The Shape Described Is One Quater Off A Solid Sphere. The Centre Is O. The Volume Of The Shape Is 576pie Cm Cubed. Find The Surface Area Give Your Answer To 3dp
Introduction
In this article, we will explore the problem of finding the surface area of a quarter of a solid sphere, given its volume. The volume of the shape is 576π cm³, and we need to find the surface area of this quarter sphere. We will use mathematical concepts and formulas to solve this problem.
Understanding the Shape
The shape described is a quarter of a solid sphere, with its center at point O. A sphere is a three-dimensional shape that is perfectly round and has no edges or corners. A quarter sphere is a portion of a sphere that is cut off by a plane passing through the center of the sphere.
Volume of a Quarter Sphere
The volume of a quarter sphere can be calculated using the formula:
V = (1/4) * (4/3) * π * r³
where V is the volume, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.
Given that the volume of the quarter sphere is 576π cm³, we can set up the equation:
576π = (1/4) * (4/3) * π * r³
Solving for the Radius
To solve for the radius, we can start by simplifying the equation:
576π = (1/4) * (4/3) * π * r³
Multiply both sides by 4 to get rid of the fraction:
2304π = (4/3) * π * r³
Now, divide both sides by π:
2304 = (4/3) * r³
Multiply both sides by 3/4 to get rid of the fraction:
1728 = r³
Now, take the cube root of both sides to solve for r:
r = ∛1728
r ≈ 12.0 cm
Surface Area of a Quarter Sphere
The surface area of a quarter sphere can be calculated using the formula:
A = (1/2) * 4 * π * r²
where A is the surface area, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.
Substitute the value of r we found earlier:
A = (1/2) * 4 * π * (12.0)²
A = (1/2) * 4 * π * 144
A = 2 * π * 144
A ≈ 904.8 cm²
Conclusion
In this article, we found the surface area of a quarter of a solid sphere, given its volume. We used mathematical concepts and formulas to solve this problem. The surface area of the quarter sphere is approximately 904.8 cm².
References
- Weisstein, E. W. (n.d.). Sphere. MathWorld--A Wolfram Web Resource.
- Weisstein, E. W. (n.d.). Volume of a Sphere. MathWorld--A Wolfram Web Resource.
- Weisstein, E. W. (n.d.). Surface Area of a Sphere. MathWorld--A Wolfram Web Resource.
The Shape Described: A Quarter of a Solid Sphere - Q&A =====================================================
Introduction
In our previous article, we explored the problem of finding the surface area of a quarter of a solid sphere, given its volume. We used mathematical concepts and formulas to solve this problem. In this article, we will answer some frequently asked questions related to the shape described.
Q: What is the shape described?
A: The shape described is a quarter of a solid sphere, with its center at point O.
Q: What is the volume of the shape?
A: The volume of the shape is 576π cm³.
Q: How do I calculate the volume of a quarter sphere?
A: The volume of a quarter sphere can be calculated using the formula:
V = (1/4) * (4/3) * π * r³
where V is the volume, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.
Q: How do I find the radius of the sphere?
A: To find the radius of the sphere, you can use the formula:
r = ∛(V * (3/4) / π)
where V is the volume of the quarter sphere.
Q: What is the surface area of a quarter sphere?
A: The surface area of a quarter sphere can be calculated using the formula:
A = (1/2) * 4 * π * r²
where A is the surface area, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.
Q: How do I calculate the surface area of a quarter sphere?
A: To calculate the surface area of a quarter sphere, you can substitute the value of r into the formula:
A = (1/2) * 4 * π * r²
Q: What is the relationship between the volume and surface area of a quarter sphere?
A: The volume and surface area of a quarter sphere are related by the formula:
V = (1/4) * (4/3) * π * r³
A = (1/2) * 4 * π * r²
These formulas show that the volume and surface area of a quarter sphere are both functions of the radius of the sphere.
Q: Can I use these formulas to find the surface area of a full sphere?
A: Yes, you can use these formulas to find the surface area of a full sphere. Simply multiply the surface area of a quarter sphere by 4:
A_full = 4 * A_quarter
where A_full is the surface area of the full sphere and A_quarter is the surface area of the quarter sphere.
Conclusion
In this article, we answered some frequently asked questions related to the shape described. We hope this Q&A article has been helpful in clarifying any doubts you may have had.
References
- Weisstein, E. W. (n.d.). Sphere. MathWorld--A Wolfram Web Resource.
- Weisstein, E. W. (n.d.). Volume of a Sphere. MathWorld--A Wolfram Web Resource.
- Weisstein, E. W. (n.d.). Surface Area of a Sphere. MathWorld--A Wolfram Web Resource.