Type The Correct Answer In The Box. Use Numerals Instead Of Words.The Surface Area Of A Sphere Is 320 Square Centimeters. What Is The Radius Of The Sphere? Round Your Answer To 2 Decimal Places.The Radius Is _____ Centimeters.

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Understanding the Surface Area of a Sphere

The surface area of a sphere is a critical concept in mathematics, particularly in geometry and trigonometry. It is defined as the total area of the surface of the sphere. The formula for the surface area of a sphere is given by:

A = 4πr^2

where A is the surface area and r is the radius of the sphere. In this problem, we are given that the surface area of the sphere is 320 square centimeters. We need to find the radius of the sphere.

Given Information

  • Surface area of the sphere (A) = 320 square centimeters
  • Formula for the surface area of a sphere: A = 4πr^2

Step 1: Substitute the Given Value into the Formula

We will substitute the given value of the surface area (A = 320) into the formula:

320 = 4πr^2

Step 2: Simplify the Equation

To simplify the equation, we will divide both sides by 4:

80 = πr^2

Step 3: Divide Both Sides by π

To isolate the term involving the radius (r), we will divide both sides by π (approximately 3.14159):

25.46 ≈ r^2

Step 4: Take the Square Root of Both Sides

To find the value of the radius (r), we will take the square root of both sides:

r ≈ √25.46

Step 5: Calculate the Square Root

Using a calculator, we can calculate the square root of 25.46:

r ≈ 5.04

Step 6: Round the Answer to 2 Decimal Places

Finally, we will round the answer to 2 decimal places:

r ≈ 5.04

The final answer is: 5.04

Conclusion

In this problem, we used the formula for the surface area of a sphere to find the radius of the sphere. We started with the given value of the surface area (A = 320 square centimeters) and substituted it into the formula. We then simplified the equation and isolated the term involving the radius (r). Finally, we took the square root of both sides and rounded the answer to 2 decimal places.

Real-World Applications

The concept of the surface area of a sphere has many real-world applications, such as:

  • Designing spherical structures: Architects and engineers use the surface area of a sphere to design spherical structures, such as domes and spheres.
  • Calculating the volume of a sphere: The surface area of a sphere is related to its volume. By using the formula for the surface area, we can calculate the volume of a sphere.
  • Understanding the behavior of fluids: The surface area of a sphere is important in understanding the behavior of fluids, such as the flow of water around a sphere.

Tips and Tricks

  • Use the correct units: When working with the surface area of a sphere, make sure to use the correct units, such as square centimeters or square meters.
  • Check your calculations: Double-check your calculations to ensure that you have the correct answer.
  • Use a calculator: A calculator can be a useful tool when working with the surface area of a sphere, especially when dealing with large numbers.

Common Mistakes

  • Forgetting to round the answer: Make sure to round the answer to the correct number of decimal places.
  • Using the wrong formula: Double-check that you are using the correct formula for the surface area of a sphere.
  • Not checking your calculations: Make sure to double-check your calculations to ensure that you have the correct answer.

Q: What is the surface area of a sphere?

A: The surface area of a sphere is the total area of the surface of the sphere. It is a critical concept in mathematics, particularly in geometry and trigonometry.

Q: What is the formula for the surface area of a sphere?

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

A = 4πr^2

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

Q: How do I calculate the surface area of a sphere?

A: To calculate the surface area of a sphere, you need to know the radius of the sphere. You can then use the formula:

A = 4πr^2

to find the surface area.

Q: What is the relationship between the surface area and the radius of a sphere?

A: The surface area of a sphere is directly proportional to the square of the radius. This means that if the radius of a sphere is doubled, the surface area will increase by a factor of 4.

Q: Can I use the surface area of a sphere to find the volume of the sphere?

A: Yes, the surface area of a sphere is related to its volume. By using the formula for the surface area, you can calculate the volume of the sphere.

Q: What are some real-world applications of the surface area of a sphere?

A: The surface area of a sphere has many real-world applications, such as:

  • Designing spherical structures: Architects and engineers use the surface area of a sphere to design spherical structures, such as domes and spheres.
  • Calculating the volume of a sphere: The surface area of a sphere is related to its volume. By using the formula for the surface area, you can calculate the volume of a sphere.
  • Understanding the behavior of fluids: The surface area of a sphere is important in understanding the behavior of fluids, such as the flow of water around a sphere.

Q: What are some common mistakes to avoid when working with the surface area of a sphere?

A: Some common mistakes to avoid when working with the surface area of a sphere include:

  • Forgetting to round the answer: Make sure to round the answer to the correct number of decimal places.
  • Using the wrong formula: Double-check that you are using the correct formula for the surface area of a sphere.
  • Not checking your calculations: Make sure to double-check your calculations to ensure that you have the correct answer.

Q: How do I use the surface area of a sphere to solve problems?

A: To use the surface area of a sphere to solve problems, you need to:

  • Understand the formula: Make sure you understand the formula for the surface area of a sphere.
  • Know the radius: You need to know the radius of the sphere to use the formula.
  • Use the formula: Use the formula to find the surface area of the sphere.
  • Check your calculations: Double-check your calculations to ensure that you have the correct answer.

Q: Can I use the surface area of a sphere to find the diameter of the sphere?

A: Yes, you can use the surface area of a sphere to find the diameter of the sphere. By using the formula for the surface area, you can calculate the radius of the sphere, and then use the radius to find the diameter.

Q: What are some tips and tricks for working with the surface area of a sphere?

A: Some tips and tricks for working with the surface area of a sphere include:

  • Use the correct units: When working with the surface area of a sphere, make sure to use the correct units, such as square centimeters or square meters.
  • Check your calculations: Double-check your calculations to ensure that you have the correct answer.
  • Use a calculator: A calculator can be a useful tool when working with the surface area of a sphere, especially when dealing with large numbers.