A catapult launches a boulder with an upward velocity of 184 feet per second. The height of the boulder, (h), in feet after t seconds is given by the function h of t equals negative 16t squared plus 184t plus 20 . How long does it take the boulder to reach its maximum height? What is the boulder’s maximum height? Round to the nearest hundredth, if necessary.
A. Reaches a maximum height of 11.6 feet after 5.75 seconds.
B. Reaches a maximum height of 549 feet after 11.5 seconds.
C. Reaches a maximum height of 549 feet after 5.75 seconds.
D. Reaches a maximum height of 23.2 feet after 11.6 seconds.
To find the time it takes for the boulder to reach its maximum height, we need to find the vertex of the parabolic function. The x-coordinate of the vertex is given by the formula -b/2a, where a, b, and c are the coefficients in the quadratic equation ax^2 + bx + c = 0.
In this case, a = -16 and b = 184, so the time it takes for the boulder to reach its maximum height is:
t = -b/2a = -(184)/(2(-16)) = 5.75 seconds
To find the maximum height, we need to plug this value of t into the equation for h(t) and simplify:
h(5.75) = -16(5.75)^2 + 184(5.75) + 20 = 549
Therefore, the boulder reaches a maximum height of 549 feet after 5.75 seconds. The answer is choice C.