Question

(100 POINTS) Buffy and Addy launch their rockets at the same time. The height of Buffy's rocket, in meters, is given by the function f(x)=−4.9x²+50x, where x is the number of seconds after the launch. The height of Addy's rocket, in meters, is given by the function g(x) =−4.9x²+32x+40 where x is the number of seconds after the launch. Graphically find the moment when the rockets are at the same height, and then use that to calculate the height. (please round to the nearest meter)

Answer 1

Around 2.22 seconds after the launch, the rockets are at the same height, which is approximately 46 meters.

Explanation:

So we have to find when the rockets are at a same height. In order to do that we have to find the x-value at which the height functions f(x) and g(x) are equal.

The equation we are using will be : f(x) = g(x)

Plugging in the info we have the equation will turn into this

-4.9x² + 50x = -4.9x² + 32x + 40

Once you work out and simply the equation you should get this answer,

18x = 40

To solve for x you have to divide by both sides by 18 to get

x = 40/18, or if you want to simplify further x ≈ 2.22

So about approximately 2.22 seconds after the launch, the rockets are at the same height.

Now that we figured out the time is about 2.22 seconds we can plug this into x, and then solve for the height.

To calculate the height, we can substitute this value of x back into either of the height functions. Let's use f(x):

f(2.22) = -4.9(2.22)² + 50(2.22)

When solving you should get:

f(2.22) ≈ 45.86

Therefore, at the moment when the rockets are at the same height, the height is approximately 45.86 meters. The question states to round to the nearest meter, which gives us an answer of 46 meters.