Question

The height in feet above the water of a diver can be modeled by the function f(x) = -3x2 + 6x + 24, where x is the time in seconds after the diver begins the dive.

1. How long does it take the diver to reach the water?
2. What is the maximum height achieved by the diver?
3. When does it occur?
4. How high is the diver after 2.4 seconds?
5. When is the diver 15 feet above the water? What is a reasonable domain for this situation?
6. What is a reasonable range?

Answer 1

The function f(x) = -3x^2 + 6x + 24 models the diver's height above the water, where the time to hit the water, maximum height, and height after specific time intervals can be calculated.

Step-by-step explanation:

The diving scenario can be described with a quadratic function, f(x) = -3x2 + 6x + 24, where x is the time in seconds and f(x) is the diver's height above the water in feet.

1. To determine how long it takes for the diver to reach the water, we set the function equal to zero and solve for x. This gives us the roots of the quadratic equation, representing the time when the diver is at the level of the water.
2. The maximum height is found at the vertex of the parabola, which occurs at x = -b/(2a) for the general form ax2 + bx + c. In this case, the vertex occurs at x = -6/(2*(-3)) = 1 second.
3. The height of the diver after 2.4 seconds can be found by substituting x = 2.4 into the function.
4. To find out when the diver is 15 feet above the water, we set f(x) to 15 and solve for x. The reasonable domain for this situation is from the moment the diver leaves the board until they hit the water.
5. The range is the set of all possible heights the diver can achieve during the dive, starting from 24 feet and going up to the maximum height.