Question

1. the grinch is throwing ornaments from your tree onto the top of his sleigh, so that its height h (in feet) after t seconds can modeled by the function: draw a picture

$h(t) = -16t^2 + 64t + 7$

a. find the maximum height of the ornament. label units appropriately.

b. how long did it take the ornament to reach its maximum height? label units appropriately.

c. how long does it take to reach the ground? show all your work, give the answer to the nearest hundredths place

d. how long is the ornament above 40 feet? show all your work, give the answer to the nearest hundredths place.

Explanation:

Part a

Step1: Identify the vertex formula

For a quadratic function \( h(t) = at^2 + bt + c \), the \( t \)-coordinate of the vertex (which gives the time of maximum height) is \( t = -\frac{b}{2a} \). Here, \( a = -16 \), \( b = 64 \), \( c = 7 \).

Step2: Find the time of maximum height

Calculate \( t = -\frac{64}{2(-16)} = -\frac{64}{-32} = 2 \) seconds.

Step3: Find the maximum height

Substitute \( t = 2 \) into \( h(t) \): \( h(2) = -16(2)^2 + 64(2) + 7 = -16(4) + 128 + 7 = -64 + 128 + 7 = 71 \) feet.

Step1: Use the vertex formula for \( t \)

As in part (a), for \( h(t) = -16t^2 + 64t + 7 \), \( a = -16 \), \( b = 64 \). The formula for the \( t \)-coordinate of the vertex is \( t = -\frac{b}{2a} \).

Step2: Calculate the time

\( t = -\frac{64}{2(-16)} = 2 \) seconds.

Step1: Set \( h(t) = 0 \)

We need to solve \( -16t^2 + 64t + 7 = 0 \) for \( t \). Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -16 \), \( b = 64 \), \( c = 7 \).

Step2: Calculate the discriminant

\( \Delta = b^2 - 4ac = 64^2 - 4(-16)(7) = 4096 + 448 = 4544 \).

Step3: Find the roots

\( t = \frac{-64 \pm \sqrt{4544}}{2(-16)} = \frac{-64 \pm \sqrt{4544}}{-32} \). Calculate \( \sqrt{4544} \approx 67.41 \). So, \( t_1 = \frac{-64 + 67.41}{-32} \approx \frac{3.41}{-32} \approx -0.11 \) (discard, time can't be negative), \( t_2 = \frac{-64 - 67.41}{-32} \approx \frac{-131.41}{-32} \approx 4.11 \) seconds.

Answer:

The maximum height is \( 71 \) feet.

Part b