The rocket achieves a maximum altitude of 144 ft, reaching this peak after 2 seconds. It then descends and touches the ground at 5 seconds, following its initial launch from an 80 ft tower with an initial velocity of 64 ft/sec.
To find the maximum altitude and the time at which the rocket hits the ground, you can use the given quadratic function for the distance s of the projectile:
s(t) = -16t^2 + v_0 t + h_0
Where:
v_0 is the initial velocity,
h_0 is the initial height, and
t is the time in seconds.
In this case, v_0 = 64 ft/sec and h_0 = 80 ft. Therefore, the function becomes:
s(t) = -16t^2 + 64t + 80
To find the maximum altitude, you need to determine the vertex of the quadratic function. The vertex of a quadratic function ax^2 + bx + c is given by the coordinates (h, k), where h = -b/(2a) and k is the maximum value of the function. In this case:
h = -64/(2 * -16) = 2
Now substitute t = 2 back into the original function to find k, the maximum altitude:
s(2) = -16(2)^2 + 64(2) + 80
s(2) = -64 + 128 + 80
s(2) = 144
So, the maximum altitude is 144 ft.
Now, to find the time at which the rocket hits the ground, set s(t) equal to 0 and solve for t:
-16t^2 + 64t + 80 = 0
You can simplify this equation by dividing by -16:
t^2 - 4t - 5 = 0
Now factor the quadratic:
(t - 5)(t + 1) = 0
This gives two possible solutions: t = 5 or t = -1. Since time cannot be negative in this context, the rocket hits the ground at t = 5 seconds.
So, the rocket's maximum altitude is 144 ft, and it hits the ground after 5 seconds.
Complete question:
The distance s of a projectile, such as a rocket, above the Earth's surface (under the influence of gravity) after t seconds is modeled by the quadratic function gt2 + vot + $0, where t > 0,9 (force of gravity acting downward and thus negative) is approximately equal to -32 ft/sec? when s is measured in feet, vo is initial velocity (when t equals zero), and so is the initial height of the rocket above the ground (when t equals zero). A rocket is fired straight upward from the top of an 80 ft tower with an initial velocity of 64 ft/sec. Use what you have learned in this unit to find the rocket's maximum altitude and the time at which the rocket hits the ground.