The volume of a sphere of radius 4.2 cm is: \(\left(\text { Use } \pi=\frac{22}{7}\right)\)
310.464 cm3
Calculating Sphere Volume Given Radius
This problem requires us to calculate the volume of a sphere when its radius is known. We are given the radius of the sphere and the value of \(\pi\) to use for the calculation. Understanding the formula for the volume of a sphere is key to solving this problem.
Formula for Sphere Volume
The volume (\(V\)) of a sphere with radius (\(r\)) is given by the formula:
where:
- \(V\) is the volume of the sphere.
- \(r\) is the radius of the sphere.
- \(\pi\) is a mathematical constant, approximately 3.14159.
Given Values
From the question, we are given:
- Radius of the sphere, \(r = 4.2\) cm.
- Value of \(\pi\) to use, \(\pi = \frac{22}{7}\).
Step-by-Step Volume Calculation
Now, we substitute the given values into the volume formula:
First, let's calculate \( (4.2)^3 \):
Performing the multiplication:
| 17.64 | \(\times\) | 4.2 |
| 3528 | (17.64 \(\times\) 0.2) | |
| 7056 | (17.64 \(\times\) 4) | |
| \(\rule{3cm}{0.4pt}\) | ||
| 74.088 | (Summing with correct decimal placement) | |
So, \( (4.2)^3 = 74.088 \).
Now substitute this back into the volume formula:
We can divide 74.088 by 21:
| \( \frac{74.088}{21} \) | \( = 3.528 \) |
Now, multiply the result by 88:
Performing the multiplication:
| 3.528 | \(\times\) | 88 |
| 28224 | (3.528 \(\times\) 8) | |
| 28224 | (3.528 \(\times\) 80) | |
| \(\rule{3cm}{0.4pt}\) | ||
| 310.464 | (Summing with correct decimal placement) | |
So, the volume of the sphere is \( 310.464 \) cm\(^3\).
Comparing with Options
Let's compare our calculated volume with the given options:
- Option 1: 310.464 cm\(^3\)
- Option 2: 312.725 cm\(^3\)
- Option 3: 278.234 cm\(^3\)
- Option 4: 297.824 cm\(^3\)
Our calculated volume, 310.464 cm\(^3\), matches Option 1.
Final Answer
The volume of the sphere of radius 4.2 cm is 310.464 cm\(^3\).
Revision Table: Sphere Geometry
| Concept | Formula | Notes |
|---|---|---|
| Area of Sphere | \( A = 4\pi r^2 \) | \(r\) is the radius |
| Volume of Sphere | \( V = \frac{4}{3}\pi r^3 \) | \(r\) is the radius |
| Circumference of Great Circle | \( C = 2\pi r \) | A great circle is a circle on the sphere's surface with the same center as the sphere. |
Additional Information: Units and Pi
When calculating volume, the units are cubic units. Since the radius is given in centimeters (cm), the volume is in cubic centimeters (cm\(^3\)). It is important to include the correct units in the final answer.
The value of \(\pi\) can be approximated as 3.14, 3.14159, or as the fraction \(\frac{22}{7}\). The question explicitly stated to use \(\pi = \frac{22}{7}\), which is often done in problems to allow for some simplification when the radius or other dimensions are multiples of 7 or 0.7.
For example, if \(r = 7\) cm, \(V = \frac{4}{3} \times \frac{22}{7} \times 7^3 = \frac{4}{3} \times \frac{22}{7} \times 343 = \frac{4}{3} \times 22 \times 49 = \frac{4312}{3}\) cm\(^3\). If \(r = 4.2 = \frac{42}{10}\) cm, using \(\pi = \frac{22}{7}\) allowed us to potentially simplify \( \frac{42}{10} \) with the \( \frac{1}{7} \) factor, although in this specific calculation, direct multiplication worked well too.
Similar Questions
The curved surface area and the volume of a cylindrical object are 88 cm 2and 132 cm 3, respectively. The height (in cm) of the cylindrical object is:
(Take π = \(\frac{{22}}{7}\) )
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(Take π = \(\frac{22}{7}\) )
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Important Questions from Solid Figures
A cone and a hemisphere have equal bases and volumes. What is the ratio of the height of the cone to the radius of the hemisphere?
A metallic solid cuboid of dimensions 36 cm × 18 cm × 12 cm is melted and recast in the form of cubes of side 6 cm. Find the number of cubes so formed.
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If the surface area of a sphere is 64 π cm 2, then the volume of the sphere is:
Find the surface area of a sphere of diameter 21 cm. (Use π = \(\frac{{22}}{7}\) )
