For a parabolic section, the enclosing rectangle has a height of 24 ft and base of 18 ft. Find the perimeter of the parabola.

To find the perimeter of the parabolic section, we need to clarify that we're finding the length of the curved section of the parabola and the perimeter of the rectangle that encloses it. However, there's no direct formula for calculating the perimeter of a parabola. We can estimate it using calculus through an arc length calculation.

### Step 1: Set up the equation of the parabola
Assume the parabola opens upwards and is symmetric about the y-axis. The vertex of the parabola is at the origin \((0, 0)\), and the parabola passes through the points \((9, 24)\) and \((-9, 24)\), since the base of the rectangle is 18 ft (from \(-9\) to 9) and the height is 24 ft.

The general form of a parabola with the vertex at the origin is:
\[
y = ax^2
\]
Using the point \((9, 24)\) to find the value of \(a\):
\[
24 = a(9^2) \quad \Rightarrow \quad 24 = 81a \quad \Rightarrow \quad a = \frac{24}{81} = \frac{8}{27}
\]
Thus, the equation of the parabola is:
\[
y = \frac{8}{27}x^2
\]

### Step 2: Arc length formula
The arc length \(L\) of a curve \(y = f(x)\) between two points \(x = a\) and \(x = b\) is given by the formula:
\[
L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]
For the parabola \(y = \frac{8}{27}x^2\), we first compute the derivative:
\[
\frac{dy}{dx} = \frac{16}{27}x
\]

### Step 3: Set up the integral
Now, substitute into the arc length formula. We integrate from \(x = -9\) to \(x = 9\):
\[
L = \int_{-9}^{9} \sqrt{1 + \left( \frac{16}{27}x \right)^2} \, dx
\]
Simplifying:
\[
L = \int_{-9}^{9} \sqrt{1 + \frac{256}{729}x^2} \, dx
\]
Since the parabola is symmetric, we can calculate the arc length from 0 to 9 and then double it:
\[
L = 2 \int_0^9 \sqrt{1 + \frac{256}{729}x^2} \, dx
\]

### Step 4: Approximate the integral
This integral doesn't have a simple antiderivative, so we approximate it numerically.

Let me compute this.The approximate length of the parabolic arc is about 52.85 feet.

### Step 5: Perimeter of the enclosing rectangle
The perimeter of the enclosing rectangle is calculated by:
\[
P_{\text{rectangle}} = 2(\text{height} + \text{base}) = 2(24 + 18) = 84 \text{ feet}
\]

### Conclusion
- The perimeter of the parabolic section is the length of the parabola, which is approximately **52.85 feet**.
- If you need the perimeter of the whole shape (including the rectangle and the curved arc), you would need to combine the arc length and any relevant linear sections (if applicable).

Would you like further details or have any questions?

Here are 5 questions to further explore:
1. How does the arc length formula apply to other curves like circles or ellipses?
2. What are the properties of a parabola in terms of focal points and directrices?
3. How do numerical methods approximate integrals like this?
4. What other shapes have an arc length formula that can be solved exactly?
5. How would the calculation change if the parabola were not symmetric?

**Tip:** For complex integrals without simple antiderivatives, numerical methods like Simpson's rule or trapezoidal rule are commonly used for approximation.

Learn how to calculate the perimeter of a parabolic section enclosed by a rectangle with a height of 24 ft and base of 18 ft. This problem involves using calculus and the arc length formula to estimate the length of the parabola. Suitable for students studying calculus or advanced geometry.

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