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2020 AMC 10B Problem 2

2020 AMC 10B Problem 2AMC 10 Step by Step

Name: Date: October 28, 2026

Solid Geometry · difficulty 1 of 5 · about 1 min · position P1-10

Carl has $5$ cubes each having side length $1$ , and Kate has $5$ cubes each having side length $2$ . What is the total volume of these $10$ cubes?

A) $24$
B) $25$
C) $28$
D) $40$
E) $45$

Answer

Key insight

A cube with side 2 has volume 2 times 2 times 2 = 8, not 2 or 4. So the total is 5 times 1 plus 5 times 8.

Solution

The volume of a cube is side $\times$ side $\times$ side.

Carl's cubes have side $1$ . Each has volume $1 \times 1 \times 1 = 1$ . Five of them give $5 \times 1 = 5$ .
Kate's cubes have side $2$ . Each has volume $2 \times 2 \times 2 = 8$ . Five of them give $5 \times 8 = 40$ .
Add the two totals: $5 + 40 = 45$ .

The answer is $\boxed{\textbf{(E)}\ 45}$ .

Why this works

When the side of a cube doubles, the volume does not double. It gets multiplied by $2 \times 2 \times 2 = 8$ , because the cube gets twice as long, twice as wide and twice as tall. Whenever a problem gives side lengths but asks about volume, turn each object into its volume first, then add.

The trap

Adding side lengths instead of volumes (5 + 10 = 15), or squaring 2 instead of cubing it (5 + 20 = 25).

Concise solution

A cube of side $1$ has volume $1^3 = 1$ , so Carl's five cubes contribute $5 \cdot 1 = 5$ .

A cube of side $2$ has volume $2^3 = 8$ , so Kate's five cubes contribute $5 \cdot 8 = 40$ .

The total volume is $5 + 40 = 45$ .

The answer is $\boxed{\textbf{(E)}\ 45}$ .

Review in the AoPS books

Introduction to Geometry, Chapter 14: Three-Dimensional Geometry
Introduction to Geometry, Chapter 15: Curved Surfaces

Techniques

Set up the equation/formula and compute; no special trick needed