To find out how much paint Ellis needs to cover all the surfaces of the 11 fenceposts, we need to calculate the surface area of a single fencepost and then multiply this by the number of fenceposts. The posts are cylindrical in shape, so we will compute the lateral surface area of the cylinder and add the area of the top and bottom circles.
Step 1: Find the surface area of a single fencepost.
A cylinder's surface area consists of two parts:
- The lateral surface area (the curved surface).
- The top and bottom areas (which are circles).
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Lateral Surface Area (A_lateral): The formula for the lateral surface area of a cylinder is: \[ A_{\text{lateral}} = 2 \pi r h \] where \( r \) is the radius and \( h \) is the height.
Given:
- Height \( h = 7 \) feet
- Diameter = 1 foot, so radius \( r = \frac{1}{2} = 0.5 \) feet
Substituting these values: \[ A_{\text{lateral}} = 2 \times 3.14 \times 0.5 \times 7 = 2 \times 3.14 \times 0.5 \times 7 = 22.0 \text{ square feet} \]
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Area of the top and bottom circles (A_circles): The area of one circle is given by: \[ A_{\text{circle}} = \pi r^2 \] The total area for both the top and bottom is: \[ A_{\text{circles}} = 2 \times \pi r^2 \]
Substituting for \( r = 0.5 \): \[ A_{\text{circles}} = 2 \times 3.14 \times (0.5)^2 = 2 \times 3.14 \times 0.25 = 1.57 \text{ square feet} \]
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Total Surface Area (A_total): Now we can add both areas together to determine the total surface area of one fencepost: \[ A_{\text{total}} = A_{\text{lateral}} + A_{\text{circles}} = 22.0 + 1.57 = 23.57 \text{ square feet} \]
Step 2: Calculate the total surface area for all 11 fenceposts. \[ \text{Total Surface Area for 11 fenceposts} = 11 \times A_{\text{total}} = 11 \times 23.57 = 259.27 \text{ square feet} \]
Conclusion: Thus, the amount of paint needed to cover all surfaces of the 11 fenceposts is approximately 259.27 square feet.
To round it to the nearest hundredth as required: \[ \text{Total paint needed} \approx 259.27 \text{ square feet} \]
Final answer: 259.27 square feet.