Algebra 2 Quadratic Word Problems
1. Christina’s garden is rectangular in shape. The length of the garden is 8 feet longer than the width. If the area of the garden is 240 square feet, what are the dimensions? Set up the quadratic equation and solve.
2. Keith kicks a soccer ball into the air with the initial upward velocity of 88 feet per second. Solve the quadratic function h(t) = -16t2 + 88t for t when h(t) = 0 to find the amount of time it takes the ball to hit the ground.
3. The quadratic function y=5t – 0.5at2 gives the height, in meters, of an astronaut after jumping vertically with an initial velocity of 5 meters per second. In the function, a is the acceleration due to gravity and t is the time in second since the astronaut jumped. The acceleration due to gravity is about 1.6 meters per second per second. How long would it take the astronaut to land?
4. A manufacturing corporation uses the function P(x) = -3x2 + 18x -15 to model the profit, in thousands of dollars, from selling x thousand items. What are the break-even points (P=0), in thousands of items?
5. The quadratic function h(x) = -x2 +1.5x +10 gives the height, h, in meters above the water, of a diver as a function of time x, in seconds.
- Graph the equation on the coordinate grid. What is the initial height of the diver when x=0?
- How many second have passed when the diver enters the water?
6. The function P(x) = -x2 +140x -4500 models the profit of a company when selling x units. The break-even points are the numbers of units for which P(x) = 0. What are the break-even points for this product?
7. In physics, the function h(t) = -16t2 + 80t can be used to model the height of a toy rocket launched up into the air with an initial velocity of 80 feet per second. The height is given in feet, and time t is given in seconds, after the launch. At what time(s) will the rocket be 84 feet above the ground? Complete the square to solve for t.
8. Roberto threw a penny up into the air from the top of a 192 foot high building. The height of the penny above the ground t seconds after he threw it is given by the function h(t) = -16t2+64t+192.
- How long does it take the penny to hit the ground (h = 0)?
- At what time(s) is the penny 240 feet above the ground (h=240)?
9. Explain how to solve the equation -3x2 + 6x = 8 using the quadratic formula.
10. How many different x-intercepts can the graph of a quadratic equation have?
11. The temperature at which water boils is affected by altitude. As the altitude increases, the boiling point of water decreases by d degrees from 212 degrees Fahrenheit. This is modeled by the equation a = d2 + 520d, where a is the altitude above sea level, in feet. Estimate the boiling point of water at 7,500 feet above sea level. Round to the nearest degree.
12. What is the relationship between the graph of a quadratic equation with no real solutions and the x-axis?
13. What is known about the solution of a quadratic equation if the discriminant is negative?
14. At the beginning of a basketball game, the referee tosses a jump ball. The referee holds the ball 4 feet above the floor and tosses the ball upward at a velocity of 26 feet per second. The acceleration of the ball if affected by gravity, which is approximately 32 feet per second squared. The situation can be described by the equation h(t) = -16t2 +26t + 4.
- If the center that wins the jump ball hits the ball at 9 feet, how long has the ball been in the air when he hits it?
- How long does it take the ball to reach its maximum height? Round to the nearest tenth.
- If the referee held the ball 3 feet above the floor, what equation models the situation?
Solutions:
1. 12 feet by 20 feet
2. 5.5 seconds
3. 6.25 seconds
4. 1000 and 5000
5. Answers are:
- 10 meters
- 4 seconds
6. 50, 90
7. 1.5 and 3.5
8. Answers are:
- 6 seconds
- 1 and 3
9. Subtract 8 from each side; let a =-3; b=6; and c=-8. Substitute into quadratic formula and solve.
10. 0, 1, 2
11. 198
12. Graph does not cross the x-axis
13. Complex or imaginary
14. Answers are:
- 0.5 and 1.125
- 0.8
15. -16t2 +26t + 3