Show that the ratio of the volume of a sphere with radius r to its surface area can be simplified to c times r, where c is a constant. State clearly the value of c.
Understand the Problem
The question is asking to show the relationship between the volume and surface area of a sphere with radius r, specifically to simplify the ratio of the volume to the surface area in the form of c times r, and to determine the value of c.
Answer
$\frac{1}{3}$
Answer for screen readers
The value of $c$ is $\frac{1}{3}$.
Steps to Solve
- Formula for Volume of a Sphere
The volume $V$ of a sphere can be calculated using the formula: $$ V = \frac{4}{3} \pi r^3 $$
- Formula for Surface Area of a Sphere
The surface area $A$ of a sphere can be calculated using the formula: $$ A = 4 \pi r^2 $$
- Setting Up the Ratio
To find the ratio of the volume to the surface area, we divide the volume formula by the surface area formula: $$ \text{Ratio} = \frac{V}{A} = \frac{\frac{4}{3} \pi r^3}{4 \pi r^2} $$
- Simplifying the Ratio
Next, simplify the expression: $$ \text{Ratio} = \frac{\frac{4}{3} \pi r^3}{4 \pi r^2} = \frac{4}{3} \cdot \frac{r^3}{4 r^2} = \frac{r}{3} $$
This shows that the ratio of the volume to the surface area is $\frac{r}{3}$.
- Identifying the Constant c
From the simplified ratio, we can express it in the form of $c \cdot r$: $$ c = \frac{1}{3} $$
The value of $c$ is $\frac{1}{3}$.
More Information
This answer demonstrates the relationship between the volume and surface area of a sphere. The derived ratio highlights how the volume grows with the cube of the radius while the surface area grows with the square, leading to their relationship of $\frac{r}{3}$.
Tips
- Forgetting to cancel terms when simplifying the ratio can lead to incorrect results. Always double-check the simplification step.
- Confusing the formulas for volume and surface area of a sphere.
