Question

Treyvon is standing 9 yards from the base of a hill that has a slope of 3/4. He throws a water balloon from a height of 2 yards. Its path is modeled by h(x)=−0. 1x^2+0. 8x+2, where h is the height of the balloon in yards and x is the distance the balloon travels in yards.

a. Write a polynomial equation to represent the situation.



b. How far from Treyvon will the balloon hit the hill? If necessary, round to the nearest tenth

Answer 1

To represent the situation, use the polynomial equation h(x) = -0.1x^2 + 0.8x + 2. To find how far from Treyvon the balloon will hit the hill, set h(x) equal to the height of the hill and solve for x.

Step-by-step explanation:

To represent the situation where Treyvon throws a water balloon from a height of 2 yards at a distance of 9 yards from the base of a hill with a slope of 3/4, we can use the equation h(x) = -0.1x^2 + 0.8x + 2. This is a polynomial equation that models the height of the balloon as a function of its distance.

To find how far from Treyvon the balloon will hit the hill, we need to find the x-coordinate where h(x) is equal to the height of the hill. We can set h(x) equal to the height of the hill and solve for x. Once we find the x-coordinate, we can round it to the nearest tenth if necessary.

For example, if the height of the hill is 5 yards, we would set -0.1x^2 + 0.8x + 2 = 5 and solve for x.

Answer 2

The polynomial equation to represent the situation and the distance from Trevon where the balloon hits the hill, are as follows;

a. h(x) = -0.1·x² + 0.8·x + 2, when x < 9

h(x) = -0.1·x² + 0.8·x + 2 - (3/4)·(x - 9) where 9 ≤ x < ∞

b. The balloon hits the hill at about 9.6 yards from Trevon

The details of the steps for finding the polynomial equation to represent the situation are as follows;

a. The function for the height of the balloon can be presented as follows;

h(x) = -0.1·x² + 0.8·x + 2

The slope of the hill = 3/4

Height from which Trevo throws the balloon = 2 yards

The polynomial that represents the situation is therefore;

h(x) = -0.1·x² + 0.8·x + 2 when 0 ≤ x < 9

h(x) = -0.1·x² + 0.8·x + 2 - (3/4)·(x - 9) when 9 ≤ x < ∞

b. The balloon hits the hill when we get;

-0.1·x² + 0.8·x + 2 = (3/4)·(x - 9)

-0.1·x² + 0.8·x + 2 - 3/4·x + (3/4)·9 = 0

-0.1·x² + 0.05·x + 8.75 = 0

(-(0.05) ± √(0.05² - 4×(-0.1)×8.75))/(2 × 0.05)

The distance x ≈ -9.1 and x ≈ 9.6

The balloon hit the hill at 9.6 yards from Treyvon