Final answer:
The maximum height Taylor's football reaches can be found by converting the quadratic equation t(x) to vertex form or calculating the vertex directly. To find the difference in maximum heights and total distances traveled, a comparison equation is needed.
Step-by-step explanation:
The equation t(x) = -0.05(x² - 50x) models the vertical height, t(x), of a football thrown by Taylor x feet from where it was thrown. To find the maximum height the football reaches, we would need to complete the square or find the vertex of the parabolic equation. The vertex form of a parabola is given by t(x) = a(x-h)² + k, where (h, k) is the vertex of the parabola. Since the coefficient of x² is negative, this parabola opens downward, and the vertex represents the maximum height. To convert the given equation to the vertex form, we can use the fact that the x-coordinate of the vertex is given by -b/(2a) in the standard form ax² + bx + c. In this case, a = -0.05 and b = -0.05 * 50 = -2.5. Thus, the x-coordinate of the vertex is -(-2.5)/(2 * -0.05) = 50/0.1 = 500/2 = 250 feet. Substituting this back into the equation, we get the maximum height as t(250) = -0.05(250² - 50 * 250). To determine the difference in total distances traveled, we identify where the football hits the ground, which occurs when t(x) = 0. We can use the quadratic formula or factor the equation to find the roots, which represent the points where the football lands, or the total horizontal distance it has traveled. However, without a second equation to compare with, we cannot calculate the difference in total distances since we only have Taylor's throw information.