Volume of a right cone with base radius r cm and height h cm :
V = 1/3 π r² h cm³
Surface area :
A = (π r² + π r √(h² + r²)) cm²
Volume of a sphere with radius r cm :
V = 4/3 π r³ cm³
Surface area :
A = 4 π r² cm²
Volume of a cuboid with dimensions r cm × r cm × h cm :
V = r² h cm³
Surface area :
A = 2 r² + 4 r h
a. If the cone and sphere have the same volume, then
1/3 π r² h = 4/3 π r³ ⇒ h = 4r
b. If the area of the cuboid is 98 cm², then
2 r² + 4 r h = 98 ⇒ r (r + 2 h) = 49
c. Since h = 4r, substituting this into the equation from (b) gives
r (r + 8 r) = 9 r² = 49 ⇒ r² = 49/9 ⇒ r = 7/3
d. With r = 7/3, we have
h = 4 × 7/3 = 28/3
and so the volume of the cuboid is
V = (7/3 cm)² (28/3 cm) = 1372/27 cm³
