Suppose a tunnel is 44 feet wide and 22 feet high is in the shape of a parabola. A) Determinate an equation, f(x), of a parabola to represent this situation and B) provide a graph to support your work. Suppose lanes pass trought the tunnel that are each 12 feet wide. C) Determinate if it is safe for a truck that is 14 feet to pass trought. D) Make a table of values to support your conclusion.
To determine the equation of the parabola representing the tunnel, we can assume that the vertex of the parabola is at the origin (0,0). Since the tunnel is 44 feet wide and 22 feet high, we can use these points to construct the equation.
A) Equation of the Parabola:
Let's consider the point (22, -22) on the parabola because the height is negative. This point lies on the right half of the parabola since we have assumed the vertex at (0, 0).
Using the general equation for a parabola, y = a(x-h)^2 + k, where (h,k) is the vertex, substituting x = 22 and y = -22, we can solve for the value of 'a'.
-22 = a(22-0)^2 + 0
-22 = 484a
a = -22/484
a = -1/22
Therefore, the equation of the parabola representing the tunnel is f(x) = -1/22 * x^2.
B) Graph:
Here is a visual representation of the parabolic tunnel with the equation f(x) = -1/22 * x^2:
^ (22,-22)
|
-40 ___|_______ 40
|
V
C) Safe Passage for a 14-foot Truck:
To determine if it is safe for a truck that is 14 feet tall to pass through the tunnel, we need to check whether the height of the tunnel at x = 0, where the truck would pass, is greater than or equal to 14 feet.
Substituting x = 0 in the equation f(x) = -1/22 * x^2, we get:
f(0) = -1/22 * 0^2
f(0) = 0
Since the height is 0 at the center of the tunnel, it is safe for a 14-foot truck to pass through.
D) Table of Values:
Here is a table showing a few x and f(x) values to support the conclusion:
| x | f(x) |
|-------|---------|
| -20 | -181.82 |
| -10 | -45.45 |
| 0 | 0 |
| 10 | -45.45 |
| 20 | -181.82 |
These values demonstrate that the height is sufficient for a 14-foot truck to pass through as the corresponding f(x) values are above 14.