Question

The archway of the main entrance of a university is modeled by the quadratic equation y = -x2 + 6x. The university is hanging a banner at the main entrance at an angle defined by the equation 4y = 21 − x. At what points should the banner be attached to the archway?

A. (1, 5.5) and (5.25, 6.56)
B. (1, 5) and (5.25, 3.94)
C. (1.5, 4.87) and (3.5, 4.37)
D. (1.5, 5.62) and (3.5, 6.12) E. There is no real solution.

Answer 1

The points where the banner should be attached to the archway are found by solving the system of equations represented by the quadratic and the linear equation. The correct attachment points are (1, 5) and (5.25, 3.94), matching option B.

Step-by-step explanation:

To find the points where the banner should be attached to the archway, we must solve a system of equations, which includes the quadratic equation of the archway, y = -x2 + 6x, and the linear equation for the angle of the banner, 4y = 21 - x. First, we solve the linear equation for y by dividing by 4, which gives us y = 5.25 - 0.25x. Next, we can substitute this expression for y into the quadratic equation to find the x coordinates where the banner crosses the archway:

• -x2 + 6x = 5.25 - 0.25x
• -x2 + 6.25x - 5.25 = 0

Using the quadratic formula, we can find the x values (rounded to two decimal places), and substitute them back into the equation y = 5.25 - 0.25x to find the corresponding y values. By doing so, we find two points of intersection, which indicate where the banner should be attached to the archway. After solving, the correct points are found to be (1, 5) and (5.25, 3.94), which corresponds to option B.