Problem 63 - Algebra A Combined Function

Question

The radius of the moon is 1080 miles. Use the formula for the radius \(r\) of a sphere given its surface area \(A\). $$ r=\sqrt{\frac{A}{4 \pi}} $$ to find the surface area of the moon. Round to the nearest square mile. (Source: National Space Science Data Center)

Step-by-Step Solution

Answer

The surface area of the moon is approximately 14,696,621 square miles.

Step 1: Identify the Known Variable

Given the radius of the moon, \( r = 1080 \) miles. We need to find the surface area \( A \) using the formula \( r=\sqrt{\frac{A}{4 \pi}} \).

Step 2: Rearrange the Formula for Surface Area

To find the surface area \( A \), rearrange the formula: \( r=\sqrt{\frac{A}{4 \pi}} \). Square both sides and then solve for \( A \): \[ r^2 = \frac{A}{4 \pi} \]\[ A = 4\pi r^2 \]

Step 3: Plug in Given Radius and Calculate

Substitute the known radius into the formula to find the surface area:\[ A = 4 \pi (1080)^2 \]Now calculate:\[ A = 4 \pi \times 1166400 \]

Step 4: Calculate the Numerical Value of the Surface Area

Use the value of \( \pi \approx 3.14159 \) to calculate:\[ A = 4 \times 3.14159 \times 1166400 \]This results in:\[ A = 14696620.8 \]

Step 5: Round the Surface Area to the Nearest Square Mile

The calculated surface area is \( 14696620.8 \) square miles. Rounded to the nearest square mile, this is \( 14696621 \) square miles.

Key Concepts

Radius Calculation

Calculating the radius of a sphere involves understanding the relationship between the sphere's radius and its surface area. Imagine the radius as the distance from the center of the sphere to any point on its surface. To find the surface area when you know the radius, you can use specific mathematical formulas.

When you are given the radius and want to know how to find the surface area of a sphere, the calculation becomes straightforward through rearranging formulas. The key formula here is derived from:

• Surface area formula: \( A = 4\pi r^2 \)
• Radius formula: \( r = \sqrt{\frac{A}{4 \pi}} \)

If you were to go in the reverse direction, given the surface area to find the radius, you acknowledge that the radius is simply a function of the surface area, involving a square root and division by \( 4 \pi \).

This understanding helps when solving for either variable based on given data, like in our moon example. The key takeaway is that knowing one measure helps you calculate the other using these formulas.

Spherical Geometry

In spherical geometry, we're pretty much dealing with the metrics of spheres. Unlike regular planar geometry, spherical geometry involves the shapes and properties on the surface of a sphere.

The surface area of a sphere depends heavily on its radius. Since the sphere is a three-dimensional object, you can visualize it as being covered with infinitely many small triangles or segments. As each side of this imaginary shape bulges or curves outward, it makes direct geometric calculations more complex than in flat geometry.

Important aspects include:

• Curved surfaces with uniform properties.
• Application in earth measurements, like mapping, where a spherical model of Earth is useful.
• Use of specialized mathematical formulas to calculate properties like area or volume.

Using the formula \( A = 4\pi r^2 \), you see that spherical geometry requires understanding how surface area scales with the square of the radius. As seen in our exercise, even small changes in radius result in large changes in surface area.

Mathematical Formulas

Mathematical formulas serve as the blueprint to decode geometric properties, like calculating the surface area of a sphere. Formulas give structure and accuracy to computations, whether they seem complex at first or not.

For spheres, important formulas include:

• Surface area formula: \( A = 4\pi r^2 \)
• Volume formula: \( V = \frac{4}{3}\pi r^3 \)
• These are derivatives of the basic circle formulas where \( 2\pi r \) calculates the circumference.

The specific formula used in the exercise is a rearranged version from the general surface area formula aimed at finding the surface area given the radius. These formulas ensure calculations are consistent and transferable across different contexts involving spheres.

Knowing how to rearrange and apply formulas simplifies many mathematical problems, providing a step-by-step path to finding unknown variables. Understanding the logic behind them is crucial for mastering spherical calculations, as seen in the example of determining the moon's surface area.