The volume of a sphere of diameter 1 unit is ________ than the volume of a cube of side 1 unit.
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Volume Comparison: Sphere vs. Cube
To determine how the volume of a sphere of diameter 1 unit compares to the volume of a cube of side 1 unit, we need to calculate the volume of each shape separately and then compare the results.
Sphere Volume Calculation
The formula for the volume of a sphere is given by:
$$V_s = \frac{4}{3} \pi r^3$$
Where \(V_s\) is the volume of the sphere and \(r\) is its radius.
- The question states the diameter of the sphere is 1 unit.
- The radius \(r\) is half of the diameter, so \(r = \frac{\text{Diameter}}{2} = \frac{1}{2}\) unit.
Substituting the value of the radius into the formula:
$$V_s = \frac{4}{3} \pi \left(\frac{1}{2}\right)^3$$
$$V_s = \frac{4}{3} \pi \left(\frac{1}{8}\right)$$
$$V_s = \frac{4\pi}{24}$$
$$V_s = \frac{\pi}{6} \text{ cubic units}$$
To get an approximate numerical value, we can use the common approximation for \(\pi \approx 3.14159\):
$$V_s \approx \frac{3.14159}{6}$$
$$V_s \approx 0.5236 \text{ cubic units}$$
Cube Volume Calculation
The formula for the volume of a cube is given by:
$$V_c = a^3$$
Where \(V_c\) is the volume of the cube and \(a\) is the length of its side.
- The question states the side of the cube is 1 unit.
- So, \(a = 1\) unit.
Substituting the value of the side into the formula:
$$V_c = (1)^3$$
$$V_c = 1 \text{ cubic unit}$$
Volume Comparison and Conclusion
Now, let's compare the calculated volumes of the sphere and the cube:
- Volume of the sphere ($V_s$) \(\approx 0.5236\) cubic units.
- Volume of the cube ($V_c$) \(= 1\) cubic unit.
Since \(0.5236 < 1\), it is clear that the volume of the sphere is numerically smaller than the volume of the cube.
| Shape | Given Dimensions | Volume Formula | Calculated Volume |
|---|---|---|---|
| Sphere | Diameter = 1 unit (Radius = 0.5 unit) | $V_s = \frac{4}{3} \pi r^3$ | $\frac{\pi}{6} \approx 0.5236$ cubic units |
| Cube | Side = 1 unit | $V_c = a^3$ | $1^3 = 1$ cubic unit |
Therefore, the volume of a sphere of diameter 1 unit is less than the volume of a cube of side 1 unit.
Important Questions from English Grammar
Directions : Item in this section has a sentence with three parts labelled (a), (b) and (c). Read each sentence to find out whether there is any error in any underlined part. Indicate your response in the Answer Sheet against the corresponding letter, i.e., (a) or (b) or (c). If you find no error, your response should be indicated as (d).
Our greatest glory is (a) / not in never falling but in (b) / raising every time we fall. (c) / No error (d)
Directions : I tem in this section has a sentence with three parts labelled (a), (b) and (c). Read each sentence to find out whether there is any error in any underlined part. Indicate your response in the Answer Sheet against the corresponding letter, i.e., (a) or (b) or (c). If you find no error, your response should be indicated as (d).
Rahim was one of that (a) / selected for (b) / the award. (c) No error (d)
Directions : I tem in this section has a sentence with three parts labelled (a), (b) and (c). Read each sentence to find out whether there is any error in any underlined part. Indicate your response in the Answer Sheet against the corresponding letter, i.e., (a) or (b) or (c). If you find no error, your response should be indicated as (d).
Sujini has met her old friend (a) / Guru before she shifted (b) / to Hyderabad two years ago. (c) / No error (d)
Directions : I tem in this section has a sentence with three parts labelled (a), (b) and (c). Read each sentence to find out whether there is any error in any underlined part. Indicate your response in the Answer Sheet against the corresponding letter, i.e., (a) or (b) or (c). If you find no error, your response should be indicated as (d).
This monument happens (a) / to be one of the old monument (b) / in the city. (c) / No error (d)
Directions : I tem in this section has a sentence with three parts labelled (a), (b) and (c). Read each sentence to find out whether there is any error in any underlined part. Indicate your response in the Answer Sheet against the corresponding letter, i.e., (a) or (b) or (c). If you find no error, your response should be indicated as (d).
Amit told to Divya (a) / that he would come (b) / with a reward for her. (c)/ No error (d)
