A Sphere Has A Diameter Of 10 In. What Is The Volume Of The Sphere?A. $V=\frac{125}{3}$ In.$^3$ B. $V=\frac{500}{3}$ In.$^3$ C. $V=\frac{500}{3} \pi$ In.$^3$ D. $V=\frac{4000}{3} \pi$
Understanding the Problem
To find the volume of a sphere, we need to use the formula for the volume of a sphere, which is given by V = (4/3)Ï€r^3, where r is the radius of the sphere. However, we are given the diameter of the sphere, which is 10 in. We need to find the radius first before we can calculate the volume.
Finding the Radius
The radius of a sphere is half of its diameter. Therefore, if the diameter of the sphere is 10 in, the radius is r = 10/2 = 5 in.
Calculating the Volume
Now that we have the radius, we can use the formula for the volume of a sphere to calculate the volume. Plugging in the value of r = 5 in into the formula, we get:
V = (4/3)Ï€(5)^3
Simplifying the Expression
To simplify the expression, we need to calculate the value of (5)^3, which is equal to 125. Therefore, the expression becomes:
V = (4/3)Ï€(125)
Evaluating the Expression
To evaluate the expression, we need to multiply (4/3) by π and then multiply the result by 125. This gives us:
V = (4/3) × π × 125
V = (4 × 125) / 3 × π
V = 500 / 3 × π
V = (500 / 3) × π
Conclusion
Therefore, the volume of the sphere is V = (500 / 3) × π in^3.
Comparison with Options
Comparing our result with the options given, we can see that the correct answer is:
C. V = (500 / 3) × π in^3
This is the only option that matches our result.
Final Answer
The final answer is C. V = (500 / 3) × π in^3.
Discussion
The problem of finding the volume of a sphere is a classic problem in mathematics. It requires the use of the formula for the volume of a sphere, which is V = (4/3)Ï€r^3. The problem also requires the use of algebraic manipulation to simplify the expression and evaluate the result.
Importance of the Problem
The problem of finding the volume of a sphere is important in many real-world applications, such as engineering, physics, and architecture. For example, in engineering, the volume of a sphere is used to calculate the volume of a tank or a container. In physics, the volume of a sphere is used to calculate the volume of a planet or a star. In architecture, the volume of a sphere is used to calculate the volume of a building or a structure.
Conclusion
In conclusion, the problem of finding the volume of a sphere is a classic problem in mathematics that requires the use of the formula for the volume of a sphere and algebraic manipulation to simplify the expression and evaluate the result. The problem is important in many real-world applications and requires a good understanding of mathematical concepts and techniques.
Related Problems
- Finding the surface area of a sphere
- Finding the volume of a cylinder
- Finding the volume of a cone
- Finding the volume of a rectangular prism
References
- "Mathematics for Engineers" by Michael Corral
- "Calculus" by Michael Spivak
- "Geometry" by Michael Artin
Tags
- mathematics
- geometry
- calculus
- algebra
- sphere
- volume
- diameter
- radius
- formula
- algebraic manipulation
- real-world applications
- engineering
- physics
- architecture
Understanding the Problem
To find the volume of a sphere, we need to use the formula for the volume of a sphere, which is given by V = (4/3)Ï€r^3, where r is the radius of the sphere. However, we are given the diameter of the sphere, which is 10 in. We need to find the radius first before we can calculate the volume.
Finding the Radius
The radius of a sphere is half of its diameter. Therefore, if the diameter of the sphere is 10 in, the radius is r = 10/2 = 5 in.
Calculating the Volume
Now that we have the radius, we can use the formula for the volume of a sphere to calculate the volume. Plugging in the value of r = 5 in into the formula, we get:
V = (4/3)Ï€(5)^3
Simplifying the Expression
To simplify the expression, we need to calculate the value of (5)^3, which is equal to 125. Therefore, the expression becomes:
V = (4/3)Ï€(125)
Evaluating the Expression
To evaluate the expression, we need to multiply (4/3) by π and then multiply the result by 125. This gives us:
V = (4/3) × π × 125
V = (4 × 125) / 3 × π
V = 500 / 3 × π
V = (500 / 3) × π
Conclusion
Therefore, the volume of the sphere is V = (500 / 3) × π in^3.
Comparison with Options
Comparing our result with the options given, we can see that the correct answer is:
C. V = (500 / 3) × π in^3
This is the only option that matches our result.
Final Answer
The final answer is C. V = (500 / 3) × π in^3.
Discussion
The problem of finding the volume of a sphere is a classic problem in mathematics. It requires the use of the formula for the volume of a sphere, which is V = (4/3)Ï€r^3. The problem also requires the use of algebraic manipulation to simplify the expression and evaluate the result.
