jane buys a cylinder plant pot with radius 15cm and height 20 cm ,she wants to paint its outside surface ,calculate the area she wants to paint?

To calculate the area that Jane wants to paint on the outside surface of a cylindrical plant pot, we need to consider both the lateral surface area (the side of the cylinder) and the area of the base (top) if it is exposed. Since the bottom is usually not painted as it sits on the ground, we will only consider the lateral surface area and the top surface area.

Here are the steps to calculate the total area Jane wants to paint:

1. **Lateral Surface Area of the Cylinder**:
\[ A_{\text{lateral}} = 2 \pi r h \]
where \( r \) is the radius and \( h \) is the height.

2. **Top Surface Area of the Cylinder**:
\[ A_{\text{top}} = \pi r^2 \]

Given:
\[ r = 15 \text{ cm} \]
\[ h = 20 \text{ cm} \]

### Calculation:

1. **Lateral Surface Area**:
\[ A_{\text{lateral}} = 2 \pi r h \]
\[ A_{\text{lateral}} = 2 \pi (15 \text{ cm}) (20 \text{ cm}) \]
\[ A_{\text{lateral}} = 2 \pi (300 \text{ cm}^2) \]
\[ A_{\text{lateral}} = 600 \pi \text{ cm}^2 \]

2. **Top Surface Area**:
\[ A_{\text{top}} = \pi r^2 \]
\[ A_{\text{top}} = \pi (15 \text{ cm})^2 \]
\[ A_{\text{top}} = \pi (225 \text{ cm}^2) \]
\[ A_{\text{top}} = 225 \pi \text{ cm}^2 \]

### Total Surface Area to be Painted:
\[ A_{\text{total}} = A_{\text{lateral}} + A_{\text{top}} \]
\[ A_{\text{total}} = 600 \pi \text{ cm}^2 + 225 \pi \text{ cm}^2 \]
\[ A_{\text{total}} = 825 \pi \text{ cm}^2 \]

To find the numerical value, we use the approximation \(\pi \approx 3.14159\):
\[ A_{\text{total}} \approx 825 \times 3.14159 \text{ cm}^2 \]
\[ A_{\text{total}} \approx 2590.80 \text{ cm}^2 \]

Therefore, the area Jane wants to paint is approximately \( 2590.80 \text{ cm}^2 \).

Would you like more details or have any questions?

### Related Questions:

1. What would be the surface area if Jane decided to paint both the top and bottom of the pot?
2. How much paint would Jane need if one liter covers 10,000 square centimeters?
3. How does the surface area change if the height of the pot is doubled but the radius remains the same?
4. What is the volume of the cylindrical plant pot?
5. How does the paint area change if the radius of the pot is increased by 5 cm?
6. How would you calculate the surface area of a similar pot with a conical top instead of a flat one?
7. What is the lateral surface area if the height is reduced by half?
8. How would the surface area change if the pot is in the shape of a truncated cone instead of a cylinder?

### Tip:

When dealing with problems involving surface area and volume of three-dimensional objects, always sketch the object and label all given dimensions to help visualize the problem clearly.

Learn how to calculate the area Jane wants to paint on the outside surface of a cylindrical plant pot with radius 15 cm and height 20 cm. This problem covers the surface area of a cylinder, including lateral surface area and top surface area calculations. Suitable for students in Grades 7-9.

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