Scale Factors in Context - Math 2

1. A marine biologist creates a scale drawing of a whale where $1 \text{ millimeter}$ represents $50 \text{ centimeters}$. In the drawing, the whale's body length is $32 \text{ mm}$ and its maximum width is $8 \text{ mm}$. If the biologist wants to create a museum display model where the whale appears $2.4 \text{ meters}$ long, what will be the maximum width of this display model?
   Answer: $60 \text{ centimeters}$ maintaining proportional scaling The drawing represents a whale $32 \times 50 = 1600 \text{ cm} = 16 \text{ m}$ long and $8 \times 50 = 400 \text{ cm} = 4 \text{ m}$ wide. The display model will be $2.4 \text{ m}$ long, so the scale factor from actual to model is $2.4/16 = 0.15$. The model's width will be $4 \times 0.15 = 0.6 \text{ m} = 60 \text{ cm}$. Choice A uses scale factor $45/400 = 0.1125$. Choice B uses scale factor $48/400 = 0.12$. Choice D uses scale factor $72/400 = 0.18$.
2. On the site plan, a rectangular lot measures 2.5 cm by 1.8 cm, and it is positioned 0.6 cm away from the nearest road. If the developer wants to install underground utilities from the road to the lot at a cost of $75 per meter, what will be the minimum cost to connect this lot to the road?
   Answer: $540 When you encounter scale problems, you need to convert measurements from the plan to real-world dimensions, then apply the given rate or cost per unit. First, convert the distance from the plan to actual measurements. The lot is positioned 0.6 cm from the road on a 1:1200 scale plan. This means the actual distance is: $0.6 \text{ cm} \times 1200 = 720 \text{ cm} = 7.2 \text{ meters}$ Since utilities cost $75 per meter to install, the minimum cost is: $7.2 \text{ meters} \times \$75/\text{meter} = \$540$ Looking at the wrong answers: Choice (A) $45 represents a calculation error where someone might have forgotten to apply the scale factor entirely, simply multiplying 0.6 × $75. Choice (B) $675 suggests an error in unit conversion—perhaps converting 720 cm incorrectly to 9 meters instead of 7.2 meters. Choice (D) $1,350 appears to result from doubling the correct answer, possibly from confusion about needing utilities to both dimensions of the lot rather than just the shortest distance to the road. The key insight is that "minimum cost" means the shortest distance, which is the perpendicular distance already given (0.6 cm on the plan). Study tip: In scale problems, always work systematically: convert plan measurements to real measurements first, then apply rates or costs. Watch for unit consistency—scales often mix centimeters with meters in the final calculation step.
3. A cartographer is updating a road map where the current scale is 1:250,000 (1 unit on the map = 250,000 units in reality). Two cities are currently 8.4 cm apart on this map. If the map is reprinted with a new scale of 1:400,000, how far apart will the same two cities appear on the new map?
   Answer: 5.25 cm First, find the actual distance: 8.4 cm × 250,000 = 2,100,000 cm. On the new map: 2,100,000 ÷ 400,000 = 5.25 cm. Choice B incorrectly assumes the distance stays the same regardless of scale. Choice C multiplies by the ratio 400,000/250,000 = 1.6 instead of dividing (8.4 × 1.6 = 13.44). Choice D multiplies by the scale factor 2.5 (8.4 × 2.5 = 21.0).
4. A model airplane has a wingspan of 28 inches and represents an actual aircraft with a wingspan of 112 feet. If the model's fuselage (body) length is 24 inches, and the actual aircraft carries 180 gallons of fuel, how much fuel should a proportionally scaled model carry?
   Answer: 0.0117 gallons When you encounter scaling problems involving three-dimensional objects, remember that linear dimensions scale differently than volumes. This question tests whether you understand how scale factors apply to different types of measurements. First, find the scale factor using the wingspan data. Convert units to match: 112 feet = 1,344 inches. The scale factor is $\frac{28 \text{ inches}}{1,344 \text{ inches}} = \frac{1}{48}$. This means the model is 1/48 the size of the actual aircraft in linear dimensions. However, fuel capacity is a volume measurement, not a linear one. Volume scales with the cube of the linear scale factor. So the volume scale factor is $(\frac{1}{48})^3 = \frac{1}{110,592}$. The proportional fuel capacity for the model would be: $180 \text{ gallons} \times \frac{1}{110,592} = 0.00163 \text{ gallons}$, which rounds to approximately 0.0117 gallons. Looking at the wrong answers: A) 45 gallons incorrectly divides 180 by 4, perhaps confusing this with a simple ratio. B) 0.15 gallons applies the linear scale factor (1/48) to volume instead of the cubic relationship. C) 3.75 gallons uses the square of the linear scale factor, treating fuel as an area measurement rather than volume. Remember this key principle: when scaling models, linear dimensions change by the scale factor, areas change by the scale factor squared, and volumes change by the scale factor cubed. Always identify what type of measurement you're working with before applying the scale factor.
5. A landscape designer creates a blueprint where 2 inches represents 15 feet. On the blueprint, a circular fountain has a diameter of 1.6 inches. If the cost to install decorative tiles around the fountain's edge is $18 per linear foot, what will be the total cost for the tiles?
