Final answer:
To find the time the swimmer's feet are in the air, set the function h(t) equal to zero and solve for t. Use the positive value to find the time. To find the highest point above the board, find the vertex of the parabolic path. To find the velocity when the swimmer's feet hit the water, find the derivative of the height function and evaluate it at the time when the height is zero.
Step-by-step explanation:
To find the time the swimmer's feet are in the air, we need to determine when the swimmer's height is zero. We can do this by setting the function h(t) equal to zero and solving for t:
0 = (8/9)t^2 - (16/3)t + 4
Next, we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula. After finding the values of t, we can use the positive value to find the time the swimmer's feet are in the air.
To find the highest point above the board, we need to find the vertex of the parabolic path. We can use the formula t = -b/(2a), where a and b are the coefficients of the quadratic equation, to find the time at the highest point. Then, substituting this value of t into the height function will give us the highest point above the board.
To find the velocity when the swimmer's feet hit the water, we can find the derivative of the height function and evaluate it at the time when the height is zero. This will give us the instantaneous velocity at that moment.