Question
During halftime of a basketball game, a sling shot launches t-shirts into the crowd. The function h = - 16t^2 + 72t + 5 models the t-shirt's height h in feet after t seconds.
Use the vertex formula to determine how many seconds it will take a t-shirt to reach the maximum height. Then find the maximum height of the t-shirt.
Round to the hundredths place if necessary.
A t-shirt will reach its maximum height at _ seconds
The maximum height of a t-shirt will be at _ feet.
Use the quadratic formula to calculate the time it will take for a t-shirt shot outside the stands to hit the ground. round to the nearest hundredth if necessary.
A t-shirt will hit the ground in about _ seconds.
Use the vertex formula to determine how many seconds it will take a t-shirt to reach the maximum height. Then find the maximum height of the t-shirt.
Round to the hundredths place if necessary.
A t-shirt will reach its maximum height at _ seconds
The maximum height of a t-shirt will be at _ feet.
Use the quadratic formula to calculate the time it will take for a t-shirt shot outside the stands to hit the ground. round to the nearest hundredth if necessary.
A t-shirt will hit the ground in about _ seconds.
Answers
GPT-4o mini
To find the time it takes for the t-shirt to reach its maximum height, we can use the vertex formula for a quadratic function given in the form \( h = at^2 + bt + c \). The time \( t \) at which the maximum height occurs is given by the formula:
\[
t = -\frac{b}{2a}
\]
In our function, \( h = -16t^2 + 72t + 5 \):
- The coefficient \( a = -16 \)
- The coefficient \( b = 72 \)
Now, substituting \( a \) and \( b \) into the vertex formula:
\[
t = -\frac{72}{2 \cdot (-16)} = -\frac{72}{-32} = \frac{72}{32} = 2.25 \text{ seconds}
\]
Next, we calculate the maximum height by substituting \( t = 2.25 \) back into the height function \( h \):
\[
h = -16(2.25)^2 + 72(2.25) + 5
\]
Calculating \( 2.25^2 \):
\[
2.25^2 = 5.0625
\]
Now substituting into the height equation:
\[
h = -16(5.0625) + 72(2.25) + 5
\]
\[
= -81 + 162 + 5
\]
\[
= 86 \text{ feet}
\]
Thus, we have:
- A t-shirt will reach its maximum height at **2.25** seconds.
- The maximum height of a t-shirt will be at **86** feet.
Now, to determine when the t-shirt hits the ground, we set the height \( h \) to 0 and solve the equation:
\[
-16t^2 + 72t + 5 = 0
\]
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- \( a = -16 \)
- \( b = 72 \)
- \( c = 5 \)
Calculating the discriminant:
\[
b^2 - 4ac = 72^2 - 4(-16)(5) = 5184 + 320 = 5504
\]
Now applying the quadratic formula:
\[
t = \frac{-72 \pm \sqrt{5504}}{2(-16)}
\]
Calculating \( \sqrt{5504} \):
\[
\sqrt{5504} \approx 74.24
\]
Now substituting back:
\[
t = \frac{-72 \pm 74.24}{-32}
\]
Calculating the two possible values for \( t \):
1. \( t = \frac{-72 + 74.24}{-32} = \frac{2.24}{-32} \) (this will yield a negative time, hence not valid)
2. \( t = \frac{-72 - 74.24}{-32} = \frac{-146.24}{-32} = 4.57 \)
Thus, rounding to the nearest hundredth, we have:
A t-shirt will hit the ground in about **4.57** seconds.
\[
t = -\frac{b}{2a}
\]
In our function, \( h = -16t^2 + 72t + 5 \):
- The coefficient \( a = -16 \)
- The coefficient \( b = 72 \)
Now, substituting \( a \) and \( b \) into the vertex formula:
\[
t = -\frac{72}{2 \cdot (-16)} = -\frac{72}{-32} = \frac{72}{32} = 2.25 \text{ seconds}
\]
Next, we calculate the maximum height by substituting \( t = 2.25 \) back into the height function \( h \):
\[
h = -16(2.25)^2 + 72(2.25) + 5
\]
Calculating \( 2.25^2 \):
\[
2.25^2 = 5.0625
\]
Now substituting into the height equation:
\[
h = -16(5.0625) + 72(2.25) + 5
\]
\[
= -81 + 162 + 5
\]
\[
= 86 \text{ feet}
\]
Thus, we have:
- A t-shirt will reach its maximum height at **2.25** seconds.
- The maximum height of a t-shirt will be at **86** feet.
Now, to determine when the t-shirt hits the ground, we set the height \( h \) to 0 and solve the equation:
\[
-16t^2 + 72t + 5 = 0
\]
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- \( a = -16 \)
- \( b = 72 \)
- \( c = 5 \)
Calculating the discriminant:
\[
b^2 - 4ac = 72^2 - 4(-16)(5) = 5184 + 320 = 5504
\]
Now applying the quadratic formula:
\[
t = \frac{-72 \pm \sqrt{5504}}{2(-16)}
\]
Calculating \( \sqrt{5504} \):
\[
\sqrt{5504} \approx 74.24
\]
Now substituting back:
\[
t = \frac{-72 \pm 74.24}{-32}
\]
Calculating the two possible values for \( t \):
1. \( t = \frac{-72 + 74.24}{-32} = \frac{2.24}{-32} \) (this will yield a negative time, hence not valid)
2. \( t = \frac{-72 - 74.24}{-32} = \frac{-146.24}{-32} = 4.57 \)
Thus, rounding to the nearest hundredth, we have:
A t-shirt will hit the ground in about **4.57** seconds.