The surface area of a closed rectangular box, which is inscribed in a sphere, is 846 sq cm, and the sum of the lengths of all its edges is 144 cm. The volume, in cubic cm, of the sphere is
Official Correct Answer: A. Given a closed rectangular box inscribed in a sphere, we know the surface area of the box is 846 sq \text{ cm} and the sum of the lengths of all its edges is 144 \text{ cm}. We need to find the volume of the sphere.
1. **Surface Area Calculation**:
The surface area (SA) of the box is given by:
$$
2(ab + bc + ac) = 846 \implies ab + bc + ac = 423
$$
2. **Sum of Edges Calculation**:
The sum of the lengths of all edges (L) is given by:
$$
4(a + b + c) = 144 \implies a + b + c = 36
$$
3. **Space Diagonal Calculation**:
Using the identity:
$$
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ac)
$$
Substituting the known values:
$$
36^2 = a^2 + b^2 + c^2 + 2 \times 423 \implies 1296 = a^2 + b^2 + c^2 + 846 \implies a^2 + b^2 + c^2 = 450
$$
The space diagonal (d) of the box is:
$$
d = \sqrt{a^2 + b^2 + c^2} = \sqrt{450} = 15\sqrt{2}
$$
4. **Volume of the Sphere**:
The radius (r) of the sphere is half the space diagonal
Answer and solution
Answer:A) 1125π2
Given a closed rectangular box inscribed in a sphere, we know the surface area of the box is 846 sq \text{ cm} and the sum of the lengths of all its edges is 144 \text{ cm}. We need to find the volume of the sphere.
1
Surface Area Calculation:
The surface area (SA) of the box is given by:
2(ab+bc+ac)=846⟹ab+bc+ac=423
2
Sum of Edges Calculation:
The sum of the lengths of all edges (L) is given by:
4(a+b+c)=144⟹a+b+c=36
3
Space Diagonal Calculation:
Using the identity:
(a+b+c)2=a2+b2+c2+2(ab+bc+ac)
Substituting the known values:
362=a2+b2+c2+2×423⟹1296=a2+b2+c2+846⟹a2+b2+c2=450
The space diagonal (d) of the box is:
d=a2+b2+c2=450=152
4
Volume of the Sphere:
The radius (r) of the sphere is half the space diagonal
