Problem 111 - College Algebra with Modeling and Visualization

Question

A window comprises a square with sides of length \(x\) and a semicircle with diameter \(x\) as shown in the figure. If the total area of the window is 463 square inches, estimate the value of \(x\) to the nearest hundredth of an inch. (PICTURE NOT COPY)

Step-by-Step Solution

Answer

The value of \(x\) is approximately 21.09 inches.

Step 1: Understand the Shapes

The window consists of two parts: a square and a semicircle. The square has sides of length \(x\) and the semicircle has a diameter of \(x\), which means its radius is \(\frac{x}{2}\).

Step 2: Compute Area of the Square

First, calculate the area of the square. Since the side length of the square is \(x\), the area is \(x^2\).

Step 3: Compute Area of the Semicircle

The area of a full circle is \(\pi r^2\). For a semicircle with radius \(\frac{x}{2}\), the area is \(\frac{1}{2} \pi \left(\frac{x}{2}\right)^2 = \frac{\pi x^2}{8}\).

Step 4: Set Up the Total Area Equation

The total area of the window, consisting of the square and the semicircle, is given by \(x^2 + \frac{\pi x^2}{8} = 463\).

Step 5: Solve the Equation for x

Combine terms to solve for \(x\):\[x^2 + \frac{\pi x^2}{8} = 463\]This simplifies to:\[x^2 \left(1 + \frac{\pi}{8}\right) = 463\]\[x^2 = \frac{463}{1 + \frac{\pi}{8}}\]

Step 6: Calculate x

Solve for \(x\) by first calculating \(\frac{463}{1 + \frac{\pi}{8}}\) and then taking the square root of the result. This computation gives \(x \approx \sqrt{\frac{463}{1 + \frac{\pi}{8}}} \approx \ 21.09\).

Key Concepts

Geometry

Geometry involves understanding and analyzing different shapes and their properties. In this exercise, the window is composed of two different geometric shapes, a square and a semicircle. Both of these shapes have unique characteristics that are crucial in determining their respective areas.
Understanding these shapes helps you properly visualize the problem and apply the relevant formulas. The square has four equal sides, and knowing the length of one side, noted as \(x\), means you can find the area easily by squaring this number.
The semicircle is essentially half of a circle, so its key attribute is its radius, which is half of its diameter. Here, the radius is \(\frac{x}{2}\) because the diameter of the semicircle is \(x\). This relationship between the diameter and radius is foundational when finding the area of a semicircle.

Area calculation

Calculating the area is a core geometry skill. The area of an object describes how much surface it covers in a plane. For our window problem, we need to find the areas of a square and a semicircle and then combine them.

• **Square Area:** The area of a square is found using the formula \(x^2\), where \(x\) is the length of a side. Since all sides are equal, the calculation is straightforward.
• **Semicircle Area:** Calculating the area for a semicircle requires a bit more work. First, you take the full circle area formula \(\pi r^2\) and adapt it for a semicircle by using its radius \(\frac{x}{2}\). Then, you divide by 2, resulting in \(\frac{\pi x^2}{8}\).

Ultimately, these calculated areas are added together to find the total area of the window, making area calculation a powerful tool for myriad applications.

Equation solving

Equation solving involves finding unknown values by manipulating equations in a logical way. In this exercise, once the area of both shapes is determined, these are combined into a single equation to find \(x\):
\[x^2 + \frac{\pi x^2}{8} = 463\]
This is a quadratic equation in terms of \(x^2\), which we must solve to determine \(x\). Here’s how it’s simplified:

• Combine like terms for all \(x^2\) parts.
• Isolate \(x^2\) by dividing both sides by the expression \(1 + \frac{\pi}{8}\).
• Finally, take the square root to solve for \(x\).

This process utilizes algebraic techniques that are foundational in solving real-world problems, providing us with an estimated \(x \approx 21.09\).