Question

A model rocket is launched from the top of a building. The height (in meters) of the rocket above the ground is given by h(t) = -6t2 +24t +14, where t is the time (in seconds) since the rocket was launched. What is the rocket’s maximum height?

Answer 1

Rocket's maximum height = 38 meters

Explanation:

Since we want to know the maximum and height represents the y-coordinate, we're essentially looking for the y-coordinate of the maximum.

Note that h(t) = -6t^2 + 24t + 14 is in standard form, whose general equation is given by y = ax^2 + bx + c.

Thus, -6 is our a value, 24 is our b value, and 12 is our c value.

We can find the y-coordinate of the maximum using the formula:

h(-b/2a) = -6(-b/2a) + 24(-b/2a) + 14.

Step 1: Use the formula -b/2a to find the x-coordinate of the maxium:

In order to make the problem simpler, we can start by finding the x-coordinate of the vertex using -b/2a:

-24 / 2(-6)

-24 / -12

2

Thus, the x-coordinate of the vertex is 2.

Step 2: Plug in 2 for t to find the y-coordinate of the maximum:

Now we can find the y-coordinate of the maximum (i.e., the rocket's maximum height) by plugging in 2 for t:

h(2) = -6(2)^2 + 24(2) + 14

h(2) = -6(4) + 48 + 14

h(2) = -24 + 48 + 14

h(2) = 24 + 14

h(2) = 38

Thus, the rocket's maximum height is 38 meters.