Question

The path of a diver is modeled by the function f(x)=−9x2+9x+1, where f(x) is the height of the diver (in meters) above the water and x is the horizontal distance (in meters) from the end of the diving board.


a. What is the height of the diving board?


The diving board is

meter(s) high.


b. What is the maximum height of the diver?


The maximum height of the diver is

meter(s).


c. Describe where the diver is ascending and where the diver is descending. Round your answers to the nearest tenth, if necessary.


The diver is ascending from

meter(s) to

meter(s) and descending from

meter(s) until hitting the water after approximately

meter(s).

Answer 1

Answer:

A) the diving board is 1m high.

B) maximum height of the diver is 3.25 meter(s)

C)i)the diver is a ascending from 0 metres to 0.5 metres.

II) the diver descends from 0.5 metres to 1 m

Step-by-step explanation:

A) We are given the function;

f(x) = −9x² + 9x + 1

Now, the height of the diving board will be when x = 0 metres. This is because at x = 0, the driver didn't jump.

Thus;

f(0) = -9(0²) + 9(0) + 1

f(0) = 1 m

Thus, the diving board is 1m high.

B) From quadratic equations, we can say that in the given function, we have as follows;

a = -9

b = 9

c = 1

Now, a < 0

This means that the parabola opens down and it's vertex formula will be used and value of x plugged into the given function to get the maximum height.

Thus, vertex is;

x = -b/2a

x = -9/(2 × -9)

x = 1/2

x = 0.5 m

Thus;

Max height = -9(0.5²) + 9(0.5) + 1

Max height = 3.25 m

The maximum height of the diver is 3.25 meter(s)

C) Since the parabola opens down, the diver is ascending from 0 metres to 0.5 metres.

Similarly, the diver descends from 0.5 metres to 1 m

Answer 2

The path of the diver is an illustration of a quadratic function.

• The height of the diving board is 1 meter
• The maximum height of the diver is 3.25 m
• The diver ascends from 0 meters to 0.5 meters

The function is given as:

$\mathbf{f(x) = -9x^2 + 9x + 1}$

(a) The height of the diving board.

This is represented by the constant term in the function.

So, we have:

$\mathbf{Height = 1}$

Hence, the height of the diving board is 1 meter

(b) The maximum height of the diver

We have:

$\mathbf{f(x) = -9x^2 + 9x + 1}$

Differentiate

$\mathbf{f'(x) = -18x + 9}$

Set to 0

$\mathbf{-18x + 9 = 0}$

Collect like terms

$\mathbf{18x = 9}$

Divide through by 18

$\mathbf{x = 0.5}$

Calculate f(0.5)

$\mathbf{f(x) = -9x^2 + 9x + 1}$

$\mathbf{f(0.5) = -9(0.5)^2 + 9(0.5) +1}$

$\mathbf{f(0.5) = 3.25}$

Hence, the maximum height of the diver is 3.25 m

(c) The depth, the diver ascends to

In (b), we have:

$\mathbf{x = 0.5}$

This means that, the diver ascends from 0 meters to 0.5 meters

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