To find the surface area of a cylindrical trash can (including its lid), we need to calculate the total surface area, which consists of:

1. The lateral surface area of the cylinder.
2. The area of the base (which is a circle).
3. The area of the lid (which is identical to the base).

The formula for the total surface area $A$ of a closed cylinder is:

$A = 2\pi r h + 2\pi r^2$

Where:

• $r$ is the radius of the base (7 inches),
• $h$ is the height of the cylinder (30 inches),
• $\pi \approx 3.1416$.

Step 1: Calculate the lateral surface area

The lateral surface area of the cylinder is given by:

$\text{Lateral Area} = 2\pi r h$

Substituting the values:

$\text{Lateral Area} = 2\pi (7)(30) = 2 \times 3.1416 \times 7 \times 30 = 1,319.47 \text{ square inches}.$

Step 2: Calculate the area of the base and the lid

The area of the base (which is also the lid since it's identical) is given by:

$\text{Area of base or lid} = \pi r^2$

Substituting the values:

$\text{Area of base or lid} = \pi (7)^2 = 3.1416 \times 49 = 153.94 \text{ square inches}.$

Since the cylinder has both a base and a lid, we multiply this by 2:

$\text{Area of base and lid} = 2 \times 153.94 = 307.88 \text{ square inches}.$

Step 3: Calculate the total surface area

Now, sum the lateral area and the area of the base and lid:

$A = 1,319.47 + 307.88 = 1,627.35 \text{ square inches}.$

Final answer:

Rounding to the nearest square inch:

$A \approx 1,627 \text{ square inches}.$

Would you like further details or explanations? Here are 5 related questions to explore further:

1. How does changing the radius affect the total surface area of the cylinder?
2. What would be the surface area if the height were doubled but the radius stayed the same?
3. How would you calculate the volume of this trash can?
4. If the trash can didn't have a lid, how would the surface area change?
5. How does the surface area of a cylinder compare to that of a cone with the same base and height?

Tip: For any solid shape, the surface area formulas typically account for all exposed surfaces, so always check whether the object has a lid or a base.