Need help asap thank you!

Question

1. Signal flares were launched from two different boats. The red flare followed the function f(x) = -(x - 15)² + 175 and the green flare followed the function f(x) = -x(x - 25), where x was the horizontal distance in meters and f(x) was the height in meters. Which flare went the highest in the air? Provide mathematical evidence to justify your answer.

2. A potato was launched in the air using a potato gun. The function for the situation was h(t) = -16t² + 100t + 10, where t was the time in seconds and h(t) was the height in feet.

a. What was the maximum height of the potato?

b. What was the lowest height of the potato?

c. What was the lowest time that applies to the actual situation?

d. What is the greatest time that applied to the actual situation?

e. Using your answers to parts a through d, what are the domain and range of the function, as they apply

to this situation? Use set builder notation.

Answer 1

1. the vertex form is a (x-h)²+ k, so red f(x) has a = -1 h = 15, k = 175. Vertex is (h,k) = (15,175) so max height = 175. green f(x) = -x (x-25). a = -1, b = 25. vertex has x coordinate -b/2a = -25 / - 2 = 12.5, f(x) = -12.5 (12.5 - 25) = 12.5²= 156.25. so red goes higher
2. here a = -16, b= 100, c = 10. vertex has x = -b/2a = 100/32. substitute that into h(t) for max height
3. lowest height, well h(t) has a<0 so facing upwards, so lowest possible height must be 0.
4. lowest time = 0
5. greatest time solve h(t)=0 and take the t value > 0. plot h(x) on Desmos and find the x intercept
6. domain - possible x values. lowest = 0, highest - answer from (d)
7. range - h(t) values. obviously minimum is 0, maximum is answer from (a)