Muriel and Connie are playing a game of catch. They decided it would be fun to stand on different sides of the building and throw the ball back and forth. Muriel can throw the ball with a velocity of 14 m/s, and Connie can throw with a velocity of 16 m/s. Each stands 10 m from the building. The paths of the ball are:

Muriel: y= -9.8/14(squared)x(squared) +x + 2

Connie: y= -9.8/16(squared)x(squared) + x + 2

Might they have broken any windows? Explain.

for Muriel height at x = 10:

y = - (9.8/14^2) (100) + 10 + 2
y = -5+12 = 7 meters high
for Connie height at x = 10
y = -(9.8/16^2)100 + 10 + 2
= -3.8 +10 + 2
= 8.2 m high

7 meters is about 21 feet. That is not very high but will do 1 story with a slightly pitched roof.

To determine if Muriel or Connie broke any windows, we need to compare the heights of their ball paths with the height of the windows.

The equations given provide the paths of the balls thrown by Muriel and Connie. Both equations are in the form of a quadratic equation, with the height (y) as a function of the horizontal distance (x).

Muriel's path equation: y = -9.8/14^2 * x^2 + x + 2
Connie's path equation: y = -9.8/16^2 * x^2 + x + 2

To find the maximum height reached by each ball, we need to determine the vertex of each quadratic equation. The vertex of a quadratic equation in the form of y = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)), where f(x) represents the equation.

For Muriel's ball:
a = -9.8/14^2
b = 1
c = 2

Using the formula, the x-coordinate of the vertex is -b/2a:
x = -1 / (2 * -9.8/14^2) = -49/14

To find the y-coordinate, we substitute the x-coordinate into the equation:
y = -9.8/14^2 * (-49/14)^2 + (-49/14) + 2
y ≈ 8.86

Therefore, Muriel's ball reaches a maximum height of approximately 8.86 meters.

For Connie's ball:
a = -9.8/16^2
b = 1
c = 2

Using the formula, the x-coordinate of the vertex is -b/2a:
x = -1 / (2 * -9.8/16^2) = -64/16

To find the y-coordinate, we substitute the x-coordinate into the equation:
y = -9.8/16^2 * (-64/16)^2 + (-64/16) + 2
y ≈ 8.91

Therefore, Connie's ball reaches a maximum height of approximately 8.91 meters.

To determine if any windows were broken, we need to compare the maximum heights reached by Muriel and Connie to the height of the windows. If the maximum height is greater than the height of the windows, then a window may have been broken.

If the height of the windows is given, we can compare it to the maximum heights we calculated. If the maximum height is higher than the height of the windows, then there is a possibility that a window may have been broken. However, if the maximum height is lower than the height of the windows, then no windows would have been broken.

Without knowing the height of the windows, it is not possible to determine definitively if any windows were broken.