High School

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Quadratic Word Problem:

If a toy rocket is launched vertically upward from ground level with an initial velocity of 128 feet per second, then its height (h) after t seconds is given by the equation [tex]h(t) = -16t^2 + 128t[/tex], assuming air resistance is neglected.

How long will it take for the rocket to return to the ground?

1) 4 seconds
2) 6 seconds
3) 8 seconds
4) 10 seconds

Answer :

Final answer:

The time it takes for the toy rocket to return to the ground level is found by setting the height equation h(f) = -16f² + 128f equal to zero. Factoring the equation, we find that the rocket returns to the ground after 8 seconds. The correct answer is option 3) 8 seconds.

Explanation:

Finding the Time for Rocket to Return to Ground Level

The student's question involves determining the time it would take for a toy rocket to return to the ground when launched vertically upward with a given initial velocity. Using the provided quadratic equation h(f) = -16f² + 128f, we set h(f) to zero to find when the rocket hits the ground after launch. Since initial velocity and gravity are involved, this is a classic quadratic word problem.

To find the time at which the rocket returns to the ground (when h=0), we use the quadratic formula, or we could factor the equation if it is factorable. In this case, we can factor the equation as follows:

h(f) = -16f² + 128f
= f(-16f + 128)

Setting this equal to zero gives us two possible times:

f = 0 seconds (the time of launch)

-16f + 128 = 0

Solving the second equation for f by adding 16f to both sides and dividing by 16 gives f = 8 seconds. Thus, the rocket will return to the ground after 8 seconds.

The correct answer from the given options is 3) 8 seconds.

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