Question

Eben, an alien from the planet Tellurango, kicks a football from field level. The equation for the football’s height in meters, h, after t seconds is h(t)=−2t2+12t.
What is the maximum height that the ball reaches, assuming nobody blocks the ball? How can you justify your answer.

Select the option that correctly answers both questions.

The ball reaches a maximum height of 18 meters. The maximum of h(t) can be found only graphically by plotting the function and locating the y-coordinate of the highest point.
The ball reaches a maximum height of 3 meters. The maximum of h(t) can be found only graphically by plotting the function and locating the x-coordinate of the highest point.
The ball reaches a maximum height of 18 meters. The maximum of h(t) can be found both graphically or algebraically, and lies at (3,18). The x-coordinate, 3, is the time in seconds it takes the ball to reach maximum height, and the y-coordinate, 18, is the max height in meters.
The ball reaches a maximum height of 3 meters. The maximum of h(t) can be found both graphically or algebraically, and lies at (3,18). The x-coordinate, 3, is the max height in meters, and the y-coordinate, 18, is the time in seconds it takes the ball to reach maximum height.

Answer 1

The maxima of a differential equation can be obtained by getting the 1st derivate dx/dy and equating it to 0.

Given the equation h = - 2 t^2 + 12 t       , taking the 1st derivative result in:

dh = - 4 t dt + 12 dt

dh / dt = 0 = - 4 t + 12   calculating for t:

t = -12 / - 4

t = 3 s

Therefore the maximum height obtained is calculated by plugging in the value of t in the given equation.

h = -2 (3)^2 + 12 (3)

h = 18 m

This problem can also be solved graphically by plotting t (x-axis) against h (y-axis). Then assigning values to t and calculate for h and plot it in the graph to see the point in which the peak is obtained. Therefore the answer to this is:

The ball reaches a maximum height of 18 meters. The maximum of h(t) can be found both graphically or algebraically, and lies at (3,18). The x-coordinate, 3, is the time in seconds it takes the ball to reach maximum height, and the y-coordinate, 18, is the max height in meters.