Sphere And Frustum Volume Calculation A Mathematical Exploration

Understanding Sphere Volume is crucial in various fields, from geometry to physics. The volume of a sphere is the amount of space it occupies, and it's calculated using a specific formula that involves the radius. Let's dive into the specifics and solve the given problem.

The formula for the volume (V) of a sphere is:

V = (4/3)πr³

Where:

• V represents the volume.
• π (pi) is a mathematical constant approximately equal to 3.14159.
• r is the radius of the sphere.

In this particular question, we are given that the radius (r) of the sphere is 3 units. Our task is to calculate the volume using the formula mentioned above.

Step-by-step Calculation

1. Substitute the value of the radius (r = 3) into the formula:

   V = (4/3)π(3)³

2. Calculate 3 cubed (3³):

   3³ = 3 * 3 * 3 = 27

3. Substitute 27 back into the formula:

   V = (4/3)π(27)

4. Multiply (4/3) by 27:

   (4/3) * 27 = 36

5. So, the volume becomes:

   V = 36π

Analyzing the Options

Now, let's compare our calculated volume (36π) with the options provided:

• A) 36 cm³
• B) 12 π cm³
• C) 12 π cm²
• D) 20 π cm²

Our calculated volume, 36π, does not directly match any of the options in the exact form. However, we need to check if any simplification or unit conversion is required. Let's re-evaluate our calculation and the options.

We calculated the volume as 36π cubic units. Since the radius is given without specific units, we'll assume the units are in centimeters (cm) for consistency with the answer choices. Thus, the volume is 36π cm³.

Revisiting the options:

• A) 36 cm³ - This option is close but missing the π factor.
• B) 12 π cm³ - This is one-third of our calculated volume and doesn't match.
• C) 12 π cm² - This has the wrong units (cm² instead of cm³) and a different numerical value.
• D) 20 π cm² - This also has the wrong units (cm² instead of cm³) and a different numerical value.

It appears there might be an error in the provided options, as none of them exactly match our calculated volume of 36π cm³. However, the closest option is A) 36 cm³, which is the numerical part of our answer. To get the correct option, we need to multiply 36 by π.

Conclusion and Corrected Options

Based on our calculation, the volume of the sphere with a radius of 3 cm is 36π cm³. If we were to provide a corrected set of options, it would look like this:

• A) 36π cm³
• B) 12 π cm³
• C) 12 π cm²
• D) 20 π cm²

With the corrected options, A) 36π cm³ would be the correct answer. In this context, it is important to recognize the correct methodology and calculation, even if the provided options contain discrepancies. Understanding the formula and applying it correctly is the key takeaway.

Understanding the Frustum of a Cone

The frustum of a cone is the portion of a cone that remains after its top has been cut off by a plane parallel to its base. Calculating the volume of a frustum is a common problem in geometry, with applications in various fields such as engineering and architecture. This problem involves finding the volume of a frustum given the radii of its top and bottom bases and its height. Let's break down the process and arrive at the solution.

The Formula for the Volume of a Frustum of a Cone

The volume (V) of a frustum of a cone is calculated using the following formula:

V = (1/3)πh(R² + Rr + r²)

Where:

• V is the volume of the frustum.
• π (pi) is the mathematical constant approximately equal to 3.14159.
• h is the height of the frustum.
• R is the radius of the bottom base.
• r is the radius of the top base.

In this problem, we are given:

• Radius of the top base (r) = 3 m
• Radius of the bottom base (R) = 5 m
• Height of the frustum (h) = 9 m

Now, we will substitute these values into the formula and calculate the volume.

Step-by-step Calculation

1. Substitute the given values into the formula:

   V = (1/3)π(9)(5² + 5*3 + 3²)

2. Calculate the squares and the product:

   5² = 25 5 * 3 = 15 3² = 9

3. Substitute these values back into the equation:

   V = (1/3)π(9)(25 + 15 + 9)

4. Add the numbers inside the parentheses:

   25 + 15 + 9 = 49

5. The equation now becomes:

   V = (1/3)π(9)(49)

6. Multiply (1/3) by 9:

   (1/3) * 9 = 3

7. The equation simplifies to:

   V = 3π(49)

8. Multiply 3 by 49:

   3 * 49 = 147

9. The volume is therefore:

   V = 147π m³

Analyzing the Options

Now, let's compare our calculated volume with the provided options:

• A) 196 π cm³
• B) 147 π cm³
• C) 441 π cm³
• D) 96 π cm³

Our calculated volume is 147π m³. Notice that the options are given in cm³, while our calculation is in m³. We need to account for this unit difference. However, let's first check if the numerical value matches any option.

We find that B) 147 π cm³ has the same numerical value (147π) as our calculated volume. However, there is a unit discrepancy (m³ vs. cm³). To correctly compare, we need to convert our result to cm³ or the options to m³.

Since 1 m = 100 cm, 1 m³ = (100 cm)³ = 1,000,000 cm³. Therefore, our calculated volume in cm³ would be:

147π m³ = 147π * 1,000,000 cm³ = 147,000,000π cm³

This conversion highlights that our initial comparison was incorrect because of the units. The value 147π corresponds to cubic meters, not cubic centimeters. Thus, option B) is incorrect due to the units provided.

Re-evaluating the Options

Given the significant difference in magnitude due to the unit conversion, it's crucial to revisit the original calculation and the options. Our calculation V = 147π m³ is correct. The discrepancy lies in the provided options, which are all in cm³.

It's highly probable that there is an error in the units provided in the options. The correct numerical value is 147π, but the units should be m³.

Conclusion and Corrected Units

The volume of the frustum of the cone is 147π m³. The correct answer, taking into account the proper units, is not explicitly listed among the options provided in cm³. The closest option in numerical value is B) 147 π, but it incorrectly states the units as cm³ instead of m³.

Therefore, the accurate representation of the volume, with the correct units, is:

V = 147π m³

In a corrected set of options, the answer B) should be:

B) 147 π m³

It’s essential to pay close attention to units in mathematical problems, as they can significantly impact the final answer and its interpretation.

Discussion Category: Mathematics

These problems fall under the category of mathematics, specifically in the subfields of geometry and mensuration. They involve the application of formulas to calculate volumes of three-dimensional shapes, namely spheres and frustums of cones. Understanding these concepts is fundamental in various areas of mathematics and physics.