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 = -5₽+ 92t + 16. How long does it take to reach maximum height? What is the boulder's maximum height? Round to the nearest hundredth, if necessary.
(1 point)
Reaches a maximum
• height of 16.00 meters in
18.4 seconds.
Reaches a maximum
• height of 18.57 meters in
9.2 seconds.
Reaches a maximum
• height of 37.14 meters in 18.4 seconds.
Reaches a maximum
• height of 439.20 meters
in 9.2 seconds.
To find the time it takes for the boulder to reach maximum height, we need to find the vertex of the quadratic function h = -5t^2 + 92t + 16. The formula for finding the x-coordinate of the vertex of a quadratic function in the form y = ax^2 + bx + c is given by x = -b/2a.
In this case, a = -5 and b = 92. Plugging these values into the formula, we get:
t = -92 / (2 * -5)
t = -92 / -10
t = 9.2 seconds
So, it takes 9.2 seconds for the boulder to reach maximum height.
To find the maximum height, we can substitute this value of t back into the original equation for h:
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 meters
Therefore, the boulder's maximum height is 439.20 meters.