Problem 96 - Algebra for College Students

Question

Set up an equation and solve each of the following problems. Suppose that a radius of a sphere is equal in length to a radius of a circle. If the volume of the sphere is numerically equal to four times the area of the circle, find the length of a radius for both the sphere and the circle.

Step-by-Step Solution

Answer

The radius for both the sphere and the circle is 3 units.

Step 1: Understand the Formulas

To solve this problem, recognize the formulas needed: the volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \) and the area of a circle is given by \( A = \pi r^2 \), where \( r \) is the radius.

Step 2: Set Up the Equation

We are given that the volume of the sphere is equal to four times the area of the circle. So, the equation becomes \( \frac{4}{3} \pi r^3 = 4 \pi r^2 \).

Step 3: Simplify the Equation

Divide both sides by \( \pi \): \( \frac{4}{3} r^3 = 4 r^2 \).

Step 4: Solve for Radius

Cancel \( r^2 \) from both sides, provided \( r eq 0 \): \( \frac{4}{3} r = 4 \). Multiply both sides by 3 to clear the fraction: \( 4r = 12 \). Finally, divide by 4: \( r = 3 \).

Key Concepts

Volume of a Sphere

The volume of a sphere is the amount of space that occupies the inside of a spherical shape. Imagine it as the three-dimensional form of a solid ball. This concept is important in understanding many real-world objects like basketballs, planets, and droplets.

To calculate the volume, we use the formula:

• \( V = \frac{4}{3} \pi r^3 \)
• Where:
  • \( V \) represents the volume.
  • \( \pi \) is a constant approximately equal to 3.14159.
  • \( r \) stands for the radius of the sphere.

The radius here is crucial as it's the distance from the center of the sphere to any point on its surface. The cube of the radius indicates how the volume quickly increases as the radius gets larger. The fraction \( \frac{4}{3} \) is a scaling factor that ensures the formula's accuracy. Understanding this formula helps in grasping the magnitude of objects' insides.

Area of a Circle

The area of a circle measures the surface enclosed within its circumference, or the total space occupied by the circle in a two-dimensional plane. Circles are very common in daily life, such as wheels, clock faces, and coins, making this concept quite useful.

To find the area, the formula is given by:

• \( A = \pi r^2 \)
• Where:
  • \( A \) is the area of the circle.
  • \( \pi \) remains the constant approximately 3.14159.
  • \( r \) is the radius of the circle.

The radius influences the area directly by being squared. This signifies that even a small change in the radius greatly impacts the area. The focus on the circle's area is important for tasks involving design, architecture, and other geometrical applications.

Equation Setup

Setting up equations is a critical step in solving algebraic problems involving geometric figures like spheres and circles. It's essentially about translating a word problem into a mathematical expression.

In our specific exercise, we're connecting the volume of a sphere with the area of a circle. The task is to find when these two quantities relate as stated: the sphere's volume equals four times the circle's area.
The initial equation is established as:

• \( \frac{4}{3} \pi r^3 = 4 \pi r^2 \)

By setting this equation, we express the relationship between the sphere's volume and the circle's area. Equations like these help in identifying mutual properties or equal values between different shapes, a fundamental step before solving for any unknowns.

Radius Calculation

Finding the radius is often the ultimate goal in problems involving circles and spheres. It involves solving the given equation for \( r \). In this exercise, simplifying the equation helps to isolate \( r \).

After dividing both sides by \( \pi \), the equation becomes:

• \( \frac{4}{3} r^3 = 4 r^2 \)

Next, the aim is to further simplify by removing \( r^2 \) from both sides (assuming \( r eq 0 \)):

• \( \frac{4}{3} r = 4 \)

Multiply both sides by 3, giving:

• \( 4r = 12 \)

And finally, divide by 4 to find:

• \( r = 3 \)

This solution shows step-by-step how algebra simplifies complex arithmetic to find precise values. Understanding how to navigate to get \( r \) is vital when connecting geometric shapes through algebra.