Some Algebra Word Problems – Part II
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.
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?
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]
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.
(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
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]
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 = 70.833333
The height of the tree is 70.833333 feet.
From Bangkok to Phnom Penh
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]
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.
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
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]
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.
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.
This blog is property of Edward Shore, 2017.