Ask a New Question
Menu

Find the surface area of a sphere with a diameter of 40 inches.

11 answers

To find the surface area of a sphere, we use the formula:

\[ A = 4\pi r^2 \]

where \( A \) is the surface area and \( r \) is the radius of the sphere.

Given the diameter of the sphere is 40 inches, we can first find the radius:

\[ r = \frac{\text{diameter}}{2} = \frac{40 \text{ inches}}{2} = 20 \text{ inches} \]

Now, we can find the surface area using the radius:

\[ A = 4\pi (20 \text{ inches})^2 \]

Calculating \( (20 \text{ inches})^2 \):

\[ (20 \text{ inches})^2 = 400 \text{ inches}^2 \]

Now substituting this back into the surface area formula:

\[ A = 4\pi \cdot 400 \text{ inches}^2 = 1600\pi \text{ inches}^2 \]

To get a numerical approximation of the surface area, we can use \( \pi \approx 3.14 \):

\[ A \approx 1600 \cdot 3.14 \text{ inches}^2 = 5024 \text{ inches}^2 \]

Therefore, the surface area of the sphere is:

\[ \boxed{1600\pi \text{ inches}^2} \quad \text{or approximately } \boxed{5024 \text{ inches}^2} \]

sorry wrong

Let’s walk through the calculation again step-by-step.

The formula for the surface area \( A \) of a sphere is given by:

\[ A = 4\pi r^2 \]

where \( r \) is the radius.

  1. Find the radius from the diameter: The diameter of the sphere is given as 40 inches. The radius \( r \) is half of the diameter:

    \[ r = \frac{40 \text{ inches}}{2} = 20 \text{ inches} \]

  2. Plug the radius into the surface area formula: Now we can calculate the surface area:

    \[ A = 4\pi (20 \text{ inches})^2 \]

  3. Calculate \( (20 \text{ inches})^2 \): \[ (20 \text{ inches})^2 = 400 \text{ inches}^2 \]

  4. Substitute back into the surface area formula: \[ A = 4\pi \cdot 400 \text{ inches}^2 = 1600\pi \text{ inches}^2 \]

Thus, the surface area of the sphere is:

\[ A = 1600\pi \text{ square inches} \]

If you need a numerical approximation of the surface area using \( \pi \approx 3.14 \):

\[ A \approx 1600 \cdot 3.14 = 5024 \text{ square inches} \]

Therefore, the correct surface area is:

\[ \boxed{1600\pi \text{ square inches}} \quad \text{or approximately } \boxed{5024 \text{ square inches}} \]

Please let me know if there’s any specific aspect you would like me to address or if you have further questions!

Find the surface area of a sphere with a radius of 3.6 ft. Use 3.14 for pi. Round your answer to the nearest hundredth, if necessary

To find the surface area \( A \) of a sphere, we use the formula:

\[ A = 4\pi r^2 \]

where \( r \) is the radius of the sphere.

Given the radius \( r = 3.6 \) ft, we can calculate the surface area using \( \pi \approx 3.14 \):

  1. Calculate \( r^2 \): \[ r^2 = (3.6 \text{ ft})^2 = 12.96 \text{ ft}^2 \]

  2. Substitute \( r^2 \) into the surface area formula: \[ A = 4 \cdot 3.14 \cdot 12.96 \text{ ft}^2 \]

  3. Calculate \( 4 \cdot 3.14 \): \[ 4 \cdot 3.14 = 12.56 \]

  4. Multiply by \( 12.96 \): \[ A = 12.56 \cdot 12.96 \text{ ft}^2 \]

    Now calculate \( 12.56 \cdot 12.96 \): \[ A \approx 162.7336 \text{ ft}^2 \]

  5. Round to the nearest hundredth: \[ A \approx 162.73 \text{ ft}^2 \]

Thus, the surface area of the sphere is:

\[ \boxed{162.73 \text{ ft}^2} \]

A cube box is 20 cm×20 cm×20 cm
. What is the surface area of the largest size sphere that can fit in this box? Leave your answer in terms of pi.

To find the surface area of the largest sphere that can fit inside a cube, we first need to determine the radius of that sphere.

