Question

Kendra dives off a diving board into the water and then comes back up to the surface. Her dive can be modeled by the equation: , where "h" is the height in feet and "x" is the horizontal distance in feet from the diving board. 1) How high is the diving board? 2) How deep does the diver dive into the water? 3) At what horizontal distance from the board does the diver enter the water? 4) At what horizontal distance from the board does the diver come to the surface of the water after the dive?

Answer 1

1. 6 feet

2. 6.25 feet

3. 1 feet

4. 6 feet

Explanation:

The equation is :
`$h(x)=x^2-7x+6$`

1. The diving board is where Kendra dives off. Here, the horizontal distance, x from the diving board is 0.

So, substituting x = 0 in the equation, we get

`$h(0)=0^2-7(0)+6$`

`$=0-0+6$`

`$=6$`

So, the diving board is 6 feet above the surface of the water.

2. From the equation, we known that it is a parabola and the vertex is minimum.

It is the minimum height which represents the depth Kendra dives into the water.

So the
`$x$` coordinate of the vertex is =
`$(-b)/(2a)$`

Here, a and b are the coefficients of linear term and the quadratic terms in the equation. Therefore,

a = 1 and b = -7

∴ x coordinate =
`$(-(-7))/(2 * 1) $`

`$(7)/(2)=3.5$`

Now substituting to find f(x),

`$h(3.5)=(3.5)^2-7(3.5)+6$`

= -6.25

Therefore, the diver dives 6.25 feet below the water surface.

3. The horizontal distance from the board the diver enters into the water.

This is the y-intercept and it is the value of x when h(x)=0.

∴
`$0=x^2-7x+6$`

Factorizing, we get
`$(x-1)(x-6)=0$`

∴
`$x=1 \text{ or}\ x=6$`

So there are two solutions that are the two x intercepts of the function. Here at x = 1 shows the horizontal distance from the board from where Kendra dives into the water.

4. We know that the equation given has
`$\text{two}$` x intercepts. These two x intercepts are the points where the parabola crosses the x-axis, which is the height
`$h(x)=0$`. The height is the water surface level.

The first x intercept represents the points where Kendra dives into the water.

And the second x intercept is the point where Kendra comes out of the water surface. This this is
`$x=6$` for
`$h(x)=0$`.

Thus Kendra dives out of the water surface at 6 feet from the board.