Study Island Help

Question

Dr. Peabody shot a rocket off the roof of a 42-foot tall building. After three seconds, the rocket reached its highest point of 96 feet.

Model this situation with a quadratic function, and use the graph to determine how many seconds it will take the rocket to hit the ground?

Answer 1

model the graph with a parabola

y=a(x-3)^2 + 96 where (3,96) is the vertex

solve for a by plugging in the point (0,42) the initial height = 42 at time zero

42 =a(0-3)^2 + 96

-54 = 9a

a = -54/9 =-6

the model is

y=-6(x-3)^2 + 96

or

h(t) = -6(t-3)^2 + 96

set it = 0 for ground level

0 =-6(t-3)^2 +96

-6(t^2 -6t +9) + 96 =0

divide by -6

t^2-6t +9 - 16 = 0

t^2 -6t -7 = 0

factor

(t-7)(t+1) = 0

set each factor = 0 and solve for t

t-7=0

t =7 seconds to reach ground level (ignore the negative time solution)

Answer 2

Based on the question, we know 2 key points on the quadratic function which are (0, 42) and vertex (3, 96).

Let's say the function is f(x)=a(x-h)²+k

Using the vertex point (h, k)=(3, 96)

So, f(x)=a(x-3)²+96.

Use point (0, 42), we get 42=a(-3)²+96.

and a=-6

So, the function will be f(x)=-6(x-3)²+96.

As the seconds for the rocket to hit ground, just solve equation f(x)=0, we can have (x-3)²=16

So, x=-1 & 7.

Obviously, the answer is 7 seconds.