III. DIRECTION: Read, analyze, and solve the following problems:

2. A cable hangs in a parabolic arch between two poles that are 800 feet apart. The poles are 40 feet high and the lowest point on the suspended cable is 15 feet above the ground. Find the height of the cable at a point 5 feet from one of the poles. Express your final answer correct to four decimal places.

To solve the problem, we need to find the height of a cable that forms a parabolic arch between two poles. Here's the step-by-step breakdown:

### Step 1: Define the coordinate system and the parabolic equation.

Let’s position the coordinate system such that:
- The lowest point of the cable is at the origin \((0, 15)\) (15 feet above the ground).
- The poles are 800 feet apart, so their horizontal distance is from \(x = -400\) feet to \(x = 400\) feet.

Since the cable forms a parabola, we assume its equation takes the form:

\[
y = ax^2 + c
\]

Where:
- \(a\) is the coefficient we need to determine.
- \(c = 15\) since the lowest point on the cable is 15 feet above the ground.

### Step 2: Use the conditions to determine the value of \(a\).

The height of the poles is 40 feet, and they are 800 feet apart. Therefore, at \(x = \pm 400\), the height of the cable \(y = 40\) feet.

Using the equation \(y = ax^2 + 15\), substitute \(x = 400\) and \(y = 40\):

\[
40 = a(400)^2 + 15
\]
\[
40 = 160000a + 15
\]
\[
160000a = 40 - 15
\]
\[
160000a = 25
\]
\[
a = \frac{25}{160000} = 0.00015625
\]

### Step 3: Find the height of the cable at a point 5 feet from one of the poles.

If the point is 5 feet from one of the poles, the horizontal distance from the center (lowest point of the cable) is \(x = 400 - 5 = 395\) feet.

Substitute \(x = 395\) into the parabolic equation \(y = 0.00015625x^2 + 15\):

\[
y = 0.00015625(395)^2 + 15
\]
\[
y = 0.00015625(156025) + 15
\]
\[
y = 24.3789 + 15
\]
\[
y = 39.3789
\]

### Final Answer:
The height of the cable at a point 5 feet from one of the poles is approximately **39.3789 feet**.

---

Do you want more details or have any questions? Here are five related questions to consider:

1. What is the height of the cable at the midpoint between the poles?
2. How does the distance between the poles affect the shape of the parabolic cable?
3. Can you express the equation of the parabola in vertex form?
4. What is the slope of the cable at a point 100 feet from the lowest point?
5. If the cable had a different lowest point, how would that change the equation?

**Tip:** When dealing with parabolic structures, the vertex form of a parabola \((y = a(x-h)^2 + k)\) is helpful to visualize its geometric properties.

A cable hangs in a parabolic arch between two poles that are 800 feet apart. The poles are 40 feet high and the lowest point on the suspended cable is 15 feet above the ground. Find the height of the cable at a point 5 feet from one of the poles. Express your final answer correct to four decimal places.

Find the height of a cable forming a parabolic arch between two poles 800 feet apart, with a step-by-step solution. The poles are 40 feet high, and the lowest point of the cable is 15 feet above the ground. This problem covers algebra, parabolic functions, and quadratic equations. Suitable for Grades 10-12.

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