AMC 8 · 2008 · #18

Grade 7 geometry-2d

Archetype Path Length Comparison & Shortest Path

perimeterarea-circlesfraction-arithmeticspatial-visualization path-length-comparisonidentify-subproblems ↑ Prerequisites: area-circlesfraction-arithmetic

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$ . How many meters does the aardvark run?

Pick an answer.

(A)

$10\pi+20$

(B)

$10\pi+30$

(C)

$10\pi+40$

(D)

$20\pi+20$

(E)

$20\pi+40$

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Toolkit + CCSS Solution

Understand

Restated: Two circles share the same center. The small one has radius $10$ meters and the large one has radius $20$ meters. An aardvark walks from $A$ at the top of the large circle to $K$ at the right of the large circle, following the path of arrows in the figure. Find the total length of that path.

Givens: Two concentric circles with radii $10$ m and $20$ m; Start point $A = (0, 20)$ on the large circle; End point $K = (20, 0)$ on the large circle; The path follows arrows that mix arcs and straight segments; Answer choices: (A) $10\pi+20$, (B) $10\pi+30$, (C) $10\pi+40$, (D) $20\pi+20$, (E) $20\pi+40$

Unknowns: The total length of the path from $A$ to $K$

Understand

Plan

Primary tool: #7 Break the Problem into Subproblems

Secondary: #1 Draw a Picture

The path is one curvy trip, but it is built from six clean pieces: three quarter-arcs and three straight segments. Tool #7 (Break the Problem into Subproblems) says to find the length of each piece by itself, then add. Tool #1 (Draw a Picture) helps us label each piece with its endpoints in coordinates so we know which circle the arc sits on and which radius the segment uses.

Execute — Answer: E

#1 Draw a Picture 6.G.A.3 Step 1

Label the path.
Following the arrows from $A=(0,20)$ to $K=(20,0)$, the aardvark traces six pieces: arc on the large circle from $(0,20)$ to $(-20,0)$; segment from $(-20,0)$ to $(-10,0)$; arc on the small circle from $(-10,0)$ to $(0,-10)$; segment from $(0,-10)$ to $(0,10)$; arc on the small circle from $(0,10)$ to $(10,0)$; segment from $(10,0)$ to $(20,0)$.

$$\text{path} = \text{arc}_1 + \text{seg}_1 + \text{arc}_2 + \text{seg}_2 + \text{arc}_3 + \text{seg}_3$$

💡 Plotting the endpoints on a coordinate grid is the Grade 6 way of seeing the shape. Each piece is either an arc on a known circle or a segment along an axis.

#7 Break the Problem into Subproblems 7.G.B.4 Step 2

Length of the big arc.
The first piece runs from the top of the large circle to the left side.
That is a quarter of the large circle, whose full circumference is $2\pi(20) = 40\pi$.

$$\text{arc}_1 = \dfrac{1}{4}(2\pi \cdot 20) = \dfrac{1}{4}(40\pi) = 10\pi$$

💡 The Grade 7 circle formula $C = 2\pi r$ gives the whole way around. A quarter turn uses a quarter of that.

#7 Break the Problem into Subproblems 7.G.B.4 Step 3

Length of the two small arcs.
Each one is a quarter of the small circle, whose full circumference is $2\pi(10) = 20\pi$.
Both quarter-arcs have the same length.

$$\text{arc}_2 = \text{arc}_3 = \dfrac{1}{4}(2\pi \cdot 10) = 5\pi$$

💡 Same circle formula, smaller radius. Two equal quarter-arcs add to $10\pi$.

#7 Break the Problem into Subproblems 6.NS.C.8 Step 4

Length of the three straight segments.
Two of them go from the large circle to the small circle along the $x$-axis, so each has length $20 - 10 = 10$.
The third runs along the $y$-axis from $(0,-10)$ to $(0,10)$ — that is a full diameter of the small circle, length $2 \cdot 10 = 20$.

$$\text{seg}_1 = 10, \quad \text{seg}_2 = 20, \quad \text{seg}_3 = 10$$

💡 Distance along an axis is just the difference in coordinates. Grade 6 number-line distance, used on the $x$- and $y$-axes.

#7 Break the Problem into Subproblems 6.EE.A.3 Step 5

Add all six pieces.
Group the $\pi$ terms and the plain numbers separately.

$$10\pi + 10 + 5\pi + 20 + 5\pi + 10 = (10\pi + 5\pi + 5\pi) + (10 + 20 + 10) = 20\pi + 40 \;\Rightarrow\; \textbf{(E)}$$

💡 Combining like terms is a Grade 6 expression move: $\pi$ terms stack together, constants stack together.

[1] #1 6.G.A.3 Label the path. Following the arrows from $A=(0,20)$ to $K=(20,0)$, the aardvark

[2] #7 7.G.B.4 Length of the big arc. The first piece runs from the top of the large circle to

[3] #7 7.G.B.4 Length of the two small arcs. Each one is a quarter of the small circle, whose f

[4] #7 6.NS.C.8 Length of the three straight segments. Two of them go from the large circle to t

[5] #7 6.EE.A.3 Add all six pieces. Group the $\pi$ terms and the plain numbers separately.

Review

Reasonableness: Sanity check the size. The path includes one big quarter-arc on the outer circle ($10\pi \approx 31.4$) plus two small quarter-arcs ($5\pi + 5\pi \approx 31.4$) plus three straight pieces ($10 + 20 + 10 = 40$). Total $\approx 62.8 + 40 \approx 102.8$ m, which matches $20\pi + 40 \approx 102.8$. The full perimeter of the large circle is only $40\pi \approx 125.7$ m, so a longer-than-half-circle answer with extra straight pieces is reasonable. Among the choices, only (E) has the $20\pi$ coefficient that comes from one big quarter-arc plus two small quarter-arcs combined.

Alternative: Tool #5 (Look for a Pattern) plus Tool #11 (Find an Invariant): the two small quarter-arcs together cover half of the small circle, which has length $\pi(10) = 10\pi$. The big quarter-arc adds another $10\pi$. The straight pieces total $10 + 20 + 10 = 40$. So the total is $10\pi + 10\pi + 40 = 20\pi + 40$, the same answer (E).

CCSS standards used (min grade 7)

6.G.A.3 Draw polygons in the coordinate plane and use coordinates to find lengths of horizontal and vertical sides (Labeling the six endpoints of the path on the coordinate plane so each piece is either an arc on a named circle or a segment along an axis.)
6.NS.C.8 Solve real-world problems by finding distances between points with the same first or second coordinate (Finding the lengths of the three straight pieces ($10$, $20$, $10$) by taking the difference of $x$- or $y$-coordinates along an axis.)
6.EE.A.3 Apply the properties of operations to generate equivalent expressions (Combining like terms in $10\pi + 10 + 5\pi + 20 + 5\pi + 10$ to get $20\pi + 40$.)
7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems (Using $C = 2\pi r$ on both circles, then taking one quarter of each circumference to get arc lengths $10\pi$ and $5\pi$.)

⭐ A curvy path becomes easy when you cut it into pieces — each arc is a quarter of its circle and each straight piece is a difference of coordinates.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 7)

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