Question

A diver jumps off a diving board. The function h = -16x² + 6x + 5 represents the height (in feet) of the diver after x seconds. What is the maximum height above the water of the diver? How many seconds did it take for the diver to reach the maximum height? Round your answers to the nearest hundredth.​

Answer 1

The maximum height of the diver is found by calculating the vertex of the parabolic equation and substituting the time back into the equation to find the corresponding height.

Step-by-step explanation:

To find the maximum height of the diver and the time to reach it, we analyze the quadratic function h = -16x² + 6x + 5. The maximum height, given a negative coefficient for the quadratic term, occurs at the vertex of the parabola represented by this function. The vertex can be found using the formula x = -b/(2a), where a and b are the coefficients from the quadratic equation.

Using the given function where a is -16 and b is 6, we calculate the time to reach maximum height:

x = -b/(2a) = -6/(2 × -16) = 6/32 = 0.1875 seconds

Now, substitute this value back into the original equation to find the maximum height:

h = -16(0.1875)² + 6(0.1875) + 5

By solving the equation, we get the maximum height the diver achieves. Then, these values should be rounded to the nearest hundredth as requested.

Answer 2

Max height above water = 5.563 feet

Time to reach max height = 0.188 seconds

Step-by-step explanation:

Given h = -16x² + 6x + 5: a= -16, b= 6, c= 5

1. How many seconds did it take for the diver to reach the maximum height?

-- find the x-coordinate of the vertex: x = -b/(2a) = -6/(2·-16) = 3/16 ≈ 0.188

2. Maximum Height of diver above the water

-- find the y-coordinate of the vertex: h = -16(3/16)² + 6(3/16) + 5

= -16(9/256) + 6(3/16) + 5

= -9/16 + 9/8 + 5

= 5 9/16 ≈ 5.563

NOTE: Always include units when there are some!!