Final answer:
The diver reaches the peak of the dive at 0.375 seconds, the springboard is 5 meters above the water, and the time at which the diver hits the water is determined by solving the quadratic equation -16t^2 + 12t + 5 = 0.
Step-by-step explanation:
The student has provided the equation h(t) = -16t^2 + 12t + 5 which models the height of an Olympic diver above the water in meters, as a function of time after they leave the springboard.
Peak of the dive
To find when the diver will reach the peak of the dive (a), we need to determine when the velocity is zero. The velocity is given by the derivative of the height function, so we calculate v(t) = h'(t) = -32t + 12. Setting v(t) to zero gives us the time at which the diver reaches the highest point: -32t + 12 = 0, solving for t gives t = 0.375 seconds.
Height of the springboard
To find the height of the springboard (b), we evaluate the height function at time t = 0, which is h(0) = -16(0)^2 + 12(0) + 5 = 5 meters above the water.
Diver hitting the water
Finally, to find when the diver will hit the water (c), we need to solve for when h(t) = 0. This requires factoring or using the quadratic formula to solve the quadratic equation -16t^2 + 12t + 5 = 0. Factoring is not straightforward in this case, so we use the quadratic formula, which yields two solutions. Only the positive solution is physically meaningful in this context, resulting in the time when the diver hits the water, which can be found by substituting the known values into the quadratic formula.