Question

ii. use what you know about quadratic equations and their graphs to answer the following questions.

1. the formula ( h(t) = -16t^2 + 32t + 180 ) gives the height in feet of a person hang gliding off a cliff.

when (in seconds) will the person reach that maximum height?
what is that maximum height (in feet)?
when (in seconds) will the person reach the ground?

Explanation:

Step1: Find time of maximum height

For a quadratic function \( h(t) = at^2 + bt + c \), the time \( t \) at which the maximum (or minimum) occurs is given by \( t = -\frac{b}{2a} \). Here, \( a = -16 \), \( b = 32 \). So, \( t = -\frac{32}{2\times(-16)} = -\frac{32}{-32} = 1 \) second.

Step2: Find maximum height

Substitute \( t = 1 \) into \( h(t) \): \( h(1) = -16(1)^2 + 32(1) + 180 = -16 + 32 + 180 = 196 \) feet.

Step3: Find time to reach ground

Set \( h(t) = 0 \), so \( -16t^2 + 32t + 180 = 0 \). Divide both sides by -4: \( 4t^2 - 8t - 45 = 0 \). Use quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4 \), \( b = -8 \), \( c = -45 \). Discriminant \( \Delta = (-8)^2 - 4\times4\times(-45) = 64 + 720 = 784 \). \( \sqrt{\Delta} = 28 \). So, \( t = \frac{8 \pm 28}{8} \). We take the positive root: \( t = \frac{8 + 28}{8} = \frac{36}{8} = 4.5 \) seconds (the other root is negative, so we discard it).

Answer:

• Time to reach maximum height: 1 second
• Maximum height: 196 feet
• Time to reach ground: 4.5 seconds