Question

these solids all have the same volume. which has the least surface area? a. solid a b. solid b c. solid c

Explanation:

Step1: Consider the property of volume and surface area for 3 - D shapes

For a given volume, the more "compact" and "symmetrical" a 3 - D shape is, the smaller its surface area. Among common 3 - D shapes (cubes, rectangular prisms, cylinders), the cube is a highly symmetrical rectangular prism.

Step2: Analyze each solid

Solid A is a cube. Let the side length of the cube be \(a\), its volume \(V=a^{3}\), and its surface area \(S_{A} = 6a^{2}\).
Solid B is a rectangular prism. Let its length \(l\), width \(w\), and height \(h\) such that \(V=l\times w\times h=a^{3}\). For a non - cube rectangular prism (where \(l
eq w
eq h\) in the non - equal sense), using the AM - GM inequality \(\frac{l + w+h}{3}\geq\sqrt[3]{lwh}\) (equality holds when \(l = w=h\)), and the surface area formula \(S_{B}=2(lw + wh+lh)\). When \(l
eq w
eq h\), \(S_{B}>S_{A}\).
Solid C is a cylinder. Let the radius of the base be \(r\) and height be \(h\), \(V=\pi r^{2}h=a^{3}\), and its surface area \(S_{C}=2\pi r^{2}+2\pi rh\). Through calculus (optimizing the surface area function \(S(r)=2\pi r^{2}+\frac{2a^{3}}{r}\) by taking the derivative \(S^\prime(r) = 4\pi r-\frac{2a^{3}}{r^{2}}\) and setting \(S^\prime(r)=0\) to find the minimum, and then comparing with the cube case), or by geometric intuition (the cube has more "regular" faces compared to the cylinder in terms of minimizing surface area for a given volume in simple geometric shapes considered here)

Answer:

A. Solid A