Question

The suspension cable that supports a footbridge hangs in the shape of a parabola. The height h, in feet, of the cable above the bridge is given by the function

h(x) = 0.25x ^ 2 - 0.8x + 30, 0 < x < 3.2

where x is the distance in feet measured from the left tower toward the right tower. What is the minimum height of the cable above the bridge? (Round your answer to two decimal places.)

Answer 1

Explanation:

The height of the cable above the bridge is given by the function:

h(x) = 0.25x^2 - 0.8x + 30

To find the minimum height of the cable above the bridge, we need to find the vertex of the parabola, which is the lowest point.

The x-coordinate of the vertex is given by:

x = -b / 2a

where a = 0.25 and b = -0.8. Substituting these values, we get:

x = -(-0.8) / 2(0.25) = 1.6

So the vertex is at x = 1.6.

To find the height of the cable at the vertex, we substitute x = 1.6 into the equation for h(x):

h(1.6) = 0.25(1.6)^2 - 0.8(1.6) + 30 = 29.48

Therefore, the minimum height of the cable above the bridge is approximately 29.48 feet (rounded to two decimal places).