Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

A belt fits tightly around the two circles, with equations \((x-1)^{2}+(y+2)^{2}=16\) and \((x+9)^{2}+(y-10)^{2}=16\) How long is this belt?

Step-by-Step Solution

Answer

The belt is approximately 7.62 units long.

Step 1: Understand Circle Equations

The given equations are of the form \((x-h)^{2}+(y-k)^{2}=r^{2}\), which represents a circle with center \((h, k)\) and radius \(r\). For the first circle, \((x-1)^{2}+(y+2)^{2}=16\), the center is \((1, -2)\) and the radius is \(4\). For the second circle, \((x+9)^{2}+(y-10)^{2}=16\), the center is \((-9, 10)\) and the radius is also \(4\).

Step 2: Calculate Distance Between Centers

To find the distance \(d\) between the centers of the two circles, we use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substituting the coordinates of the centers \((1, -2)\) and \((-9, 10)\), we get:\[d = \sqrt{((-9 - 1)^2 + (10 + 2)^2)}\]\[d = \sqrt{(-10)^2 + (12)^2} = \sqrt{100 + 144} = \sqrt{244}\]

Step 3: Determine Length of Belt

The length of the belt is equal to the distance between the two centers minus two times the radius of one of the circles, since the belt fits tightly around both circles only touching at one point of each circle. Therefore, the length of the belt is:\[ \text{Length} = \sqrt{244} - 2(4) = \sqrt{244} - 8 \]

Step 4: Simplify Belt Length

We simplify \(\sqrt{244}\) to a numerical approximation if needed. The calculated value of \(\sqrt{244}\) is approximately \(15.62\). Thus, the length of the belt is:\[ \text{Length} \approx 15.62 - 8 = 7.62 \]

Key Concepts

Circle Equations

Understanding circle equations is pivotal in solving many geometry problems, especially those involving circles and tangential lines or belts. A circle's equation in standard form is

• \((x-h)^2 + (y-k)^2 = r^2\)

Here, - \((h, k)\) represents the center of the circle, and- \(r\) is the radius.For instance, the equation \((x-1)^2 + (y+2)^2 = 16\) describes a circle with a center at \((1, -2)\) and a radius of \(4\), as \(16\) is the square of the radius \(r^2\). Similarly, the second circle \((x+9)^2 + (y-10)^2 = 16\) has the center at \((-9, 10)\) and an identical radius of \(4\). Being able to identify these parameters quickly is key to tackling related problems in geometry.

Distance Formula

The distance formula is like a map for points in a coordinate plane. It helps to find the straight line or 'as-the-crow-flies' distance between two points. The formula is derived from the Pythagorean theorem and is given as:

• \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

In our exercise, identifying centers of the circles as \((1, -2)\) and \((-9, 10)\), the distance \(d\) can be computed with ease. First, substitute the x-values of the points: * \((-9 - 1)^2 = (-10)^2 = 100\)Next, substitute the y-values:* \((10 - (-2))^2 = (10 + 2)^2 = 12^2 = 144\)Add the squares of these differences:* \(100 + 144 = 244\)Finally, take the square root:* \(d = \sqrt{244}\).This yields the direct distance between the two circle centers, a critical step for determining the belt's length.

Simplifying Radicals

Simplifying radicals is like cleaning up our math expressions. It involves breaking down square roots or other radicals into simpler or exact forms. In the context of our problem, we found the distance between the circle centers as \(\sqrt{244}\).Firstly, let's spot any perfect square factors:

• 244 can be broken down as \(4 \times 61\)

Since \(4\) is a perfect square, simplify:

• \(\sqrt{4} \times \sqrt{61} = 2\sqrt{61}\)

This simplification helps in cleaner calculations, reducing expression clutter. It provides exact results, rather than relying on a numerical approximation.

Geometric Problem Solving

Geometric problem solving often involves visualizing components of a problem and understanding how they fit together. In this belt and circle problem: * **Find crucial distances.** Use the distance formula to determine the space between circles. * **Identify key dimensions.** Determine how the geometric elements (circles and belt) interact. Remember they only touch at a single tangent point per circle. * **Apply logic to geometry.** Recognize how the belt wraps tightly around the circles with its length realized through strategic subtraction of the combined radii from center distance. The length of the belt is calculated by subtracting the diameters that do not contribute to the outer part of the belt. Knowing these problem components translates into solving more abstract problems, such as designing structures or mechanical arrangements.