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To solve the problem, we need to estimate the radius of a concrete sphere given its mass and density.

The key formula to use here is the relationship between mass, density, and volume:

\[
\text{Mass} = \text{Density} \times \text{Volume}
\]

For a sphere, the volume is given by:

\[
V = \frac{4}{3} \pi r^3
\]

Where:
- \(V\) is the volume
- \(r\) is the radius

We are given:
- Mass = 1400 kg
- Density = 2400 kg/m³

First, rearrange the formula for mass to solve for the volume:

\[
V = \frac{\text{Mass}}{\text{Density}} = \frac{1400}{2400} = 0.5833 \, \text{m}^3
\]

Now use the volume of the sphere formula to find the radius:

\[
0.5833 = \frac{4}{3} \pi r^3
\]

Solve for \(r^3\):

\[
r^3 = \frac{0.5833 \times 3}{4\pi} = \frac{1.7499}{12.566} \approx 0.1392
\]

Now solve for \(r\):

\[
r = \sqrt[3]{0.1392} \approx 0.52 \, \text{m}
\]

### Answers:
- Vertical stretch by a factor: The vertical stretch corresponds to the relationship between the volume of the original function (parent cubic function) and the given mass. The stretch factor is about **0.58** (the ratio of the given mass and the density-derived value).
  
- Radius: The estimated radius is **0.52 meters**.

Would you like further details on the steps? Here are 5 questions to extend the concept:

1. How does the formula for volume change if the shape is not a sphere?
2. What would happen to the radius if the mass of the sphere doubled but the density remained the same?
3. How does the concept of vertical stretch apply in transformations of functions?
4. Why is the cubic root involved when solving for the radius of a sphere?
5. How would the process differ if you were estimating the radius of a hollow sphere?

**Tip:** Always ensure that the units are consistent when solving problems involving physical quantities like mass, density, and volume!

What transformation of the parent cubic function can you use to estimate the radius of a concrete sphere with a mass of 1400 kilograms and a density of 2400 kg/m³? What is the estimate of the radius? Round to the nearest whole number for vertical stretch and to the nearest hundredth for radius.

Learn how to estimate the radius of a concrete sphere using its mass (1400 kg) and density (2400 kg/m³). This solution involves applying the volume of a sphere formula and transforming cubic functions, ideal for high school students.

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