Kendra dives off of a diving board into the water and then comes back up to the surface. Her dive can be modeled by the equation

Question

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Kendra dives off of a diving board into the water and then comes back up to the surface. Her dive can be modeled by the equation

: h(x) = x^2 - 7x + 6, 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?

Mathematics

2 answers:

il63 [147K] · Answer 1 · 2022-08-12T21:09:50+03:00

Answer:

1) How high is the diving board?:  6 feet above the water surface

2) How deep does the diver dive into the water?: 6.5feet into the water

3) At what horizontal distance from the board does the diver enter the water? 1 feet from the board

4) At what horizontal distance from the board does the diver come to the surface of the water after the dive? 6 feet from the board

Explanation:

The dive is modeled by the equation h(x) = x² - 7x + 6.

Two important features that you must notice quickly are:

That is a quadratic function, which is a parabola.

Since the leading term, x², is positive, the parabola open upwards, and the vertex is a minimum.

As for the questions:

1) How high is the diving board?

The diving board is the place from which Kendra dives off. At this point, the horizontal distance, x, from the diving board is 0.

Then, the height will be given by h(x) with x = 0. Substitute into the equation and compute:

h(0) = 0² - 7(0) + 6

h(0) = 0 - 0 + 6

h(0) = 6

Therefore, the diving board is 6 feet above the water surface.

2) How deep does the diver dive into the water?

This is the minimum heigth given by the function h(x).

As explained above, this is the vertex of the parabola.

You can find the x-coordinate of the vertex with the equation:

x-coordinate of the vertex = -b/(2a)

Where b and a are the coefficients of the linear term and the quadratic terms in the equation for h(x), respetively:

a = 1
b = -7

x-coordinate = - (-7) / (2 × 1) = 7/2 = 3.5

Now you can find h(x) for x = 3.5

h(3.5) = (3.5)² - 7(3.5) + 6 = 12.25 - 24.5 + 6 = - 6.25

That is 6.25 feet below the water surface.

3) At what horizontal distance from the board does the diver enter the water?

That is the x-intercept of the function. This is, the value of x when h(x) = 0.

Thus, solve the equation 0 = x² - 7x + 6.

You can factor the equation:

Find two numbers that add up -7 and their product is +6. Those are - 6 and -1:

0 = (x - 1) (x - 6)

Using the zero product property:

(x - 1) = 0 or (x - 6) = 0

x = 1 or x = 6

Thus, you have two solutions, which are the two x-intercepts of the funtion.

x = 1 represents the horizontal distance from the board where the diver enters the water.

The other solution is the answer to the next question.

4) At what horizontal distance from the board does the diver come to the surface of the water after the dive?

As explained above, the equation has two x-intercepts. They both are the points where the parabola crosses the x-axis, which is when the height, h(x) is 0. That height is the surface of the water level.

The first x-intercept is the point (horizontal distant from the board) where the diver enters the water, such as asserted above.

The other x-intercept is where the diver comes to the surface of water after the dive and after being submerged in the water: the diver enters the water, dives 6.5feet into the water, and returns to the surface.

Thus, this is the solution x = 6 for h(x) = 0.

Hence, Kendra comes to the surface of water at 6 feet from the board.