These word problems involve situations I've discussed in other word problems: The area of a rectangle, motion (time, speed, and distance), and work. However, these problems lead to quadratic equations. You can solve them by factoring or by using the Quadratic Formula.
Example. One number is the square of another. Their sum is 132. Find the numbers.
Let A and B be the numbers. The first sentence says one is the square of the other, so I can write
The sum is 132, so
Plug into and solve for B:
The possible solutions are and .
If , then .
If , then .
So two pairs work: -12 and 144, and 11 and 121.
Example. The difference of two numbers is 2 and their product is 224. Find the numbers.
Let x and y be the numbers. Their difference is 2, so I can write
Their product is 224, so
From , I get . Plug this into and solve for y:
If , then .
If , then .
So two pairs work: -14 and -16, and 14 and 16.
Example. The area of a rectangle is 560 square inches. The length is 3 more than twice the width. Find the length and the width.
Let L be the length and let W be the width. The length is 3 more than twice the width, so
The area is 560, so
Plug in and solve for W:
Use the Quadratic Formula:
Since the width can't be negative, I get . The length is .
Example. Calvin and Bonzo can eat 1260 hamburgers in 12 hours. Eating by himself, it would take Calvin 7 hours longer to eat 1260 hamburgers than it would take Bonzo to eat 1260 hamburgers. How long would it take Bonzo to eat 1260 hamburgers by himself?
Let c be Calvin's rate, in hamburgers per hour. Let b be Bonzo's rate, in hamburgers per hour.
The last equation gives
The second equation gives
Plug into :
Plug into the first equation and solve for t:
The solution doesn't make sense, since time can't be negative. The solution is 21. Bonzo takes 21 hours, and Calvin takes hours.
Example. If Calvin and Bonzo eat together, they can eat 480 hot dogs in 6 hours.
Eating alone, Bonzo takes 16 hours longer than Calvin would to eat 480 hot dogs.
How long does it take Calvin to eat 480 hot dogs?
Let x be Calvin's rate (in hot dogs per hour), let y be Bonzo's rate, and let t be the time it takes Calvin to eat 480 hot dogs.
The second equation says , so .
The third equation says , so .
Plug these into the first equation and solve for t:
The solutions are and . Since t can't be negative, the answer is hours.
Example. Calvin rides his power boat up and down a drainage ditch. The water in the drainage ditch flows at 6 miles per hour. Calvin takes 5 hours longer to travel 360 miles against the current than he does to travel 360 miles with the current. What is the speed of Calvin's boat in still water?
Let x be the speed of Calvin's boat in miles per hour in still water, and let t be the time in hours it takes him to travel 360 miles with the current. Thus, it takes him hours to travel 360 miles against the current.
The equations are
Solve the second equation for t:
Plug this into the first equation and solve for x:
The solutions are . Since the speed can't be negative, the answer is 30 miles per hour.
Example. The hypotenuse of a right triangle is 4 times the smallest side. The third side is . Find the hypotenuse and the smallest side.
Let s be the smallest side and let h be the hypotenuse. By Pythagoras,
The hypotenuse is 4 times the smallest side, so
Plug into and solve for s:
Since doesn't make sense, the solution is . Then .
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