When calculating the effect of Earth's gravitational pull on a launched projectile, the expression −16t2 is used to represent that the projectile falls at a rate of 16 feet per second squared. If a toy rocket is launched vertically upward from the ground at an initial velocity of 132 feet per second, what will be its maximum height? Round to the nearest hundredth.
To find the maximum height reached by the toy rocket, we first need to find the time it takes for the rocket to reach its maximum height.
The equation for the height of the rocket at any time t is given by:
h(t) = -16t^2 + 132t
To find the time it takes for the rocket to reach its maximum height, we need to find the vertex of the parabola represented by the equation h(t). The t-coordinate of the vertex of a parabola in the form ax^2 + bx + c is given by:
t = -b / 2a
In our case, a = -16 and b = 132. Plugging these values into the formula, we get:
t = -132 / 2(-16)
t = -132 / -32
t ≈ 4.125 seconds
Now that we have found the time it takes for the rocket to reach its maximum height, we can find the maximum height by plugging this time back into the equation h(t):
h(4.125) = -16(4.125)^2 + 132(4.125)
h(4.125) = -16(17.015625) + 544.5
h(4.125) = -272.25 + 544.5
h(4.125) = 272.25 feet
Therefore, the maximum height reached by the toy rocket is approximately 272.25 feet.