If h(t) represents the height of an object above ground level at time t and h(t) is given by h(t)=-16t2+11t+1 find the height of the object at the time when the speed is zero.
Speed is the fist derivative of your function, so
velocity = -32t + 11
set that equal to zero, solve for t
plug that t value back in the original height equation
To find the height of the object when the speed is zero, we need to find the time t when the velocity is zero. The velocity is the derivative of the height function h(t).
Let's calculate the velocity function v(t) by taking the derivative of h(t):
v(t) = h'(t) = -32t + 11
To find the time when the velocity is zero, we set v(t) = 0 and solve for t:
-32t + 11 = 0
32t = 11
t = 11/32
Now that we have found the time when the velocity is zero, we can substitute this value into the height function h(t) to find the height:
h(t) = -16t^2 + 11t + 1
Plugging in t = 11/32:
h(11/32) = -16(11/32)^2 + 11(11/32) + 1
Now we can simplify the expression to find the height at that time.
To find the height of the object at the time when the speed is zero, we need to find the t-value(s) where the derivative of the height function h(t) is equal to zero.
First, let's find the derivative of h(t) with respect to t. The derivative of -16t^2 + 11t + 1 is obtained by applying the power rule for derivatives:
h'(t) = d/dt(-16t^2 + 11t + 1)
= -32t + 11
Next, we need to find the values of t for which h'(t) = -32t + 11 = 0. This means we need to solve the equation:
-32t + 11 = 0
To solve this equation, we isolate t by moving 11 to the other side:
-32t = -11
Dividing both sides by -32 gives:
t = -11 / -32
t = 11/32
So, the time at which the speed of the object is zero is t = 11/32.
Now that we have the value of t, we can substitute it back into the height function h(t) to find the height of the object at that time:
h(t) = -16t^2 + 11t + 1
= -16(11/32)^2 + 11(11/32) + 1
= -16(121/1024) + 121/32 + 1
= -1936/1024 + 3872/1024 + 1024/1024
= 968/1024
= 121/128
Therefore, the height of the object at the time when the speed is zero is 121/128 units above ground level.