Quadratic Word Problems

Show your work for each problem.

1.
A ball is thrown upward. Its height in meters after seconds is . When does the ball hit the ground?
1. a)After 5 seconds
2. b)After 10 seconds
3. c)After 4 seconds
4. d)After 2 seconds
Answer: After 4 seconds
Setting : , so . This gives (start) or (landing).
2.
The product of two consecutive positive integers is 72. What are the integers?
1. a)8 and 9
2. b)9 and 10
3. c)6 and 12
4. d)7 and 8
Answer: 8 and 9
gives . Factoring: , so . The integers are 8 and 9.
3.
A rectangular garden has length 3 meters more than its width. If the area is 40 square meters, what is the width (in meters)?
Answer: 5
becomes . Factoring: , so meters (since width must be positive).
4.
The sum of a number and its square is 56. What is the positive number?
Answer: 7
factors as . Since we need positive, .
5.
Which type of quadratic word problem is this? 'A rocket's height is . Find the maximum height.'
1. a)Optimization problem
2. b)Consecutive integer problem
3. c)Area problem
4. d)Number problem
Answer: Optimization problem
Finding maximum or minimum values is an optimization problem. Use the vertex formula to find when the maximum occurs.
6.
A ball is thrown upward with initial velocity 30 m/s from ground level. Its height is . Find the maximum height.
Answer: 45
1. What is the value of in the equation? -5
2. What is the value of ? 30
3. At what time does maximum height occur? Use 3
4. Calculate when : 45
7.
A rectangle has perimeter 30 cm. What width (in cm) maximizes the area?
Answer: 7.5
Area = . Maximum at cm. A square maximizes area for a given perimeter.
8.
The height of a ball is . At what time does it reach maximum height?
1. a)1 second
2. b)2 seconds
3. c)4 seconds
4. d)3 seconds
Answer: 2 seconds
seconds.
9.
Find two consecutive even integers whose product is 168.
Answer: 12 and 14
1. If the first even integer is , what is the next consecutive even integer? n + 2
2. Write the equation: . Expand it. n^2 + 2n = 168
3. Rewrite in standard form: . Factor it. (n + 14)(n - 12) = 0
4. What is the positive value of ? 12
10.
A picture frame is 2 cm wide on all sides. The picture inside is 8 cm by 6 cm. What is the total area of the frame (including picture) in square cm?
Answer: 120
Total dimensions: square cm.
11.
A company's profit (in euros) is where is units sold. What is the break-even point (where )?
1. a)10 and 60 units
2. b)25 and 75 units
3. c)20 and 80 units
4. d)30 and 70 units
Answer: 20 and 80 units
factors as , giving or units.
12.
A ball is thrown from a 20-meter building with initial velocity 15 m/s upward. Its height is . When does it hit the ground?
Answer: 4
1. Set . What equation do you get? -5t^2 + 15t + 20 = 0
2. Divide by -5 to simplify. What is the equation? t^2 - 3t - 4 = 0
3. Factor the quadratic. (t - 4)(t + 1) = 0
4. What is the positive solution for ? 4
13.
A rocket's height is . What is its maximum height in meters?
Answer: 125
Time at max: . Height: meters.
14.
A theater charges 25 euros per ticket and sells 400 tickets. For each 5 euro increase, they sell 20 fewer tickets. What price maximizes revenue?
Answer: 75
1. Let = number of 5-euro increases. Express the price. 25 + 5n
2. Express the number of tickets sold. 400 - 20n
3. Revenue = price × tickets. Expand . 10000 + 1500n - 100n^2
4. Find at maximum revenue using . 7.5
5. Calculate the optimal price: . 62.5
15.
A ball thrown from height 10 m follows . When does it reach 40 meters?
1. a)Never reaches 40 m
2. b)At 1 second only
3. c)At 2 and 3 seconds
4. d)At 1.5 and 4 seconds
Answer: At 2 and 3 seconds
gives , or . Factoring: , so or seconds.
16.
Two numbers differ by 5. Their product is 126. Find the larger number.
Answer: 14
1. Let the smaller number be . What is the larger number? x + 5
2. Write the equation for their product. x(x + 5) = 126
3. Expand and rearrange to standard form. x^2 + 5x - 126 = 0
4. Factor the quadratic. (x + 14)(x - 9) = 0
5. If (taking the positive value), what is the larger number? 14