Sphere and Cube Volume Ratio Calculation
This problem requires finding the ratio between the volume of a sphere and a cube when their surface areas are equal. We'll use the standard geometric formulas and algebraic manipulation.
Sphere Formulas
For a sphere with radius r:
- Surface Area ($SA_S$): $SA_S = 4 \pi r^2$
- Volume ($V_S$): $V_S = \frac{4}{3} \pi r^3$
Cube Formulas
For a cube with side length a:
- Surface Area ($SA_C$): $SA_C = 6 a^2$
- Volume ($V_C$): $V_C = a^3$
Equal Surface Area Condition
The problem states that the surface area of the sphere equals the surface area of the cube ($SA_S = SA_C$). Setting the formulas equal:
$$ 4 \pi r^2 = 6 a^2 $$This equality allows us to establish a relationship between the sphere's radius ($r$) and the cube's side length ($a$). Let's solve for a in terms of r:
$$ a^2 = \frac{4 \pi r^2}{6} $$ $$ a^2 = \frac{2 \pi r^2}{3} $$ $$ a = \sqrt{\frac{2 \pi r^2}{3}} $$ $$ a = r \sqrt{\frac{2 \pi}{3}} $$Deriving the Volume Ratio
Our goal is to find the ratio of the volume of the sphere to the volume of the cube ($\frac{V_S}{V_C}$). First, let's express the cube's volume ($V_C$) using the relationship we found between a and r:
$$ V_C = a^3 = \left( r \sqrt{\frac{2 \pi}{3}} \right)^3 $$ $$ V_C = r^3 \left( \frac{2 \pi}{3} \right)^{3/2} $$Now, we can set up the ratio $\frac{V_S}{V_C}$ using the volume formulas:
$$ \frac{V_S}{V_C} = \frac{\frac{4}{3} \pi r^3}{r^3 \left( \frac{2 \pi}{3} \right)^{3/2}} $$The $r^3$ terms cancel out:
$$ \frac{V_S}{V_C} = \frac{\frac{4}{3} \pi}{\left( \frac{2 \pi}{3} \right)^{3/2}} $$Let's simplify this expression:
$$ \frac{V_S}{V_C} = \frac{4 \pi}{3} \times \frac{1}{\left( \frac{2 \pi}{3} \right)^{3/2}} $$ $$ = \frac{4 \pi}{3} \times \left( \frac{3}{2 \pi} \right)^{3/2} $$ $$ = \frac{4 \pi}{3} \times \frac{3^{3/2}}{(2 \pi)^{3/2}} $$ $$ = \frac{4 \pi}{3} \times \frac{3 \sqrt{3}}{2 \sqrt{2} \pi \sqrt{\pi}} $$Now, we cancel common factors to simplify further:
$$ = \frac{4 \times 3 \sqrt{3}}{3 \times 2 \sqrt{2} \sqrt{\pi}} $$ $$ = \frac{12 \sqrt{3}}{6 \sqrt{2 \pi}} $$ $$ = \frac{2 \sqrt{3}}{\sqrt{2 \pi}} $$To express the ratio in the format $\sqrt{X} : \sqrt{Y}$, we can square the numerator and place it under the square root sign:
$$ = \sqrt{\frac{(2 \sqrt{3})^2}{2 \pi}} $$ $$ = \sqrt{\frac{4 \times 3}{2 \pi}} $$ $$ = \sqrt{\frac{12}{2 \pi}} $$ $$ = \sqrt{\frac{6}{\pi}} $$Therefore, the ratio of the volume of the sphere to the volume of the cube is $\sqrt{6} : \sqrt{\pi}$.
Similar Questions
- Find the surface area of a ball of radius 21 mm. (Use $\pi = \frac{22}{7}$)
- The dimensions of a pond are 20 m, 14 m and 6 m. Find the capacity of the pond.
Important Questions from Mensuration
In a circular garden of radius 15 m, a path of 2 m wide has to be made inside the garden at the rate of ₹ 24 per sq. m. The cost of making the path is: (Take π = \(\frac{22}{7}\) )
A hollow cylinder with outer radius 4 cm and height 2 cm is made up of 1 cm thick metal sheet. What is the volume of metal used? (Take π = \(\frac{22}{7}\))
The length and breadth of a rectangular field are in the ratio 4 : 3. If the cost of cultivating the field at ₹2 per m 2is ₹600, then the length of the field is:
A cylindrical tank has a capacity of 5632 m3. If the diameter of its base is 8 m, what is the depth of the cylindrical tank? (Use π = \(\frac{22}{7}\))
Find the surface area of a sphere whose diameter is equal to 28 cm.
