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A cook wants to cover the side of a cone-shaped funnel with parchment paper to avoid any leaks or spills. The funnel has a diameter of 6 inches and a slant height of 4 inches. How many square inches of parchment paper is needed to cover the side of the funnel? Use 3.14 for pi and round your answer to the nearest hundredth

11 answers

To find the area of the side of the cone-shaped funnel that needs to be covered with parchment paper, we can calculate the lateral surface area of the cone. The formula for the lateral surface area \( A \) of a cone is given by:

\[ A = \pi r l \]

where \( r \) is the radius of the base of the cone, and \( l \) is the slant height.

Step 1: Calculate the radius of the cone.

Given that the diameter of the funnel is 6 inches, we can find the radius by dividing the diameter by 2:

\[ r = \frac{6}{2} = 3 \text{ inches} \]

Step 2: Use the formula to calculate the lateral surface area.

We know the slant height \( l \) is given as 4 inches. Plugging in the values into the formula:

\[ A = \pi r l \] \[ A = 3.14 \times 3 \times 4 \]

Step 3: Calculate the area.

Now perform the multiplication:

\[ A = 3.14 \times 3 = 9.42 \] \[ A = 9.42 \times 4 = 37.68 \]

Thus, the lateral surface area, which is the amount of parchment paper needed to cover the side of the funnel, is:

\[ A = 37.68 \text{ square inches} \]

Final Result:

When rounded to the nearest hundredth, the area of parchment paper needed is

\[ \boxed{37.68} \text{ square inches} \]

A waffle cone is in the shape of a cone with a slant height of 8 inches and a radius of 3 inches. A baker wants to know the surface area of the cone in order to cover the cone with melted chocolate. What is the surface area of the side of the cone in square inches? Use 3.14 for pi and round your answer to the nearest hundredth

To calculate the lateral surface area of the cone (which is the area of the side that the baker wants to cover with melted chocolate), we can use the formula for the lateral surface area of a cone, given by:

\[ A = \pi r l \]

where:

  • \( r \) is the radius of the base of the cone,
  • \( l \) is the slant height of the cone.

Step 1: Identify the values.

  • The radius \( r \) is given as 3 inches.
  • The slant height \( l \) is given as 8 inches.

Step 2: Plug the values into the formula. Using \( \pi \approx 3.14 \):

\[ A = 3.14 \times 3 \times 8 \]

Step 3: Perform the multiplication.

  1. Calculate \( 3.14 \times 3 \):

\[ 3.14 \times 3 = 9.42 \]

  1. Now multiply this result by the slant height \( l \):

\[ A = 9.42 \times 8 \]

Calculating that:

\[ 9.42 \times 8 = 75.36 \]

Final Result:

The lateral surface area of the cone, which is the amount of melted chocolate needed to cover the cone, is:

\[ \boxed{75.36} \text{ square inches} \]

This is rounded to the nearest hundredth.

What is the surface area, in square inches, of a cone with a radius of 10 inches and a slant height of 15 inches? Use 3.14 for pi

To calculate the surface area of a cone, we need to find both the lateral surface area and the base area. The total surface area \( A \) of a cone is given by the formula:

\[ A = \text{lateral surface area} + \text{base area} \]

The lateral surface area \( A_{\text{lateral}} \) is calculated using the formula:

\[ A_{\text{lateral}} = \pi r l \]

where:

  • \( r \) is the radius,
  • \( l \) is the slant height.

The base area \( A_{\text{base}} \) of the cone is calculated using the formula:

\[ A_{\text{base}} = \pi r^2 \]

Step 1: Identify the given values.

  • Radius \( r = 10 \) inches
  • Slant height \( l = 15 \) inches

Step 2: Calculate the lateral surface area.

Using the lateral surface area formula:

\[ A_{\text{lateral}} = \pi r l = 3.14 \times 10 \times 15 \]

Calculating that:

\[ A_{\text{lateral}} = 3.14 \times 10 = 31.4 \]

Then multiply by 15:

\[ A_{\text{lateral}} = 31.4 \times 15 = 471 \]

Step 3: Calculate the base area.

Using the base area formula:

\[ A_{\text{base}} = \pi r^2 = 3.14 \times (10^2) = 3.14 \times 100 = 314 \]

Step 4: Calculate the total surface area.

Now, adding the lateral surface area and the base area together:

\[ A = A_{\text{lateral}} + A_{\text{base}} = 471 + 314 = 785 \]

Final Result:

The total surface area of the cone is:

\[ \boxed{785} \text{ square inches} \]

What is the surface area of the cone? Use 3.14 for pi.

