Comparing Value of Cylindrical Popcorn Containers
To determine which container is the better value, we need to calculate the volume of popcorn that each container can hold and then compare it to their costs.
The containers are both cylindrical, so we can use the formula for the volume of a cylinder \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height of the cylinder.
Let's start with the first container:
- The height (h) is 19 cm.
- The diameter is 12 cm, so the radius (r) is half that, which is 6 cm.
Volume for the first container \( V_1 \) is:
\[ V_1 = \pi (6\,cm)^2 (19\,cm) \]
\[ V_1 = 3.14 \times 36\,cm^2 \times 19\,cm \]
\[ V_1 = 3.14 \times 684\,cm^3 \]
\[ V_1 = 2148.56\,cm^3 \]
Now for the second container:
- The height (h) is 15 cm.
- The diameter is 8 cm, so the radius (r) is half that, which is 4 cm.
Volume for the second container \( V_2 \) is:
\[ V_2 = \pi (4\,cm)^2 (15\,cm) \]
\[ V_2 = 3.14 \times 16\,cm^2 \times 15\,cm \]
\[ V_2 = 3.14 \times 240\,cm^3 \]
\[ V_2 = 753.6\,cm^3 \]
Now, let's calculate the cost per cubic centimeter for each container:
- First container cost per cubic centimeter is \(\frac{$6.75}{2148.56\,cm^3}\).
- Second container cost per cubic centimeter is \(\frac{$6.25}{753.6\,cm^3}\).
Let's do the math:
- First container cost per cubic centimeter: \(\frac{6.75}{2148.56} \approx 0.00314\,\text{$/cm}^3\)
- Second container cost per cubic centimeter: \(\frac{6.25}{753.6} \approx 0.00829\,\text{$/cm}^3\)
Since the first container has a lower cost per cubic centimeter, it is the better value.
