Section 4.5
Word Problems
Examples:
a. The second of two numbers is 5 larger than the first. The sum of the squares of the two numbers is 125. What are the two numbers? (There are two solutions.)
Solution:
Let $x$ be one of the numbers. Then $x+5$ is the second number. So,
\begin{align}
x^2+(x+5)^2&=125\\
x^2+x^2+10x+25&=125\\
2x^2+10x-100&=0\\
2(x^2+5x-50)&=0\\
2(x+10)(x-5)&=0
\end{align}
So, $x=5$ or $x=-10$. Since $x$ is the first number, if $x=5$ then the second number is $10$. If $x=-10$ is the first number, the second number is $-5$.
b. Find two consecutive numbers with the property that the square of the larger, minus the smaller, is 31. (There are two solutions.)
Solution:
Let $x$ be one of the numbers. Then $x+1$ is the second number. So,
\begin{align}
(x+1)^2-x&=31\\
x^2+2x+1-x&=31\\
x^2+x-30&=0\\
(x+6)(x-5)&=0
\end{align}
So, $x=-6$ or $x=5$. Since $x$ is the first number, if $x=-6$ then the second number is $-5$. If $x=5$ is the first number, the second number is $6$.
Exercises:
1. The second of two numbers is 5 larger than the first. The sum of the squares of the two numbers is 125. What are the two numbers? (There are two solutions.)2. The second of two numbers is 7 larger than the first. The square of the larger number, increased by twice the smaller number, is 21. What are the two numbers? (There are two solutions.)
3. Find two consecutive numbers with the property that the square of the larger, minus the smaller, is 31. (There are two solutions.)
4. Find two consecutive integers with the property that the sum of the squares of the numbers, increased by the sum of both the numbers, is 242. (There are two solutions.)
5. Find two consecutive even integers with the property that twice the square of the larger, minus the smaller, is 122.
6. The tens' digit of a two-digit number is 3 less than the units' digit. The sum of the squares of the digits is 18 more than the number itself. What is the number?
7. Al Pine jumps off the Lake Placid ski jump with an initial upward velocity of 18 meters per second.
a. How long does it take Al to reach 9 meters above the jump?
b. How far below the lip of the ski jump does Al land if he is in the air for 5 seconds?
8. The tens' digit of a two-digit number is eight less than the square of the units' digit. If the digits are reversed, the resulting number is 36 less than the original number. What is the original number?
9. Find two positive numbers that are in the ratio of $2:3$ and such that the square of the smaller one added to twice the larger one gives 88.
10. Find two positive numbers that are in the ratio of $2:5$ and such that the square of the smaller one decreased by four times the larger one is 24.
11. The tens' digit of a two-digit number is one more than the units' digit. The number formed by adding the squares of both the tens' digit and the units' digit is four less than the number. What is the number?
12. The tens' digit of a two-digit number is two less than the units' digit. The square of the tens' digit added to twice the square of the units' digit is four less than twice the original number. What is that number?
13. Find (by experimentation, not by algebraic means) three different three-digit numbers such that the sum of the squares of the digits is 25.
14. Determine all the numbers with the property that the square of the number is 34 more than 15 times the number.
15. A rectangle is 8 inches by 10 inches. There is a border of uniform width around the rectangle. The area of the combined rectangle-plus-border is 99 square inches. How wide is the border?
16. A rectangle is 9 inches by 11 inches. There is a border of uniform width around the rectangle. The area of the combined rectangle-plus-border is 130 square inches. How wide is the border?
17. Jack and Jill plan to race to the top of the hill, 1.5 miles from where they are right now. Jack walks, but Jill jogs. She can do 2 miles per hour faster than he. They leave the base of the hill at the same time, and she beats him to the top by 12 minutes. How fast do Jack and Jill go?
18. On his most recent trip from Lowell to Hartford and back, a distance of 96 miles each way, Chris averaged 12 mph faster on the return than going. As a result, the return trip took 24 minutes less time. What was Chris's average speed on the return trip?
19. A fast train from Washington, DC, to Danbury, CT, a distance of 288 miles, runs 12 miles per hour faster than a slow train. The fast train makes the trip in hour less time than the slow train. What is the rate of the fast train?
20. Lois, kidnapped by the Joker at noon, is being flown to a hideout in South America. At 12:15, Clark finally realizes that Lois is missing, and sets off in pursuit. Clark flies 200 mph faster than the Joker's Lear jet. How fast do they both go if Clark catches up to the master crook at 1:00?
21. A 9 by 12 room has a rectangular carpet on the floor. The carpet covers half the floor area. In addition, the same width of bare floor shows all the way around the carpet. What are the dimensions of the carpet?
22. John made a 480-mile car trip at one speed. On his return, his average speed was 10 miles per hour faster. That return trip took 1 hour 36 minutes less time than the trip going. What were the average speeds for the two parts of the trip?
23. The length of a rectangle is 5 more than the width. If the area of the rectangle is 60, then what are the dimensions?
24. Find two consecutive numbers the sum of whose squares is 181.
25. Find three consecutive numbers such that the square of the largest number, plus twice the middle-sized number, minus the smallest, is 420.
26. In the figure to the right, there is a rectangle with a square hole in it. The square is $x-2$ on a side. If the area of the shaded region is 37, then what is $x$?
27. It is 80 miles from Andover to Deerfield. The team bus for the big doubleheader averaged a reasonable speed for the trip. One of the players (a day student) missed the bus by 20 minutes, and took off in pursuit at a speed that was 8 miles per hour faster. The bus and the day student arrived at the same time. How fast did the bus go?
28. Two students work together on a project, and finish it in one hour. If each of the students had done the project alone, one would have taken one hour longer than the other to complete the task. How long would each have taken alone?