Question

Write a quadratic inequality to represent the area of a stained-glass mosaic that sits under a parabolic arch 10 feet tall and 8 feet wide at the base, if the left side of the base is the origin.

Answer 1

To represent the area of a stained-glass mosaic that sits under a parabolic arch, we can use the equation for a parabolic arch, which is of the form

y = a(x - h)^2 + k

where (h,k) is the vertex of the parabola. The width of the parabolic arch at the base is 8 feet, so the vertex of the parabola is located at (4,0). Therefore, the equation for the parabolic arch is

y = a(x - 4)^2

The height of the parabolic arch is 10 feet, so we can represent the area of the stained-glass mosaic as the region where y is between 0 and 10. This can be expressed as the inequality

0 ≤ a(x - 4)^2 ≤ 10

Solving this inequality for x gives us the solution

-2 ≤ x - 4 ≤ 2

which simplifies to

2 ≤ x ≤ 6

Therefore, the quadratic inequality that represents the area of the stained-glass mosaic is

2 ≤ x ≤ 6

Explanation:

Answer 2

• y ≤- 5/8(x - 4)² + 10 and
• y ≥ 0

-------------------------------------------------------

According to given details we have:

• The vertex at (8/2, 10) = (4, 10),
• The parabola opens down,
• It passes through the origin.

The equation for the parabola is y = a(x - h)² + k, where (h, k) is vertex.

Substitute h = 4, k = 10:

• y = a(x - 4)² + 10

We know y = 0 at x = 0, substitute to find the value of a:

• 0 = a(0 - 4)² + 10
• 0 = a(16) + 10
• 16a = - 10
• a = - 10/16 = - 5/8

So the parabola is:

• y = - 5/8(x - 4)² + 10

We are looking for the area between the parabola and the x-axis, therefore we need two inequalities:

• y ≤- 5/8(x - 4)² + 10 and
• y ≥ 0

See attached for reference.