A toy rocket is launched vertically into the air from a launching pad 8 feet above the ground with an initial velocity of 80 feet per second. The height \( h(t) \), in feet, of the rocket above the ground at \( t \) seconds after launch is given by the function:

\[ h(t) = -16t^2 + 80t + 8 \]

How long will it take the rocket to reach its maximum height? What is the maximum height?

Question

20-02-2024
Mathematics

College

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A toy rocket is launched vertically into the air from a launching pad 8 feet above the ground with an initial velocity of 80 feet per second. The height \( h(t) \), in feet, of the rocket above the ground at \( t \) seconds after launch is given by the function:

\[ h(t) = -16t^2 + 80t + 8 \]

How long will it take the rocket to reach its maximum height? What is the maximum height?

Answer :

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JaymaMays JaymaMays · Answer 1 · 2024-05-26T21:37:29-04:00

Final answer:

The toy rocket will take 2.5 seconds to reach its maximum height, which is 108 feet above the ground.

Explanation:

The question asks about the time it takes for a toy rocket to reach its maximum height and what that maximum height is, given the equation of its height h(t) = -16t2 + 80t + 8. This is a quadratic equation representing the height of the toy rocket as a function of time, which is a typical problem in physics or mathematics dealing with projectile motion. To solve for the time when the rocket reaches its maximum height, we need to find the vertex of the parabola shaped by the quadratic equation where the coefficient of t2 is negative indicating that the parabola opens downwards.

The formula to calculate the time to reach maximum height in a quadratic equation like this is t = -b/(2a), where a is the coefficient of t2 (-16) and b is the coefficient of t (80). Thus, t = -80/(2*(-16)) = 2.5 seconds. To find the maximum height, we plug this time into the height equation: h(2.5) = -16*(2.5)2 + 80*(2.5) + 8 = 108 feet.