Question
A catapult launches a boulder with an upward velocity of 92 m/s. The height of the boulder, h, in meters after t seconds is given by the function h=−5t^2+92t+16
. Round to the nearest hundredth, if necessary.
How long does it take to reach maximum height? _ meters
What is the boulder's maximum height? _ seconds
Show your work
. Round to the nearest hundredth, if necessary.
How long does it take to reach maximum height? _ meters
What is the boulder's maximum height? _ seconds
Show your work
Answers
GPT 3.5
To find the time it takes to reach the maximum height, we first need to find the t-coordinate of the vertex of the parabola given by the equation h=-5t^2+92t+16.
The t-coordinate of the vertex can be found using the formula t = -b/2a, where a = -5 and b = 92 in this case.
t = -92 / 2*(-5)
t = -92 / -10
t = 9.2
Therefore, it takes 9.2 seconds to reach the maximum height.
To find the maximum height, we substitute this time back into the equation h=-5t^2+92t+16:
h = -5(9.2)^2 + 92(9.2) + 16
h = -5(84.64) + 846.4 + 16
h = -423.2 + 846.4 + 16
h = 439.2
Therefore, the boulder reaches a maximum height of 439.2 meters.
The t-coordinate of the vertex can be found using the formula t = -b/2a, where a = -5 and b = 92 in this case.
t = -92 / 2*(-5)
t = -92 / -10
t = 9.2
Therefore, it takes 9.2 seconds to reach the maximum height.
To find the maximum height, we substitute this time back into the equation h=-5t^2+92t+16:
h = -5(9.2)^2 + 92(9.2) + 16
h = -5(84.64) + 846.4 + 16
h = -423.2 + 846.4 + 16
h = 439.2
Therefore, the boulder reaches a maximum height of 439.2 meters.