Modeling with Quadratic Equations Answers

When using a quadratic equation in the form y = ax2 + bx + c to model the height of a projectile (y) over time (x), which of the following is always represented by the constant term?

A baseball is thrown into the air from a height of 5 feet. The ball reaches a maximum height of 43.5 feet and spends a total of 3.2 seconds in the air. Which equation models the height of the baseball? Assume that acceleration due to gravity is –16 ft/s2.

A quarterback throws an incomplete pass. The height of the football at time t is modeled by the equation h(t) = –16t2 + 40t + 7. Rounded to the nearest tenth, the solutions to the equation when h(t) = 0 feet are –0.2 s and 2.7 s. Which solution can be eliminated and why?

The Davidson family wants to expand its rectangular patio, which currently measures 15 ft by 12 ft. They want to extend the length and width the same amount to increase the total area of the patio by 160 ft2. Which quadratic equation best models the situation?A = lw

Both Rachel and Dominique throw tennis balls into the air. At any time, t, the height, h, of Rachel’s ball is modeled by the equation h = –16t2 + 30t + 5. Dominique throws his tennis ball with the same acceleration, a, from the same initial height, , but with an initial velocity, v, double that of Rachel’s. Which equation best models the height of Dominique’s tennis ball?h(t) = at2 + vt + h0

A skydiver jumps from an airplane at an altitude of 2,500 ft. He falls under the force of gravity until he opens his parachute at an altitude of 1,000 ft. Using the initial velocity of zero, approximately how long does the jumper fall before he opens his chute?For this quadratic model we will let the y-axis be the axis of symmetry.h(t) = -16t2 + vt + h0

A projectile is launched straight up from the ground with an initial velocity of 120 ft/s. If acceleration due to gravity is –16 ft/s2, after about how many seconds will the object reach a height of 200 ft?h(t) = at2 + vt + h0

A firework is launched into the air from ground level with an initial velocity of 128 ft/s. If acceleration due to gravity is –16 ft/s2, what is the maximum height reached by the firework?h(t) = at2 + vt + h0

A soccer ball is kicked into the air from the ground. If the ball reaches a maximum height of 25 ft and spends a total of 2.5 s in the air, which equation models the height of the ball correctly? Assume that acceleration due to gravity is –16 ft/s2.h(t) = at2 + vt + h0

The total area of two square windows is 1,025 in.2. Each side of the larger window is 5 in. longer than the sides of the smaller window. How long are the sides of the smaller window?A = s2

The owner of an office building is expanding the length and width of a parking lot by the same amount. The lot currently measures 120 ft by 80 ft, and the expansion will increase its area by 4,400 ft2. By how many feet should the length of the parking lot be increased?A = lw

Two children are throwing a baseball back and forth. The ball is 4 ft above the ground when it leaves one child’s hand with an upward velocity of 36 ft/s. If acceleration due to gravity is –16 ft/s2, how high above the ground is the ball 2 s after it is thrown?h(t) = at2 + vt + h0

Carmen is using the quadratic equation (x + 15)(x) = 100 where x represents the width of a picture frame. Which statement about the solutions x = 5 and x = –20 is true?

A car’s stopping distance in feet is modeled by the equation where v is the initial velocity of the car in miles per hour and f is a constant related to friction. If the initial velocity of the car is 47 mph and f = 0.34, what is the approximate stopping distance of the car?

The Davidson family wants to expand its rectangular patio, which currently measures 15 ft by 12 ft. They want to extend the length and width the same amount to increase the total area of the patio by 160 ft2. Which quadratic equation best models the situation?A = lw

The main cable of a suspension bridge forms a parabola modeled by the equation y = a(x – h)2 + k where y is the height in feet of the cable above the road, x is the horizontal distance in feet from the right bridge support, a is a constant, and (h, k) is the parabola’s vertex. What is the maximum and minimum height of the bridge modeled by the equation y = 0.005(x – 60)2 + 8?

Both Rachel and Dominique throw tennis balls into the air. At any time, t, the height, h, of Rachel’s ball is modeled by the equation h = –16t2 + 30t + 5. Dominique throws his tennis ball with the same acceleration, a, from the same initial height, , but with an initial velocity, v, double that of Rachel’s. Which equation best models the height of Dominique’s tennis ball?h(t) = at2 + vt + h0

A softball pitcher tosses a ball straight upward from a height of 3 feet above the ground with an initial velocity of 50 feet per second. The acceleration due to gravity is –16 ft/s².Which quadratic equation represents the height of the ball above the ground after t seconds?h(t) = at2 + vt + h0