Question

Suppose you throw a ball straight up from the ground with a velocity of 224 feet per second. as the ball moves up, gravity slows it. eventually, the ball begins to fall back to the ground. the height h of thr ball after t seconds in the air is given by the equation h=-16t^2 + 224t. use the given information to answer.
a. find the height of the ball after four seconds. (type a whole number)

b. how long will it take the ball to reach a height of 768 ft?

c. how long after you throw the ball will it return to the ground? (type w whole number)

Answer 1

Answer:

  a) 640 ft

  b) 6 seconds (the first time)

  c) 14 seconds

Step-by-step explanation:

These problems are solved by using the given equation for the height of the thrown ball with the given values, and solving for the unknown value.

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a. height after a time

To find the height at a specific time, substitute the time value for t in the height formula and do the arithmetic.

For t = 4, we have ...

  h = -16t² +224t = (-16(4) +224)(4) = 160(4) = 640

The height after 4 seconds is 640 ft.

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b. time for a height

The time required to reach a specific height is found by solving the quadratic equation with h equal to the height value. It can be easier if the equation is in vertex form.

  h = -16t² +224t = -16(t² -14t) = -16(t² -14t +49) +784 = -16(t -7)² +784

We want the time to a height of 768 ft, so ...

  768 = -16(t -7)² +784

  -16/-16 = (t -7)² . . . . . . . . subtract 784, divide by -16

  ±√1 = t -7 . . . . . . . . . take the square root

  t = 7 ±1 = 6 or 8 . . . . . . add 7

The ball will be at a height of 768 ft after 6 seconds, and again after 8 seconds.

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c. time to reach the ground

The equation from part b applies.

  0 = -16(t -7)² +784 . . . . . . . . the height is 0 when the ball is on the ground

  -784/-16 = (t -7)² = 49 . . . . . subtract 784, divide by -16

  √49 = t -7 . . . . . . . . . . . . . take the positive square root

  t = 14 . . . . . . . . . . . . . add 7

The ball will return to the ground after 14 seconds.