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Let's solve the problem step by step.

### Given:
- The arch is shaped like a parabola.
- The height at the center is 14 feet.
- The base width of the arch is 20 feet.

We will use a parabolic equation in the form \( h(x) = ax^2 + bx + c \), where \( h(x) \) is the height at position \( x \).

### (a) Finding \( h(x) \):
1. **Set up a coordinate system:** Let the center of the parabola be at the origin, \( (0, 14) \), so that \( h(0) = 14 \).
2. **The parabola goes through the points \( (-10, 0) \) and \( (10, 0) \) since the base is 20 feet wide (from -10 to 10 on the x-axis).**

The equation simplifies to the form:
\[
h(x) = a(x - 10)(x + 10)
\]
\[
= a(x^2 - 100)
\]

Since \( h(0) = 14 \), we have:
\[
14 = a(0^2 - 100)
\]
\[
14 = -100a
\]
\[
a = -\frac{14}{100} = -\frac{7}{50}
\]

Thus, the equation for \( h(x) \) becomes:
\[
h(x) = -\frac{7}{50}x^2 + 14
\]

**Domain of \( h(x) \)**: \( -10 \leq x \leq 10 \).

### (b) Checking if a truck 8 feet wide and 12 feet high can pass:
The truck spans from \( -4 \leq x \leq 4 \) since it's 8 feet wide. We need \( h(x) \geq 12 \) for \( |x| \leq 4 \).

Calculate \( h(4) \):
\[
h(4) = -\frac{7}{50}(4^2) + 14 = -\frac{7}{50}(16) + 14 = -\frac{112}{50} + 14 = 14 - 2.24 = 11.76
\]

Since \( 11.76 < 12 \), the truck cannot pass.

### (c) Tallest truck (8 feet wide) that can pass:
To find the maximum height \( h(4) \), we already calculated \( h(4) = 11.76 \). So, the tallest truck that can pass is 11.76 feet high.

### (d) Widest 12-foot-high truck that can pass:
Set \( h(x) = 12 \):
\[
12 = -\frac{7}{50}x^2 + 14
\]
\[
-2 = -\frac{7}{50}x^2
\]
\[
x^2 = \frac{100}{7} \approx 14.29
\]
\[
x \approx \pm \sqrt{14.29} \approx \pm 3.78
\]

So, the truck can be about 7.56 feet wide.

Let me know if you need further details or explanations.

### Related Questions:
1. How would the equation change if the parabola's height was increased to 20 feet?
2. What would be the maximum height of a 6-foot-wide truck that can pass through?
3. How does changing the base width of the arch affect the equation of \( h(x) \)?
4. Can a 10-foot-high truck that is 15 feet wide pass through the arch?
5. How would you determine the vertex form of the parabola from the given information?

### Tip:
Remember that a parabola's equation can always be adjusted based on its vertex and width. Adjusting these parameters changes its curvature and maximum height.

The arch of a bridge is in the shape of a parabola 14 feet high at the center and 20 feet wide at the base. (a) Express the height of the arch h(x) in terms of x and state its domain. (b) Can a truck that is 8 feet wide and 12 feet high pass through the arch? (c) What is the tallest 8-ft.-wide truck that can pass through the arch? (d) What (to two decimal places) is the widest 12-ft.-high truck that can pass through the arch?

Solve a parabolic arch problem involving truck clearance. Learn how to express the height of a bridge arch in terms of x, determine whether a truck can pass, and calculate the widest and tallest truck dimensions for clearance. This detailed solution is suitable for Grades 9-11 and covers parabolas and quadratic equations.

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