Question

Use the following information to answer each part.

An acorn is tossed upwards with an initial velocity of 8 ft/sec from the top of a 24 foot building. The model given describes the height, h, of the acorn as a function of time, t, in seconds.

h(t) = -16t² + 8t + 24

Make sure to include all necessary work in order to receive full credit.

a) Find the maximum height that the acorn reaches. After how many second does the acorn reach this maximum height?

b) How many seconds would it take the acorn to reach the ground?

Answer 1

The maximum height that the acorn reaches is 28 ft, and it reaches this height after 1/4 second. It takes the acorn 3/2 seconds to reach the ground.

Step-by-step explanation:

To find the maximum height that the acorn reaches, we need to find the vertex of the quadratic equation. The equation is in the form h(t) = -16t² + 8t + 24. The vertex of a quadratic equation in the form h(t) = at² + bt + c is (t, h), where t = -b/2a and h = -D/4a. In this case, a = -16, b = 8, and c = 24.

Substituting these values into the formulas, we find that t = -8/(-32) = 1/4. To find the height at this time, we substitute t = 1/4 into the equation: h(1/4) = -16(1/4)² + 8(1/4) + 24 = 28 ft.

Therefore, the maximum height that the acorn reaches is 28 ft and it reaches this height after 1/4 second.

To find the time it takes for the acorn to reach the ground, we need to find the value of t when h(t) = 0. We set -16t² + 8t + 24 = 0 and solve for t. Factoring the equation, we get (-4t + 6)(4t + 4) = 0. So, -4t + 6 = 0 or 4t + 4 = 0. Solving these equations, we find that t = 3/2 or t = -1. However, the time cannot be negative, so we discard t = -1. Therefore, it takes the acorn 3/2 seconds to reach the ground.

Answer 2

The maximum height that the acorn reaches is 25 feet, and it reaches this height after 0.25 seconds. It takes 3 seconds for the acorn to reach the ground.

Step-by-step explanation:

To find the maximum height that the acorn reaches, we need to determine the time at which the height is maximum. The height function is given by h(t) = -16t² + 8t + 24. To find the maximum height, we need to find the vertex of the parabolic function. The vertex can be found using the formula t = -b/2a, where a = -16 and b = 8. Plugging in these values, we get t = -8/(2*-16) = 0.25 seconds.

To find the maximum height, we substitute this time back into the height function: h(0.25) = -16(0.25)² + 8(0.25) + 24 = 25 feet.

Therefore, the maximum height that the acorn reaches is 25 feet and it reaches this maximum height after 0.25 seconds.

To find the time it takes for the acorn to reach the ground, we set the height function equal to zero and solve for t: -16t² + 8t + 24 = 0. This is a quadratic equation that can be solved using factoring, completing the square, or the quadratic formula. Solving the equation, we get t = 3 seconds and t = -0.5 seconds. Since time cannot be negative, we take the positive value, which means it takes 3 seconds for the acorn to reach the ground.