Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.
Solution:
Equation of a sphere is given by x2 + y2 + z2 = r2 ……(1)
Volume of the box, VB = 2x. 2y. 2z = 8xyz ……(2)
From (1), z2 = r2 - x2 - y2
z = √(r2 - x2 - y2) ……(3)
Substituting the value of z (from (3)) in (2),
VB = 8x y √(r2 - x2 - y2) …….(4)
Partially differentiating (4) w.r.t x,
\(\frac{\partial V_{B} }{\partial x} = 8 y \sqrt{(r^{2} - x^{2} - y^{2})} + 8xy \frac{1}{2}(r^{2} - x^{2} - y^{2})^{-1/2} .(-2x)\)
\(\frac{\partial V_{B} }{\partial x} = 8 y \sqrt{(r^{2} - x^{2} - y^{2})} - \frac{8x^{2}y}{\sqrt{(r^{2} - x^{2} - y^{2})}}\)
\(\frac{\partial V_{B} }{\partial x} = \frac{8y(r^{2} - x^{2} - y^{2}) - 8x^{2}y}{\sqrt{(r^{2} - x^{2} - y^{2})}}\)
\(\frac{\partial V_{B} }{\partial x} = \frac{8y(r^{2} - 2x^{2} - y^{2})}{\sqrt{(r^{2} - x^{2} - y^{2})}}\)
Similarly partially differentiating (4) w.r.t y,
\(\frac{\partial V_{B} }{\partial y} = \frac{8x(r^{2} - x^{2} - 2y^{2})}{\sqrt{(r^{2} - x^{2} - y^{2})}}\)
→ Critical points occur where partial derivatives are equal to zero.
→ If denominator =0, fractions will be undefined. x and y cannot be equal to zero.
∴ Taking r2 - 2x2 - y2 = 0 and r2 - x2 - 2y2 = 0, solving as simultaneous equations we get,
r2 - 2x2 - y2 = 0 ⇒ 2x2 + y2 = r2…..(5)
r2 - x2 - 2y2 = 0 ⇒ x2 + 2y2 = r2…..(6)
Solving equations (5) and (6), we get
2x2 + y2 = r2
2x2+ 4y2 = 2r2
- 3y2 = -r2
y2 = (1/3)r2 ⇒ y = + r/√3 (negative cannot be considered)
Substituting the value of y in one of the simultaneous equations (5) we get,
x = r/√3
Substituting the value of x and y in equation (2),
VB = 8xyz
VB = 8 × r/√3 × r/√3 × √(r2 -( r/√3)2 -( r/√3)2
VB = 8r2/3 × √(3r2/3) -2( r/√3)2
VB = 8r2/3 × √r2/3
VB = 8r2/3 × r/√3
VB = 8r3/3√3
This is the maximum value of a rectangular box that is inscribed in a sphere of radius r.
Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.
Summary:
The maximum volume of a rectangular box that is inscribed in a sphere of radius ‘r’ is \(\frac{8r^3}{3}\sqrt3\)
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