(edited 5/31/2017)

**Some Algebra Word Problems – Part II**
Source: Blitzer, Robert

*Introductory & Intermediate Algebra for College Students*3^{rd}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.

Here are five
more problems. For the previous
installment, click this link: http://edspi31415.blogspot.com/2017/05/some-algebra-word-problems.html

**What size is the Garden?**

Problem:

The
length of a rectangular garden is 5 feet greater than its width. The garden’s area is 300 square feet. Find the dimensions of the garden. [pg. 452, number 82 – Blitzer]

Discussion:

The area of a rectangle
is:

Area = Length *
Width

In this case,
Width = x, Length = x + 5, and Area = 300, with width and length in feet, area
in square feet.

Then:

(x + 5) * x =
300

x^2 + 5 * x =
300

What we have is
a quadratic equation. Subtract 300 from
both sides yields

x^2 + 5*x – 300
= 0

We can use the
quadratic equation or try to factor. I’m
going to factor:

(x – 15) * (x +
20) = 0

Which leads to
x = 15 and x = -20. In this case,
negative lengths do not make sense, hence the solution we are looking for x =
15.

To conclude,
the Width is 15 feet, while the Length is 20 feet (15 + 5).

**Trees and Triangles**

Problem:

A
person who is 5 feet tall is standing 80 feet from a tree. At the time, the tree casts an 86-foot
shadow, while the person casts a 6-foot shadow.
What is the tree’s height? [pg.
527, number 32 – Blitzer]

Discussion:

We are working
with similar triangles. Please see the
diagram. The ratio of height/length is:

x/86 = 5/6

x = 86 * 5/6

x = 71.666666

The height of
the tree is 71.666666 feet.

*

*

*I previously had the incorrect answer of 70.83333. I thank Gianfranco Cazzaro for alerting me on my mistake. The answer is now correct. - EWS 5/31/2017***From Bangkok to Phnom Penh**

Problem:

On
a map, Bangkok are given coordinates (-115, 170) and Phnom Penh has the
coordinates (65, 70). Units are in
miles. How long will it take a plane
flying 400 miles an hour (mph) on a direct flight between Bangkok and Phnom
Penh? Round off the answer to the nearest tenth of an hour. Assume the flight takes place in a straight
line. [pg. 750, number 121 – Blitzer]

Discussion:

We can use the
following two formulas to help us solve the problem:

Distance
between points (x1, y1) and (x2, y2):

Distance = √(
(x1 – x2)^2 + (y1 – y2)^2 )

Relationship
between Distance, Speed, and Time:

Time = Distance
/ Speed

Let Point 1
(x1, y1) represent Bangkok (-115, 170) and Point 2 represents Phnom Penh (65,
70). Hence, Distance =√( (-115 – 65)^2 +
(170 – 70)^2 ) , and Speed = 400.

Then:

Time = √( (-115
– 65)^2 + (170 – 70)^2 ) / 400

Time = √( 180^2
+ 100^2 ) / 400

Time ≈ 0.5

The plane will
take about a half an hour to travel from Bangkok to Phnom Penh.

**Making It Rain**

Problem:

A
rain gutter (as shown above) is made from aluminum sheet 20 inches wide. Edges that are x inches wide are turned up on
each side to form the gutter. Find x to
allow a cross sectional area of 13 square inches. Round off the answer to the nearest tenth of
an inch. [pg. 764, number 81 – Blitzer]

Discussion:

If you fold the
edges up, you get a rectangle with sides x in. and 20 – 2*x in. We want to find x such that the area is 13
in^2. Then:

x * (20 – 2 *
x) = 13

20 * x – 2 *x^2
= 13

Bringing all
terms to one side:

0 = 2*x^2 – 20
*x + 13

We have a
quadratic equation. This time, we’ll use
the quadratic formula:

x = ( 20 ±
√(20^2 – 4 * 2 * 13) ) / (2 * 2)

With a
calculator the results are (results rounded to tenths):

x ≈ 9.3 in and
x ≈ 0.7 in.

There is no
specification which measure is preferable, so both answers are acceptable.

**Plot Enclosure**

Problem:

You
have 600 feet of fencing to enclose a rectangular plot (see figure). The side facing the river will not be
enclosed. What is the largest area that
can be enclosed? [pg. 782, number 65 –
Blitzer]

The area of the
enclosure is:

A = x * (600 –
2*x)

A = 600 * x – 2
* x^2 (I)

The equation is
in the form of a quadratic equation y = a*x^2 + b*x + c. In general, if a < 0, the quadratic
equation will have a maximum at x = -a / (2*b).

From (I), a =
-2, b = 600, and c = 0. Since a < 0,
we will have a maximum at x = -600 / (2 * -2) = 150.

The sides will
have 150 feet by 300 feet. The maximum
area is 150 ft * 300 ft = 45,000 ft^2.

Eddie

This blog is property of Edward Shore, 2017.

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