A catapult launches a boulder with an upward velocity of 92 ft/s. The height of the boulder, h, in feet after t seconds is given by the function h=-16t^2+92t+30. How long Does it take the boulder to reach it’s maximum height? What is the boulders maximum height
the max height is on the axis of symmetry of the parabola
tmax = -b / (2 a) = -92 / (2 * -16)
find tmax , then substitute into the original equation to find hmax
a. Y = yo+g*Tr = 0.
92 + (-32)Tr = 0
Tr = 2.9 s. = Rise time (time to reach max. ht.).
b. Y^2 = yo^2+2g*h = 0.
92^2+(-64)h = 0
h = 132 Ft. above launching point.
ho+h = 30+132 = 162 Ft. above gnd. = h max.
To find the time it takes for the boulder to reach its maximum height, we need to determine when the velocity of the boulder becomes zero. When the velocity is zero, the boulder reaches its highest point.
The velocity of an object can be obtained by taking the derivative of the position function with respect to time. In this case, the position function is given by h = -16t^2 + 92t + 30.
To find the derivative of the position function, we differentiate each term separately using the power rule:
h' = -32t + 92
Now, set the velocity function equal to zero and solve for t:
-32t + 92 = 0
Simplifying the equation, we have:
-32t = -92
Divide both sides by -32:
t = 92/32
Evaluating the expression, we get:
t ≈ 2.875 seconds
Hence, it takes approximately 2.875 seconds for the boulder to reach its maximum height.
To find the maximum height, substitute the value of t into the position function h:
h = -16(2.875)^2 + 92(2.875) + 30
Simplifying the equation, we get:
h ≈ 158.125 feet
Therefore, the boulder's maximum height is approximately 158.125 feet.