Question

A fountain on a lake sprays water in a parabolic arch modeled by the equation ( y = -0.4x² + 3x ), where y is the height in feet and x is the horizontal distance. A beam of light is passed through the fountain to create a rainbow effect. If the beam is directed at an angle modeled by the equation ( -2.2x + 4.9y = 8.18 ), at what distance from the ground will the beam first touch the water spray? What is the solution set of ( y = x² + 2x + 7 ) and ( y = x + 7 )?

a. 2 units
b. 4 units
c. 6 units
d. 8 units

Answer 1

The beam will first touch the water spray at a distance of approximately 5.7 units from the ground. The solution set of the given equations is {0}.

Step-by-step explanation:

To find the distance from the ground where the beam first touches the water spray, we need to solve the system of equations:

y = -0.4x² + 3x

-2.2x + 4.9y = 8.18

Substituting the value of y from the first equation into the second equation, we get:

-2.2x + 4.9(-0.4x² + 3x) = 8.18

Simplifying the equation and solving for x, we find that x ≈ 5.7 units.

Therefore, the beam will first touch the water spray at a distance of approximately 5.7 units from the ground.

Now let's solve the system of equations:

y = x² + 2x + 7

y = x + 7

Substituting the value of y from the second equation into the first equation, we get:

x + 7 = x² + 2x + 7

Simplifying the equation and solving for x, we find that x = 0.

Therefore, the solution set of the given equations is {0}.