A catapult launches a boulder with an upward velocity of 92 m/s. The height of the boulder, h, in meters after 1 seconds is given in the function h= -5t^2+92t+16. How many seconds does it take to reach maximum height? What is the boulders maximum height?
Sorry
I think that it is 439.20 meters in 9.2 seconds
the t of the vertex is -b/(2a) = -92/-10 = 9.2
you had that
sub in t = 9/2,
h = 439.2
you were right, good job
To find the time it takes for the boulder to reach its maximum height, we need to find the time at which the velocity becomes zero. At the maximum height, the velocity becomes zero because the boulder momentarily stops before starting to fall back down.
The velocity function is obtained by differentiating the height function with respect to time:
v = h'(t) = -10t + 92,
where h'(t) denotes the derivative of h(t) with respect to t.
To find the time it takes for the velocity to become zero, we set v(t) = 0 and solve for t:
-10t + 92 = 0,
-10t = -92,
t = -92 / -10,
t = 9.2 seconds.
Therefore, it takes the boulder 9.2 seconds to reach its maximum height.
To find the maximum height, we substitute the value of t = 9.2 into the height function:
h(9.2) = -5(9.2)^2 + 92(9.2) + 16.
Evaluating this expression:
h(9.2) = -5(84.64) + 846.4 + 16,
h(9.2) = -423.2 + 846.4 + 16,
h(9.2) = 439.2 meters.
Hence, the boulder reaches a maximum height of 439.2 meters.