During halftime of a basketball game, a sling shot launches T-shirts at the crowd. A T-shirt is launched with an initial upward velocity of 78 ft/s. The height of the T-shirt (ℎ) in feet after t seconds is given by the function ℎ = −16t^2+ 78t + 5. How long will it take the T-shirt to reach its maximum height? What is its maximum height?
To find the time it takes for the T-shirt to reach its maximum height, we first need to determine the vertex of the parabolic function ℎ = -16t^2 + 78t + 5.
The t-coordinate of the vertex can be found using the formula t = -b/(2a), where the function is in the form ℎ = at^2 + bt + c. In this case, a = -16 and b = 78.
t = -78 / (2*(-16))
t = -78 / (-32)
t = 2.4375
So, the T-shirt will reach its maximum height after approximately 2.44 seconds.
To find the maximum height, we substitute this value of t back into the function:
ℎ = -16(2.4375)^2 + 78(2.4375) + 5
ℎ = -16(5.9492) + 78(2.4375) + 5
ℎ = -94.3875 + 190.125 + 5
ℎ = 100.7375
Therefore, the T-shirt will reach a maximum height of approximately 100.74 feet.
the answers are either
a. 1.22 s, 76.35 ft
b. 2.44 s, 146.28 ft
c. 2.44 s, 100.06 ft
d. 2.44 s, 131 ft
To find the time it takes for the T-shirt to reach its maximum height, we first need to determine the vertex of the parabolic function ℎ = -16t^2 + 78t + 5.
The time at which the T-shirt reaches its maximum height can be found using the formula t = -b/(2a), where the function is in the form ℎ = at^2 + bt + c. In this case, a = -16 and b = 78.
t = -78 / (2*(-16))
t = -78 / (-32)
t = 78 / 32
t = 2.4375
So, the T-shirt will reach its maximum height after approximately 2.44 seconds.
To find the maximum height, we substitute this value of t back into the function:
ℎ = -16(2.4375)^2 + 78(2.4375) + 5
ℎ = -16(5.9492) + 78(2.4375) + 5
ℎ = -95.1875 + 190.125 + 5
ℎ = 100.9375
Therefore, the T-shirt will reach a maximum height of approximately 100.94 feet.
Therefore, the correct answer is c. 2.44 s, 100.06 ft.