Final answer:
Using the discriminant from the quadratic equation that models Eva's jump from the trampoline, we determined that there are no real solutions because the discriminant is negative. This means Eva will not reach the height of 10 feet needed to dunk the basketball.
Step-by-step explanation:
To determine if Eva will reach a height of 10 feet to dunk the basketball, we can model her jump using projectile motion equations. The height h after time t can be described by the quadratic equation h(t) = -16t^2 + v0t + s0, where -16 represents half the acceleration due to gravity in ft/s2, v0 is the initial velocity, and s0 the starting height above the ground. Since we are only interested in whether she can reach 10 feet, we set the equation to equal 10 feet and solve for t.
The equation becomes -16t^2 + 22.3t + 2 = 10. Simplifying, we get -16t^2 + 22.3t - 8 = 0. To check if there are real solutions for t, indicating that she will reach 10 feet, we use the discriminant D = b^2 - 4ac of the quadratic equation at^2 + bt + c = 0. Here, a = -16, b = 22.3, and c = -8. So, the discriminant is D = 22.3^2 - 4(-16)(-8).
Discriminant calculation: D = 22.32 - 4(16)(8) = 497.29 - 512 = -14.71. Since the discriminant is negative, there are no real solutions to the equation, meaning that Eva will not reach a height of 10 feet on her jump from the trampoline.