Question

The path traveled by a bottlenose dolphin as it jumps out of water is modeled by the equation y = −0.4x2 + 3x, where y is the height above water and x is the horizontal distance in feet. If a beam of light is shone upward at an angle modeled by the equation x + y = 10, the beam of light will hit the dolphin at what height from the surface of the water?

A.) 2.7 feet
B.) 3 feet
C.) 5 feet
D.) 5.6 feet

Answer 1

C.5 feet

Explanation:

We are given that path traveled by a bottlenose dolphin it jumps out of water is modeled by the equation

`y=-0.4 x^2+3x` (Equation I)

Where y is the height above water and x is the horizontal distance in feet

If a beam of light is shown upward at angle modeled by the equation

`x+y=10` (Equation II)

We have to find the height of dolphin from the surface of the water when the beam of light will hit the dolphin

Substitute the value of y in equation II

Then we get

`x-0.4x^2+3x=10`

`0.4x^2-4x+10=0`

`4x^2-40x+100=0`

`x^2-10 x+25=0`

`x^2-5x-5x+25=0`

`x(x-5)-5(x-5)=0`

`(x-5)(x-5)=0`

x-5=0 and x-5=0

x=5 and x=5

Then , substitute x=5 in equation y

Then we get

`y=-0.4(5)^2+3(5)`

`y=-10+15`

`y=5`

Hence, the height of dolphin from the surface of water when the beam of light hit the dolphin=5 feet

Option C: 5 feet

Answer 2

We can find the answer by solving the two equations simultaneously.

equation 1: y = -0.4x^2 +3xequation 2: x + y = 10
We need to write equation to with y as the subject:equation 2: y = 10 - x

equation 1 = equation 2:
10 - x = -0.4x^2 + 3x0.4x^2 - 4x + 10 = 02x^2 - 20 + 50 = 0(2x - 10)(x - 5) = 0x = -10/2 or x = 5

But x >= 0 (the question says it's a height above the water), so x = 5

Substituting x = 5 into one of the equations:
y = 10 - x
y = 10 - 5
y = 5

So the answer is C