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 = −16𝑡2 + 78𝑡 + 5. How long will it take the T-shirt to reach its maximum height? What is its maximum height?
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
The answer is C.
they didnt explain it very well. the formula of the problem is;
ax^2 + bx + c.
a = -16
b = 78
c = 2
the formula to find the variable “t” is;
-b/2a.
if you plug in the variables, it is;
-78/2( -16)
which in decimals is 2.4375 (or 2.44 for the answer)
now if you put the value of t in the equation it looks like;
-16(2.4375)^2 + 78(2.4375) + 5 = h
if you solve it, 100.0625 = h
The answers are t, h. If you round their values, you get 2.44s and 100.06ft.
the time of max height (tmax) is on the axis of symmetry of the parabola
tmax = -b / 2a = -78 / (2 * -16) = 2.44 s
to find hmax , plug tmax into the height equation
Thanks @i+just+got+100 you are 100% correct and helpful!!!!!!!
To find the time it takes for the T-shirt to reach its maximum height, we need to determine when the upward velocity becomes zero. In other words, we need to find the time at which the derivative of the height function is equal to zero.
The height function is given by h = -16t^2 + 78t + 5.
To find the derivative, we can apply the power rule to each term separately:
h' = -32t + 78.
Now, set h' equal to zero and solve for t:
-32t + 78 = 0.
-32t = -78.
t = (-78)/(-32).
t = 2.44 seconds.
Therefore, it will take the T-shirt 2.44 seconds to reach its maximum height.
To find the maximum height, we substitute this value of t back into the height function:
h = -16(2.44)^2 + 78(2.44) + 5.
h ≈ 146.28 feet.
Therefore, the T-shirt will reach a maximum height of approximately 146.28 feet.
The correct answer is b. 2.44 s, 146.28 ft.