Question

If the volume of a sphere is doubled, what happens to the radius? A. It increases by a factor of 2 . B. It increases by a factor of 1.414 . C. It increases by a factor of 1.26 . D. It increases by a factor of 2.24 .

Short answer

It increases by a factor of 1.26.

Step-by-step solution

Understand the Problem

We need to determine the factor by which the radius of a sphere increases if its volume is doubled.

Write the Volume Formula of a Sphere

The volume (\text{V}) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \]where \( r \) is the radius.

Express the New Volume

If the volume is doubled, then the new volume \( V' \) is:\[ V' = 2V \]

Set Up the Equation

We can write the new volume in terms of the new radius \( r' \): \[ V' = \frac{4}{3} \pi (r')^3 \]

Equate the Volumes

Since \( V' = 2V \), we have:\[ \frac{4}{3} \pi (r')^3 = 2 \left( \frac{4}{3} \pi r^3 \right) \]

Simplify the Equation

Cancel out the common terms: \[ (r')^3 = 2r^3 \]

Solve for the New Radius

Take the cube root of both sides:\[ r' = \sqrt[3]{2} r \]Using approximation:\[ \sqrt[3]{2} \approx 1.26 \]

Conclude

The radius increases by a factor of approximately 1.26, which matches option \( C \).

Key concepts

sphere volume

To start with, let's understand the volume of a sphere. The volume of a sphere is determined by a specific formula:
\[ V = \frac{4}{3} \ \text{pi} r^3 \]where \( V \) represents the volume and \( r \) represents the radius of the sphere. This formula is essential as it tells us the amount of space inside the sphere.
Knowing this formula makes it easier to solve various problems related to spheres. Remember that any time you see a problem about the volume of a sphere, try to think of this formula.

radius calculation

Now that we understand the volume, let's talk about calculating the radius when the volume of a sphere changes.
If the volume of a sphere is doubled, we need to figure out how the radius changes. We'll work through this step-by-step, using the relationship between volume and radius.
When the original volume \( V \) is doubled, the new volume \( V' \) is given by: \[ V' = 2V \]Next, we express this new volume in terms of a new radius \( r' \), leading to:\[ V' = \frac{4}{3} \ \text{pi} (r')^3 \]
By setting these equal and solving for \( r' \), we can understand how the radius scales with the volume.

mathematical reasoning

Good mathematical reasoning is essential for solving complex problems. In this problem, it means understanding how doubling the volume affects the radius.
Let's use the formula for volume to equate the doubled volume with new radius \( r' \). We have:\[ 2V = \frac{4}{3} \ \text{pi} (r')^3 \]After canceling common terms, we get: \[ (r')^3 = 2r^3 \]Taking the cube root of both sides, we simplify to find the new radius: \[ r' = \root{3} \ {2} \ r \]Using an approximation, we get: \[ \root{3} \ {2} \ \approx 1.26 \]Hence, the radius increases by approximately a factor of 1.26.

step-by-step solution

The key to mastering any mathematical concept is to break it down into simple, clear steps. Let's recap the steps of our solution:
1. **Understand the Problem:** Determine the factor by which the radius increases when the volume is doubled.
2. **Recall the Volume Formula:** Use the volume formula \( V = \frac{4}{3} \ \text{pi} r^3 \).
3. **Double the Volume:** Set the new volume as \( V' = 2V \).
4. **Setup the Equation:** Express the new volume as \( V' = \frac{4}{3} \ \text{pi} (r')^3 \).
5. **Equate and Simplify:** Link the original volume to the new one and cancel out common terms: \( (r')^3 = 2r^3 \).
6. **Solve for New Radius:** Take the cube root of both sides to find: \( r' = \root{3} \ {2} \ r \).7. **Approximate the Increase:** Simplify the cube root to find the radius factor is approximately 1.26.
By following these steps, you can solve similar problems and understand how different factors influence one another.