Answer:
To find the time it takes for the object to reach its maximum height, we can use the following equation for the vertical motion of the object:
s(t) = -5t^2 + 102t + 730
The object will reach its maximum height when its vertical velocity becomes zero. In other words, at the maximum height, the rate of change of height with respect to time (velocity) is zero.
To find the time at which this occurs, we can take the derivative of s(t) with respect to t and set it equal to zero:
s'(t) = 0
The derivative of s(t) is:
s'(t) = -10t + 102
Now, set this equal to zero and solve for t:
-10t + 102 = 0
Add 10t to both sides:
102 = 10t
Divide both sides by 10:
t = 10.2
So, it takes 10.2 seconds for the object to reach its maximum height.
To find the maximum height, substitute this time back into the original equation for s(t):
s(10.2) = -5(10.2)^2 + 102(10.2) + 730
Now, calculate the value:
s(10.2) = -5(104.04) + 1040.4 + 730
s(10.2) = -520.2 + 1040.4 + 730
s(10.2) = 1250.2 feet
So, the maximum height reached by the object is 1250.2 feet.
Explanation:
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