Question

the path traveled by a bottlenose dolphin as it jumps out of water is modeled by the equation ( y = - 0.4 x ^ { 2 } + 3 x ), 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 ), at what height from the waters surface will the beam of light hit the dolphin?
a. 2.7 feet
b. 3 feet
c. 5 feet
d. 5.6 feet

Explanation:

Step1: Solve the system of equations

We have two equations:

1. \(y = - 0.4x^{2}+3x\)
2. \(y=10 - x\)

Set them equal to each other: \(-0.4x^{2}+3x=10 - x\)

Step2: Rearrange the equation

Rearrange to get a quadratic equation: \(-0.4x^{2}+3x+x - 10=0\)

\(-0.4x^{2}+4x - 10 = 0\)

Multiply through by \(- 10\) to clear the decimal: \(4x^{2}-40x + 100=0\)

Divide by \(4\): \(x^{2}-10x + 25=0\)

Step3: Factor the quadratic equation

Factor: \((x - 5)^{2}=0\)

So \(x = 5\)

Step4: Find the value of \(y\)

Substitute \(x = 5\) into \(y=10 - x\)

\(y=10-5\)

Answer:

B. 3 feet