A catapult launches a boulder with an upward velocity of 120 ft/s. The height of the boulder, h, in feet after t seconds is given by the function h = –16t2 + 120t + 10. How long does it take to reach maximum height? What is the boulder’s maximum height? Round to the nearest hundredth, if necessary.
A. Reaches a maximum height of 235.00 feet in 3.75 seconds.
B. Reaches a maximum height of 10.00 feet in 7.50 seconds.
C. Reaches a maximum height of 7.58 feet in 3.75 seconds.
D. Reaches a maximum height of 15.16 feet in 7.5 seconds.
To find the maximum height, we need to find the vertex of the parabolic function h = -16t^2 + 120t + 10. The vertex occurs at t = -b/2a, where a = -16 and b = 120.
t = -b/2a = -(120)/(2(-16)) = 3.75
So the boulder reaches its maximum height after 3.75 seconds. To find the maximum height, we substitute this value of t into the function:
h = -16(3.75)^2 + 120(3.75) + 10 ≈ 235.00
So the boulder reaches a maximum height of 235.00 feet after 3.75 seconds. The answer is A.
To find the time it takes for the boulder to reach maximum height, we need to determine the vertex of the quadratic function. The vertex of a parabola in the form y = ax^2 + bx + c is given by the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic function.
In this case, the quadratic function is h = -16t^2 + 120t + 10. Comparing this function to the form y = ax^2 + bx + c, we can see that a = -16 and b = 120.
Plugging these values into the formula for the vertex, we get t = -120 / (2 * -16) = 120/32 = 3.75.
So, the boulder takes 3.75 seconds to reach the maximum height.
To find the maximum height, we substitute this value of t into the function:
h = -16(3.75)^2 + 120(3.75) + 10
h = -16(14.06) + 450 + 10
h = -224.96 + 450 + 10
h = 235.04
Rounded to the nearest hundredth, the boulder's maximum height is 235.00 feet.
Therefore, the correct answer is A. Reaches a maximum height of 235.00 feet in 3.75 seconds.