Importance of the Problem
The problem of finding the volume of a sphere is important in many real-world applications, such as engineering, physics, and architecture. For example, in engineering, the volume of a sphere is used to calculate the volume of a tank or a container. In physics, the volume of a sphere is used to calculate the volume of a planet or a star. In architecture, the volume of a sphere is used to calculate the volume of a building or a structure.
Conclusion
In conclusion, the problem of finding the volume of a sphere is a classic problem in mathematics that requires the use of the formula for the volume of a sphere and algebraic manipulation to simplify the expression and evaluate the result. The problem is important in many real-world applications and requires a good understanding of mathematical concepts and techniques.
Related Problems
- Finding the surface area of a sphere
- Finding the volume of a cylinder
- Finding the volume of a cone
- Finding the volume of a rectangular prism
References
- "Mathematics for Engineers" by Michael Corral
- "Calculus" by Michael Spivak
- "Geometry" by Michael Artin
Tags
- mathematics
- geometry
- calculus
- algebra
- sphere
- volume
- diameter
- radius
- formula
- algebraic manipulation
- real-world applications
- engineering
- physics
- architecture
Q: What is the formula for the volume of a sphere?
A: The formula for the volume of a sphere is V = (4/3)Ï€r^3, where r is the radius of the sphere.
Q: How do I find the radius of a sphere if I know its diameter?
A: The radius of a sphere is half of its diameter. Therefore, if the diameter of the sphere is 10 in, the radius is r = 10/2 = 5 in.
Q: How do I calculate the volume of a sphere using the formula?
A: To calculate the volume of a sphere using the formula, you need to plug in the value of the radius into the formula. For example, if the radius of the sphere is 5 in, the volume is V = (4/3)Ï€(5)^3.
Q: What is the volume of a sphere with a diameter of 10 in?
A: The volume of a sphere with a diameter of 10 in is V = (500 / 3) × π in^3.
Q: What is the surface area of a sphere with a diameter of 10 in?
A: The surface area of a sphere with a diameter of 10 in is A = 4Ï€r^2, where r is the radius of the sphere. Plugging in the value of r = 5 in, we get A = 4Ï€(5)^2 = 100Ï€ in^2.
Q: What is the volume of a cylinder with a radius of 5 in and a height of 10 in?
A: The volume of a cylinder is V = πr^2h, where r is the radius and h is the height. Plugging in the values of r = 5 in and h = 10 in, we get V = π(5)^2(10) = 250π in^3.
Q: What is the volume of a cone with a radius of 5 in and a height of 10 in?
A: The volume of a cone is V = (1/3)Ï€r^2h, where r is the radius and h is the height. Plugging in the values of r = 5 in and h = 10 in, we get V = (1/3)Ï€(5)^2(10) = 83.33Ï€ in^3.
Q: What is the volume of a rectangular prism with a length of 10 in, a width of 5 in, and a height of 2 in?
A: The volume of a rectangular prism is V = lwh, where l is the length, w is the width, and h is the height. Plugging in the values of l = 10 in, w = 5 in, and h = 2 in, we get V = 10(5)(2) = 100 in^3.
Q: What is the formula for the surface area of a sphere?
A: The formula for the surface area of a sphere is A = 4Ï€r^2, where r is the radius of the sphere.
Q: What is the formula for the volume of a cylinder?
A: The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.
Q: What is the formula for the volume of a cone?
A: The formula for the volume of a cone is V = (1/3)Ï€r^2h, where r is the radius and h is the height.
Q: What is the formula for the volume of a rectangular prism?
A: The formula for the volume of a rectangular prism is V = lwh, where l is the length, w is the width, and h is the height.
Q: What is the formula for the surface area of a rectangular prism?
A: The formula for the surface area of a rectangular prism is A = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.
Q: What is the formula for the surface area of a cylinder?
A: The formula for the surface area of a cylinder is A = 2Ï€rh + 2Ï€r^2, where r is the radius and h is the height.
Q: What is the formula for the surface area of a cone?
A: The formula for the surface area of a cone is A = πr^2 + πrl, where r is the radius and l is the slant height.
Q: What is the formula for the surface area of a sphere?
A: The formula for the surface area of a sphere is A = 4Ï€r^2, where r is the radius of the sphere.
Q: What is the formula for the volume of a sphere?
A: The formula for the volume of a sphere is V = (4/3)Ï€r^3, where r is the radius of the sphere.
Q: What is the formula for the volume of a cylinder?
A: The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.
Q: What is the formula for the volume of a cone?
A: The formula for the volume of a cone is **