let x be the
number of minutes. We can describe the
two plans as equations:
Plan A: C = 15 + 0.08*x
Plan B: C = 5 + 0.10*x
Note that I
assigned C as the cost. Also, both
equations take the form of C = fixed cost + variable cost * x.
The problem
asks us how many minutes will make both plans have the same cost. For this, equate the costs of both plans A
and B.
15 + 0.08 * x =
5 + 0.10 * x
15 + 0.08 * x –
5 = 5 + 0.10 * x – 5
10 + 0.08 * x =
0.10 * x
10 + 0.08 * x –
0.08 * x = 0.10 * x – 0.08 * x
10 = 0.02 *
x (note that 0.02 = 2/100)
10 * 100/2 =
2/100 * x * 100/2
500 = x
As a
check: 15 + 0.08 * 500 = 55 and 5 + 0.10
*500 = 55
The amount of
minutes of long distance calls that will make both plans cost the same is 500
minutes, and the cost will be $55.00.
Please let me
know if you find this helpful, especially students who are in algebra
classes. This is something different I
want to do with the blog in addition to all the calculator programming.
Eddie
This blog is
property of Edward Shore, 2017.
