Questions: 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)=-0.1 x^2+0.8 x+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.

Solution Steps

Calculate the height of the hill at the point where Treyvon is standing, which is 9 yards from the base. The slope of the hill is given as \( \frac{3}{4} \). Thus, the height of the hill can be calculated as: \[ \text{Hill Height} = \frac{3}{4} \times 9 = 6.75 \text{ yards} \]

Evaluate the height of the balloon when it has traveled 9 yards. The height of the balloon is modeled by the function: \[ h(x) = -0.1x^2 + 0.8x + 2 \] Substituting \( x = 9 \): \[ h(9) = -0.1(9)^2 + 0.8(9) + 2 = 1.1 \text{ yards} \]

Set up the polynomial equation that represents the situation where the height of the balloon equals the height of the hill at the distance of 9 yards. The equation can be expressed as: \[ -0.1x^2 + 0.8x + 2 = \frac{3}{4}(x - 9) + 6.75 \] This simplifies to: \[ -0.1x^2 + 0.8x + 2 = 0.75x \]

Rearranging the equation gives: \[ -0.1x^2 + 0.8x - 0.75x + 2 = 0 \] This simplifies to: \[ -0.1x^2 + 0.05x + 2 = 0 \] Solving this quadratic equation yields the solutions: \[ x \approx -4.229 \quad \text{and} \quad x \approx 4.729 \]

Final Answer