Question

A gymnast who can stretch her arms up to reach 6 feet jumps straight up on a trampoline. The height of her feet about the trampoline can be modeled by the equation (h = -16x² + 12x), where (x) is the time in seconds after her jump. Do the gymnast’s hands reach a height of 10 feet above the trampoline? Use the discriminant to explain.

a. Yes, the hands reach a height of 10 feet.
b. No, the hands do not reach a height of 10 feet.
c. It cannot be determined from the given information.
d. The discriminant is not relevant to the problem.

Answer 1

By setting the quadratic equation h = -16x² + 12x equal to 10 feet and using the discriminant from the quadratic formula, we determine that the discriminant is positive. This result indicates that the gymnast's hands reach a height of 10 feet above the trampoline at two points in time during her jump.

Step-by-step explanation:

To determine if the gymnast's hands reach a height of 10 feet above the trampoline, we use the given quadratic equation h = -16x² + 12x and set it equal to 10 feet to solve for x (the time in seconds). This gives us the following equation: -16x² + 12x = 10

We can use the discriminant from the quadratic formula to find out if there are real solutions to this equation, which would mean the gymnast's hands can reach that height. The discriminant is given by b² - 4ac, where a, b, and c are the coefficients from the quadratic equation ax² + bx + c = 0.

Substituting our coefficients into the discriminant gives us (12²) - 4(-16)(10) which simplifies to 144 + 640, equal to 784. Since the discriminant is positive, we have two real solutions, which means the height of the gymnast's hands will reach 10 feet above the trampoline at two points in time during her jump. Therefore, the correct answer is: (a) Yes, the hands reach a height of 10 feet.