Question

The following function gives the height, h metres, of a fired rocket as a function of time, t seconds, since the rocket was fired.

h = − t2 + t +4.91

a) What was the maximum height reached by the rocket? At what time did this occur?




b) What was the initial height of the rocket?



c) Determine the maximum height of the rocket



d) Determine the height of the rocket after 2 seconds

Answer 1

(a.) max height = 5.16 ft at 0.5 seconds

(b.) 4.91 ft

(c.) max height = 5.16 ft

(d.) height at 2 seconds = 2.91 ft

Explanation:

(a.) The maximum height is also known as the vertex.

We can find it by using the formula

`x =(-b)/(2a)`, which gives us the x coordinate of the vertex (or in this case, the time at which the rocket reaches its maximum height).

Then, we can plug this value in the quadratic equation to find the y coordinate of the vertex (i.e., the maximum height the rocket reaches.

The equation is currently in the standard form, which is:

`y = ax^2+bx + c`.

Thus, our a is -1 and b is 1:

`x=(-1)/(2(-1))=(1)/(2)=0.5 \\\\y=-(0.5)^2+0.5+4.91\\y=-0.25+0.5+4.91\\y=0.25+4.91\\y=5.16`

Thus, the max height is 5.16 ft and the rocket reaches it at 0.5 seconds.

(b). As mentioned in the explanation for part a, the equation is currently in standard form and c is the constant or y-intercept. Thus, the initial height is 4.91 ft

(c.) We found that the maximum height of the rocket is 5.16 ft.

(d.) We simply plug in 2 for x in the equation to find h:

`h(2)=-(2)^2+2+4.91\\h(2)=-4+2+4.91\\h(2)=-2+4.91\\h(2)=2.91`

As shown, the height of the rocket at 2 seconds is 2.91 ft.