Question: The figure above is rolled up and folded to make a cylinder of volume 10 cm^3. Which of the following statements about the figure
Graph/shape description: In the top-left/center area there is a horizontal rectangle labeled $A_R$, with a circle labeled $A_1$ above it and a circle labeled $A_2$ below it; a double-headed horizontal arrow above the rectangle is labeled $L_R$, and a vertical dimension arrow to the right is labeled $2$ cm. This cylinder has a volume of $10$ cm$^3$. What statement about the figure must be true? The answer is: The area of each circular section, $A_1$ and $A_2$, is $5$ cm$^2$. Please explain how the area of the circles was found assuming we didn't know that
1. **State the problem:** We have a cylinder formed by rolling and folding a figure. The volume of the cylinder is given as $10$ cm$^3$. The height of the cylinder is $2$ cm (given by the vertical dimension). We want to find the area of each circular base, $A_1$ and $A_2$.
2. **Recall the formula for the volume of a cylinder:**
$$V = A_{base} \times h$$
where $V$ is the volume, $A_{base}$ is the area of the circular base, and $h$ is the height of the cylinder.
3. **Identify known values:**
- Volume $V = 10$ cm$^3$
- Height $h = 2$ cm
4. **Use the volume formula to find the base area:**
$$A_{base} = \frac{V}{h}$$
Substitute the known values:
$$A_{base} = \frac{10}{2}$$
5. **Simplify the fraction:**
$$A_{base} = \frac{\cancel{10}}{\cancel{2}} \times \frac{5}{1} = 5$$
6. **Interpretation:** The area of each circular base, $A_1$ and $A_2$, is $5$ cm$^2$.
**Final answer:** The area of each circular section $A_1$ and $A_2$ is $5$ cm$^2$.