Question

Algebra 2: help please

a scuba diver enters the water 1m away from her boat. She dives down and then resurfaces 12m away from her boat. Her diving path id in the shape of a quadratic function, where x is the distance away from the boat and y is the distance away from the surface of the water, and negative values of y are below the water. She reaches a maximum depth of 30.25m below the surface of the water.

Her diving partner wants to estimate the diver’s depths at different points away from the boat. What is the diver’s depth when she is 4m away from the boat?

Answer 1

24 m depth

Explanation:

The quadratic function modeling depth will have zeros at x=1 and x=12, the distances in meters from the boat where the diver is at the surface.

Quadratic function

The factored form of a quadratic function can be written as ...

y = a(x -p)(x -q)

where p and q are zeros of the function. The value 'a' is a vertical scale factor.

Here, we are given y=0 at the surface, at points where x=1 and x=12. This means the function can be written as ...

y = a(x -1)(x -12)

Scale factor

To find the value of 'a', we can use the maximum depth value. That depth will be halfway between the function zeros, at x = (1+12)/2 = 6.5

-30.25 = a(6.5 -1)(6.5 -12)

30.25 = 5.5²·a = 30.25a ⇒ a=1

Model of depth

Then the function modeling the diver's depth in meters is ...

y = (x -1)(x -12)

And the depth at x=4 will be ...

y = (4 -1)(4 -12) = 3(-8) = -24 . . . . meters

The diver's depth is 24 meters when she is 4 m away from the boat.