Final answer:
To find how long it will take Ben to reach the water, the quadratic equation h = -16t^2 + 27t + 10 is solved for t when h equals zero, using the quadratic formula. Two solutions may be found, but only the positive time value relevant to the dive scenario should be used.
Step-by-step explanation:
The student's question is about finding the time it takes for Ben to reach the water while diving from a 10-foot platform with the provided equation h = -16t2 + 27t + 10. To determine when Ben reaches the water, we need to solve this quadratic equation for when the height h equals zero because h represents the height above the water, and it is zero when Ben is at the water's surface.
To do this, we can either factor the equation if possible, complete the square, or use the quadratic formula, which is t = (-b ± √(b2 - 4ac)) / (2a). Here, a is -16, b is 27, and c is 10. Using the quadratic formula, we might find two solutions, and we need to choose the one that makes physical sense—i.e., the positive time value.
The solutions to the quadratic equation may resemble those in a related problem where a ball reaches a certain height twice during its trajectory—once on the way up and once on the way down. In such a scenario, use of the quadratic formula might yield two times: t = 3.79 s and t = 0.54 s. We would then select the longer solution for the time it takes to reach a particular point, discounting any negative time values or times that don't apply to the situation described.