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.

We are given that a cable hangs in a parabolic shape between two poles that are 800 feet apart. The poles are 40 feet high, and the lowest point of the cable is 15 feet above the ground.

### Problem Breakdown:
- The two poles are 800 feet apart.
- The cable forms a parabolic shape, so we can represent the cable by a quadratic equation \( y = ax^2 + bx + c \).
- The lowest point of the cable is at \( y = 15 \) feet (vertex of the parabola).
- The poles are at \( x = -400 \) feet and \( x = 400 \) feet, and their height is 40 feet.
- We are asked to find the height of the cable 5 feet from one of the poles.

### Step 1: Define the coordinate system
We will set the origin of our coordinate system at the lowest point of the cable. Hence:
- The lowest point (vertex) is at \( (0, 15) \), since the vertex of the parabola is 15 feet above the ground.
- The two poles are at \( x = -400 \) and \( x = 400 \), with \( y = 40 \).

### Step 2: Equation of the parabola
Since the parabola is symmetric and the vertex is at \( (0, 15) \), we can use the standard form of a parabola:
\[
y = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex. In this case, \( h = 0 \) and \( k = 15 \), so the equation becomes:
\[
y = ax^2 + 15
\]

### Step 3: Use known points to solve for \( a \)
We know that at \( x = 400 \), \( y = 40 \) (the height of the poles). Substitute these values into the equation to find \( a \):
\[
40 = a(400)^2 + 15
\]
\[
40 = 160000a + 15
\]
\[
25 = 160000a
\]
\[
a = \frac{25}{160000} = \frac{1}{6400}
\]

### Step 4: Equation of the cable
Thus, the equation of the cable is:
\[
y = \frac{1}{6400}x^2 + 15
\]

### Step 5: Find the height of the cable 5 feet from one pole
We are asked to find the height at a point 5 feet from one of the poles. Since the poles are at \( x = 400 \) and \( x = -400 \), we will evaluate the height at \( x = 395 \) (which is 5 feet away from \( x = 400 \)).

Substitute \( x = 395 \) into the equation:
\[
y = \frac{1}{6400}(395)^2 + 15
\]
Now, let's calculate the height.

\[
y = \frac{1}{6400}(156025) + 15
\]
\[
y = \frac{156025}{6400} + 15
\]
\[
y = 24.3789 + 15
\]
\[
y = 39.3789
\]

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

---

Do you want any further explanation on these steps? Here are five related questions to consider:

1. How would the equation change if the poles were a different height?
2. What if the distance between the poles was increased? How would it affect the parabola?
3. Can you derive the formula for the height of the cable at any point between the poles?
4. How would the parabola look if the lowest point of the cable was at ground level (0 feet)?
5. How would this problem change if the cable didn't form a perfect parabola?

**Tip:** When solving quadratic problems, always try to find the vertex and symmetry properties to simplify calculations.

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.

Solve the problem of finding the height of a cable hanging in a parabolic arch between two poles 800 feet apart, with the lowest point 15 feet above the ground. Learn to apply quadratic functions, coordinate geometry, and the vertex form of a parabola. Suitable for high school students in Grades 9-12.

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