Since the sphere fits perfectly inside the cube, its diameter will be equal to the side length of the cube. Given that the cube has dimensions \(20 , \text{cm} \times 20 , \text{cm} \times 20 , \text{cm}\), the diameter of the sphere is \(20 , \text{cm}\).

  1. Calculate the radius \(r\): \[ r = \frac{\text{diameter}}{2} = \frac{20 , \text{cm}}{2} = 10 , \text{cm} \]

  2. Use the surface area formula for a sphere: The formula for the surface area \(A\) of a sphere is:

    \[ A = 4\pi r^2 \]

  3. Substituting the radius into the formula: \[ A = 4\pi (10 , \text{cm})^2 \]

  4. Calculate \( (10 , \text{cm})^2 \): \[ (10 , \text{cm})^2 = 100 , \text{cm}^2 \]

  5. Now substitute back into the surface area formula: \[ A = 4\pi \cdot 100 , \text{cm}^2 = 400\pi , \text{cm}^2 \]

Therefore, the surface area of the largest sphere that can fit in the cube is:

\[ \boxed{400\pi , \text{cm}^2} \]

wrong

Let's re-evaluate the problem step-by-step to find the surface area of the largest sphere that can fit within a cube with dimensions \(20 , \text{cm} \times 20 , \text{cm} \times 20 , \text{cm}\).

  1. Understanding the dimensions of the cube: The cube has a side length of \(20 , \text{cm}\).

  2. Finding the diameter of the sphere: The largest sphere that can fit inside the cube will have a diameter equal to the side length of the cube. \[ \text{Diameter of the sphere} = 20 , \text{cm} \]

  3. Calculating the radius \(r\) of the sphere: To find the radius: \[ r = \frac{\text{Diameter}}{2} = \frac{20 , \text{cm}}{2} = 10 , \text{cm} \]

  4. Using the surface area formula for a sphere: The formula for the surface area \(A\) of a sphere is given by: \[ A = 4\pi r^2 \]

  5. Substituting the radius into the formula: Now substitute \(r = 10 , \text{cm}\): \[ A = 4\pi (10 , \text{cm})^2 \]

  6. Calculating \((10 , \text{cm})^2\): \[ (10 , \text{cm})^2 = 100 , \text{cm}^2 \]

  7. Calculating the surface area: Now calculate \(A\): \[ A = 4\pi \cdot 100 , \text{cm}^2 = 400\pi , \text{cm}^2 \]

Thus, the surface area of the largest sphere that can fit in the cube is indeed:

\[ \boxed{400\pi , \text{cm}^2} \]

If there's any specific misunderstanding or detail you would like to clarify further, please let me know!

A rectangular box is 12 in. × 14 in. × 18 in.
What is the surface area of the largest size sphere that could fit in this box? Leave your answer in terms of pi

To find the surface area of the largest sphere that can fit inside a rectangular box, we need to determine the diameter of the sphere. The largest sphere that can fit in the box will have a diameter equal to the smallest dimension of the box.

Given the dimensions of the box are:

  • Length = 12 in.
  • Width = 14 in.
  • Height = 18 in.

  1. Determine the smallest dimension: The smallest dimension is: \[ \text{Smallest dimension} = 12 \text{ in.} \]

  2. Calculate the diameter of the sphere: The diameter of the largest sphere that can fit in the box is equal to this smallest dimension: \[ \text{Diameter} = 12 \text{ in.} \]

  3. Calculate the radius \(r\): The radius is half the diameter: \[ r = \frac{\text{Diameter}}{2} = \frac{12 \text{ in.}}{2} = 6 \text{ in.} \]

  4. Use the surface area formula for a sphere: The formula for the surface area \(A\) of a sphere is given by: \[ A = 4\pi r^2 \]

  5. Substituting the radius into the formula: \[ A = 4\pi (6 \text{ in.})^2 \]

  6. Calculate \((6 \text{ in.})^2\): \[ (6 \text{ in.})^2 = 36 \text{ in.}^2 \]

  7. Calculate the surface area: \[ A = 4\pi \cdot 36 \text{ in.}^2 = 144\pi \text{ in.}^2 \]

Therefore, the surface area of the largest sphere that can fit in the box is:

\[ \boxed{144\pi \text{ in.}^2} \]