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
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Important Questions from Mensuration
A solid cube is painted yellow, blue and black such that opposite faces are of same colour. The cube is then cut into 36 cubes of two different sizes such that 32 cubes are small and the other four cubes are Big. None of the faces of the bigger cubes is painted blue. How many cubes have only one face painted?
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A gardener increased the area of his rectangular garden by increasing its length by 40% and decreasing its width by 20%. The area of the new garden
A village having a population of 4000 requires 150 liters of water per head per day. It has a tank measuring 20 m x 15 m x 6 m. The water of this tank will last for
The centroid of an equilateral triangle ABC is G. If AB is 6 cms, the length of AG is