(1 point)
Responses

2,383.26 square centimeters
2,383.26 square centimeters

3,689.5 square centimeters
3,689.5 square centimeters

1,354.125 square centimeters
1,354.125 square centimeters

863.5 square centimeters

To calculate the surface area of a cone, we will need the radius and the slant height of the cone. The formula for the surface area of a cone is:

\[ A = \pi r l + \pi r^2 \]

Where:

  • \( A \) is the total surface area,
  • \( r \) is the radius of the base,
  • \( l \) is the slant height.

To accurately determine which answer choice is correct from the provided options, we need specific values for \( r \) (radius) and \( l \) (slant height) for the cone. However, in your question, those values are not provided.

Please provide the radius and slant height of the cone so we can perform the calculation. Once we have that data, we can compute the surface area and find the matching option from the answers you've listed.

sorry diamiter is 25 side linght is 25

No problem! Let's calculate the surface area of the cone with a diameter of 25 cm and a slant height of 25 cm.

Step 1: Calculate the radius. The radius \( r \) is half of the diameter:

\[ r = \frac{25}{2} = 12.5 \text{ cm} \]

Step 2: Calculate the lateral surface area.

Using the formula for the lateral surface area:

\[ A_{\text{lateral}} = \pi r l = 3.14 \times 12.5 \times 25 \]

Calculating that:

  1. \( 3.14 \times 12.5 = 39.25 \)
  2. Then, multiply by 25:

\[ A_{\text{lateral}} = 39.25 \times 25 = 981.25 \text{ square centimeters} \]

Step 3: Calculate the base area.

Using the formula for the base area:

\[ A_{\text{base}} = \pi r^2 = 3.14 \times (12.5^2) = 3.14 \times 156.25 \]

Calculating that:

\[ A_{\text{base}} = 3.14 \times 156.25 \approx 490.625 \text{ square centimeters} \]

Step 4: Calculate the total surface area.

Now, adding the lateral surface area and the base area together:

\[ A = A_{\text{lateral}} + A_{\text{base}} = 981.25 + 490.625 = 1471.875 \text{ square centimeters} \]

Final Result:

The total surface area of the cone is approximately:

\[ \boxed{1471.88} \text{ square centimeters} \]

Given your options:

  • 2,383.26 square centimeters
  • 3,689.5 square centimeters
  • 1,354.125 square centimeters
  • 863.5 square centimeters

None of these match the calculated surface area. Please double-check the diameter and slant height values you provided. If those values are indeed correct, then there could be a possibility of a difference in the expected results or options.

no side length is 22 sorry

No problem! Let’s recalculate the surface area of the cone using the updated values.

You have:

  • Diameter = 25 cm, hence radius \( r = \frac{25}{2} = 12.5 \) cm.
  • Slant height \( l = 22 \) cm.

Step 1: Calculate the Lateral Surface Area

The lateral surface area \( A_{\text{lateral}} \) is given by:

\[ A_{\text{lateral}} = \pi r l \] Substituting the values we have:

\[ A_{\text{lateral}} = 3.14 \times 12.5 \times 22 \]

Calculating this:

  1. Calculate \( 3.14 \times 12.5 \):

\[ 3.14 \times 12.5 = 39.25 \]

  1. Now multiply this by the slant height \( l \):

\[ A_{\text{lateral}} = 39.25 \times 22 = 864.5 \text{ square centimeters} \]

Step 2: Calculate the Base Area

The base area \( A_{\text{base}} \) is given by:

\[ A_{\text{base}} = \pi r^2 \]

Substituting the radius:

\[ A_{\text{base}} = 3.14 \times (12.5^2) = 3.14 \times 156.25 \]

Calculating this:

\[ A_{\text{base}} = 3.14 \times 156.25 \approx 490.625 \text{ square centimeters} \]

Step 3: Calculate the Total Surface Area

Now, adding the lateral surface area and the base area together:

\[ A = A_{\text{lateral}} + A_{\text{base}} = 864.5 + 490.625 = 1355.125 \text{ square centimeters} \]

Final Result:

The total surface area of the cone is approximately:

\[ \boxed{1355.13} \text{ square centimeters} \]

Comparing with your options:

  • 2,383.26 square centimeters
  • 3,689.5 square centimeters
  • 1,354.125 square centimeters
  • 863.5 square centimeters

The closest option is 1,354.125 square centimeters.