Wiley jumps straight up in the air from the edge of a cliff so that when he comes down he falls off the cliff. The cliff is 40 feet off the ground. His height t seconds after he jumps is s= -16t2 + 12t + 40.What is the maximum height? What is his maximum velocity?
ds/dt = -32t + 12
= 0 for a max of s
32t = 12
t = 12/32 sec = 3/8 sec
for that time
s = -16(9/64) + 12(3/8) + 40
= 169/4 or 42.25 ft above the base of the cliff
when he hits the ground, s = 0
-16t^2 + 12t +40=0
divide by -4
4t^2 - 3t - 10 = 0
(t-2)(4t+5) = 0
t = 2 or t = -5/4, but t > 0
so he will hit the ground after 2 seconds
ds/dt = -32t+12
when t=2
ds/dt = -32(2) + 12 = -52 ft/sec
the negative indicates a downward direction, he is falling
To find the maximum height and maximum velocity, we need to analyze the quadratic equation that represents Wiley's height at any given time t.
Given that the equation for Wiley's height is s = -16t^2 + 12t + 40, we can determine the maximum height by finding the vertex of the parabolic function.
To find the vertex, we can use the formula t = -b/(2a), where a, b, and c are the coefficients from the quadratic equation in standard form.
In this case, the coefficient of t^2 is -16, and the coefficient of t is 12. Plugging these values into the formula, we have:
t = -(12) / (2*(-16))
t = -12 / (-32)
t = 3/8
Therefore, the time when Wiley reaches his maximum height is t = 3/8 seconds.
To find the maximum height, substitute the value of t into the equation:
s = -16(3/8)^2 + 12(3/8) + 40
s = -16(9/64) + 36/8 + 40
s = -9/4 + 9/2 + 40
s = -9/4 + 18/4 + 40
s = (9 + 18 + 160) / 4
s = 187/4
s = 46.75
Therefore, Wiley's maximum height is 46.75 feet.
To determine the maximum velocity, we need to find the derivative of the position function s(t). The derivative will give us the velocity function v(t), and to find the maximum velocity, we need to find when v(t) = 0.
Taking the derivative of s(t) = -16t^2 + 12t + 40, we get:
v(t) = -32t + 12
Setting v(t) to 0, we have:
0 = -32t + 12
32t = 12
t = 12/32
t = 3/8
This means the time when Wiley reaches his maximum velocity is t = 3/8 seconds.
To find the maximum velocity, substitute the value of t into the velocity function:
v = -32(3/8) + 12
v = -96/8 + 12
v = -12 + 12
v = 0
Therefore, Wiley's maximum velocity is 0.
In summary:
Wiley's maximum height is 46.75 feet, and his maximum velocity is 0 (indicating that he momentarily stops before falling).