Modeling Word Problem -- Quadratic Functions #53

Question

A toy rocket (not internally powered) is launched straight up from the top of a building 50 ft tall at an initial velocity of 200 ft per sec. 

 

a). give the function that describes the height of the rocket in terms of time t. 

b). determine the time at which the rocket reaches its maximum height and the maximum height in feet. 

c). for what time interval will the rocket be more than 300 ft above ground level?

d). after how many seconds will it hit the ground?

Answer 1

The function is

 

h(t) = -16t² + 200t + 50, where t is the time in seconds and h(t) is measured in feet.

 

This answers (a).

 

To answer (b), find the vertex (h,k) of the graph of this equation:

 

h = -b/(2a) = -200/(2(-16)) = 200/32 = 6.25 secs.

 

Then, k = h(6.25) = 675 ft.

 

The time to the max height is 6.25 secs and the max height is 625 ft.

 

(c) Solve the inequality h(t) > 300:

 

-16t² + 200t + 50 > 300

-16t² + 200t - 250 > 0

16t² - 200t + 250 < 0

8t² - 100t + 125 < 0

 

Solve 8t² - 100t + 125 = 0 for t:

 

t = (100 ± √[(-100)² - 4(8)(125)])/(2 · 8) = (100 ± √(10000 - 4000))/16 = 100/16 ± (√6000)/16.

 

The two solutions are (rounded to two decimal places):

 

t ≈ 11.09 and t ≈ 1.41 secs. Since the parabola opens upward, the interval is between these two numbers.

 

The time interval is (1.41, 11.09).

 

(d) Set h(t) = 0 and solve for t. One answer will be negative (why?) and should be discarded as nonsense. The other answer is the correct one.

 

-16t² + 200t + 50 = 0

8t² - 100t - 25 = 0

 

Solve using the quadratic formula again.

 

t = (100 ± √[(-100)² - 4(8)(-25)])/(2 · 8) = (100 ± √10800)/16 = 100/16 ± (√10800)/16

 

The + solution gives t ≈ 12.75 secs; the minus solution gives t ≈ -0.25 secs. Again the correct flight time is 12.75 secs.

 

Notice that this is NOT twice the time to the maximum height. There IS symmetry to the graph, but since the ball started at a height of 50 feet (and not zero feet), it is a little MORE than twice the time to the max height.