Adrian is painting the outside of a cylinder that he plans to use as a coffee table. The cylinder has a radius of 4 feet and a height of 3 feet. Adrian wants to paint all around the outside of the cylinder, including the top and bottom faces. In order to understand how much paint is needed, he wants to know the surface area of the outside of the cylinder. What is the surface area of the outside of the cylinder, measured in square feet? Use 3.14 for pi and round your answer to the nearest tenth.

3 answers

To find the surface area of the outside of a cylinder, we need to calculate the lateral surface area (the area around the sides) and the area of the two circular bases (top and bottom).

  1. Calculate the lateral surface area (A_l): The formula for the lateral surface area of a cylinder is:

    \[ A_l = 2 \pi r h \]

    where:

    • \( r \) is the radius
    • \( h \) is the height

    Given:

    • \( r = 4 \) feet
    • \( h = 3 \) feet

    Plugging in the values:

    \[ A_l = 2 \times 3.14 \times 4 \times 3 \]

    First, calculate \( 2 \times 3.14 \):

    \[ 2 \times 3.14 = 6.28 \]

    Now, compute:

    \[ 6.28 \times 4 = 25.12 \]

    Finally:

    \[ 25.12 \times 3 = 75.36 \text{ square feet} \]

  2. Calculate the area of the top and bottom faces (A_b): The area of one circular base is given by:

    \[ A_b = \pi r^2 \]

    Since there are two bases, we multiply this area by 2:

    \[ A_b = 2 \times \pi r^2 = 2 \times 3.14 \times (4)^2 \]

    First, calculate \( (4)^2 \):

    \[ (4)^2 = 16 \]

    Now calculate:

    \[ A_b = 2 \times 3.14 \times 16 \]

    Compute \( 2 \times 3.14 \):

    \[ 2 \times 3.14 = 6.28 \]

    Now, find:

    \[ 6.28 \times 16 = 100.48 \text{ square feet} \]

  3. Add the lateral surface area and the area of the bases to find the total surface area (A_total):

    \[ A_{total} = A_l + A_b \]

    Now, add:

    \[ A_{total} = 75.36 + 100.48 = 175.84 \text{ square feet} \]

  4. Round to the nearest tenth:

    The total surface area rounded to the nearest tenth is:

    \[ \boxed{175.8} \text{ square feet} \]

are you 100% right?

Let's double-check the calculations to ensure everything is correct.

Step 1: Lateral Surface Area (A_l)

The formula for the lateral surface area of a cylinder is:

\[ A_l = 2 \pi r h \]

Given:

  • Radius \( r = 4 \) feet
  • Height \( h = 3 \) feet
  • Using \( \pi \approx 3.14 \)

Calculate:

\[ A_l = 2 \times 3.14 \times 4 \times 3 \]

Calculating \( 2 \times 3.14 \):

\[ 2 \times 3.14 = 6.28 \]

Now calculate \( 6.28 \times 4 \):

\[ 6.28 \times 4 = 25.12 \]

Finally evaluate:

\[ 25.12 \times 3 = 75.36 \text{ square feet} \]

Step 2: Area of the Top and Bottom Faces (A_b)

The area of one circular base is given by:

\[ A_b = \pi r^2 \]

For two bases:

\[ A_b = 2 \times \pi r^2 = 2 \times 3.14 \times (4)^2 \]

Calculating \( (4)^2 \):

\[ (4)^2 = 16 \]

Thus,

\[ A_b = 2 \times 3.14 \times 16 \]

Calculating \( 2 \times 3.14 \):

\[ 2 \times 3.14 = 6.28 \]

Now calculate:

\[ 6.28 \times 16 = 100.48 \text{ square feet} \]

Step 3: Total Surface Area (A_total)

Now we add the lateral area and the area of the bases:

\[ A_{total} = A_l + A_b \]

Combining these results:

\[ A_{total} = 75.36 + 100.48 \]

Calculating:

\[ A_{total} = 175.84 \text{ square feet} \]

Step 4: Rounding

Rounding \( 175.84 \) to the nearest tenth:

\[ A_{total} \approx 175.8 \text{ square feet} \]

Conclusion

After checking all steps carefully, the calculations are correct. Thus, the total surface area of the outside of the cylinder, rounded to the nearest tenth, is indeed:

\[ \boxed{175.8} \text{ square feet} \]

If you have any specific concerns or details you'd like to verify further, please let me know!