AMC 10 · 2009 · #11
Grade 7 geometry-3dOne dimension of a cube is increased by 1, another is decreased by 1, and the third is left unchanged. The volume of the new rectangular solid is 5 less than that of the cube. What was the volume of the cube?
Pick an answer.
AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A cube has all three edges the same length. Build a new box from it: make one edge longer by $1$, make another edge shorter by $1$, and leave the third edge alone. The new box holds $5$ less volume than the cube. Find the cube's volume.
Givens: The starting solid is a cube, so its length, width, and height are all equal.; One edge is increased by $1$, a second edge is decreased by $1$, and the third edge is unchanged.; The new box's volume is exactly $5$ less than the cube's volume.
Unknowns: The volume of the original cube.
Understand
Restated: A cube has all three edges the same length. Build a new box from it: make one edge longer by $1$, make another edge shorter by $1$, and leave the third edge alone. The new box holds $5$ less volume than the cube. Find the cube's volume.
Givens: The starting solid is a cube, so its length, width, and height are all equal.; One edge is increased by $1$, a second edge is decreased by $1$, and the third edge is unchanged.; The new box's volume is exactly $5$ less than the cube's volume.
Plan
Primary tool: #4 Introduce a Variable
Secondary: #13 Convert to Algebra, #3 Eliminate Possibilities
The cube's edge length is the one thing that ties both solids together, but it is unknown, so tool #4 (Introduce a Variable) names it $s$ and lets us describe the cube and the new box with the same symbol. Tool #13 (Convert to Algebra) then turns the sentence "$5$ less" into an equation we can simplify, where the two opposite changes ($+1$ and $-1$) collapse neatly. Once we solve for $s$, tool #3 (Eliminate Possibilities) matches the cube's volume to exactly one of the five answer choices.
Execute — Answer: D
6.EE.B.6 Step 1 Name the cube's edge
- Let $s$ stand for the length of one edge of the cube.
- Because it is a cube, all three edges equal $s$.
- Every length in the problem can now be written in terms of $s$.
💡 Giving the unknown edge a name lets you describe both solids with a single symbol.
5.MD.C.5 Step 2 Write both volumes
- The cube's volume is $s \times s \times s = s^3$.
- The new box keeps one edge at $s$, makes another $s+1$, and makes the third $s-1$, so its volume is $(s+1)(s-1)(s)$.
💡 Volume is just the three side lengths multiplied together.
7.EE.A.1 Step 3 Simplify the box's volume
- Multiply the two changed edges first.
- Since $+1$ and $-1$ are opposites, $(s+1)(s-1) = s^2 - 1$.
- Multiplying by the untouched edge $s$ gives $V_{\text{box}} = s(s^2 - 1) = s^3 - s$.
💡 The two opposite tweaks, $+1$ and $-1$, cancel into a clean $s^2 - 1$.
6.EE.B.7 Step 4 Set up and solve for the edge
- The box's volume is $5$ less than the cube's, so $V_{\text{cube}} - V_{\text{box}} = 5$.
- That is $s^3 - (s^3 - s) = 5$.
- The $s^3$ terms cancel, leaving $s = 5$.
💡 Because the cube parts cancel, the whole $5$-unit drop is exactly one edge length.
6.EE.A.1 Step 5 Find the cube's volume
- The cube has edge $s = 5$, so its volume is $5^3 = 125$.
- Matching this to the answer choices gives $\textbf{(D)}$.
💡 Cubing the edge you found gives the volume the question asked for.
6.EE.B.6 Let $s$ stand for the length of one edge of the cube. Because it is a cube, all 5.MD.C.5 The cube's volume is $s \times s \times s = s^3$. The new box keeps one edge at 7.EE.A.1 Multiply the two changed edges first. Since $+1$ and $-1$ are opposites, $(s+1)( 6.EE.B.7 The box's volume is $5$ less than the cube's, so $V_{\text{cube}} - V_{\text{box 6.EE.A.1 The cube has edge $s = 5$, so its volume is $5^3 = 125$. Matching this to the an Review
Reasonableness: Check the numbers directly: an edge of $5$ makes a cube of volume $5^3 = 125$, and the new box measures $6 \times 4 \times 5 = 120$. The difference is $125 - 120 = 5$, exactly as stated. The volume $125$ is answer choice (D), so the result is consistent with both the problem and the choices.
Alternative: Every answer choice is a perfect cube, so their edges are $2, 3, 4, 5, 6$. Guess and check each: for edge $5$, the box is $6 \times 4 \times 5 = 120$, which is $5$ less than $125$. No other edge gives a gap of exactly $5$ (for edge $4$ the gap is $4$, for edge $6$ it is $6$), confirming the cube's volume is $125$.
CCSS standards used (min grade 7)
6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Letting $s$ stand for the cube's edge so both solids can be described with one symbol.)5.MD.C.5Relate volume to the operations of multiplication and addition (Writing the cube's volume $s^3$ and the box's volume as a product of its three edges.)7.EE.A.1Apply properties of operations to add, subtract, factor, and expand linear expressions (Expanding $(s+1)(s-1)$ to $s^2 - 1$ and simplifying the box volume to $s^3 - s$.)6.EE.B.7Solve real-world problems by writing and solving equations of the form px = q (Turning "$5$ less" into $s^3 - (s^3 - s) = 5$ and solving to get $s = 5$.)6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Evaluating $5^3 = 125$ to get the cube's volume.)
⭐ When you stretch one side by $1$ and shrink another by $1$, the volume drops by exactly the length of the side you left alone.
⭐ When you stretch one side by $1$ and shrink another by $1$, the volume drops by exactly the length of the side you left alone.
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