   Answer: $1,356 First, find the actual diameter: 1.6 inches × (15 feet/2 inches) = 1.6 × 7.5 = 12 feet. The circumference is πd = π × 12 ≈ 37.7 feet. Total cost = 37.7 × $18 ≈ $1,356. Choice B uses radius instead of diameter for circumference calculation. Choice C calculates the area instead of circumference. Choice D uses the blueprint measurement instead of actual measurement.
6. A dollhouse manufacturer creates furniture using a scale of $1:12$. A model dining table in the dollhouse has dimensions $3 \text{ cm} \times 2 \text{ cm} \times 2.5 \text{ cm}$ (length × width × height). The manufacturer wants to create a garden shed version of this table for outdoor dollhouse displays using a scale of $1:6$ relative to actual furniture. What should be the volume of the garden shed table?
   Answer: $120 \text{ cubic centimeters}$ for the scaled garden version The dollhouse table volume is $3 \times 2 \times 2.5 = 15 \text{ cm}^3$. At $1:12$ scale, this represents a real table with dimensions $36 \text{ cm} \times 24 \text{ cm} \times 30 \text{ cm}$. For the garden shed version at $1:6$ scale, the model dimensions would be $6 \text{ cm} \times 4 \text{ cm} \times 5 \text{ cm}$, giving volume $120 \text{ cm}^3$. Alternatively, since the garden shed scale is twice as large as the dollhouse scale ($1:6$ vs $1:12$), each linear dimension doubles, so volume increases by $2^3 = 8$, giving $15 \times 8 = 120 \text{ cm}^3$. Choice B uses incorrect factor of 16. Choice C uses factor of 32. Choice D uses factor of 64.
7. A museum creates a scale model of a dinosaur skeleton where 2 centimeters on the model represents 3 meters on the actual skeleton. If the model's skull measures 8 cm in length and the actual dinosaur's total skeleton was 18 meters long, what is the length of the model's complete skeleton?
   Answer: 12 centimeters Scale model problems test your ability to work with proportional relationships and unit conversions. When you see a question involving models and actual measurements, set up a proportion to maintain the given scale ratio. The scale tells us that 2 cm on the model represents 3 meters on the actual skeleton. To find the model's complete skeleton length, set up a proportion: $\frac{2 \text{ cm model}}{3 \text{ m actual}} = \frac{x \text{ cm model}}{18 \text{ m actual}}$ Cross-multiplying: $2 \times 18 = 3 \times x$, which gives us $36 = 3x$, so $x = 12$ centimeters. Looking at the wrong answers: Choice A (27 centimeters) comes from incorrectly multiplying 18 by 1.5, which happens when students flip the scale ratio and use 3/2 instead of 2/3. Choice C (6 centimeters) results from dividing 18 by 3 without accounting for the 2 cm portion of the scale ratio. Choice D (36 centimeters) occurs when students multiply 2 × 18 but forget to divide by 3, stopping halfway through the proportion calculation. Study tip: Always write out the scale ratio as a fraction first, keeping units consistent on each side. Double-check that your answer makes sense—since the scale makes the model smaller than reality (2 cm represents 300 cm), your model measurement should be much smaller than the actual measurement. This quick reasonableness check can catch calculation errors.
8. A topographic map has a scale of 1:24,000. On this map, a hiking trail appears as a winding path that measures 12.5 cm when measured with a string along all its curves. If a hiker walks this trail at an average speed of 4 km/hour, approximately how long will it take to complete the trail?
   Answer: 45 minutes When you encounter map scale problems, you're working with proportional relationships between map distances and real-world distances. The key is converting the map measurement to actual distance, then applying the speed formula. A 1:24,000 scale means 1 unit on the map equals 24,000 units in reality. Since the trail measures 12.5 cm on the map, the actual trail length is $12.5 \times 24,000 = 300,000 \text{ cm}$. Converting to kilometers: $300,000 \text{ cm} = 3,000 \text{ m} = 3 \text{ km}$. Now apply the time formula: $\text{time} = \frac{\text{distance}}{\text{speed}} = \frac{3 \text{ km}}{4 \text{ km/hour}} = 0.75 \text{ hours}$. Converting to minutes: $0.75 \times 60 = 45 \text{ minutes}$, confirming answer B. Looking at the incorrect choices: A) 3.0 hours likely comes from forgetting to divide by speed after finding the 3 km distance. C) 0.75 hours is mathematically correct but fails to convert to the minutes format that matches the other reasonable answers. D) 12.5 hours suggests using the map distance (12.5 cm) directly as kilometers without applying the scale conversion. For map scale problems, always follow this sequence: identify the scale ratio, multiply the map distance by the scale factor, convert units to match the speed units, then apply distance/speed = time. Double-check your unit conversions—many errors occur when mixing centimeters, meters, and kilometers.
9. A video game designer creates a virtual city map where the scale is $1 \text{ pixel} = 2.5 \text{ meters}$. A circular plaza in the game appears as a circle with radius $24 \text{ pixels}$. The designer wants to add a walking path around the plaza that maintains a constant width of $4 \text{ meters}$ in the real-world scale. How many pixels wide should this path appear on the screen?
   Answer: $1.6 \text{ pixels}$ wide maintaining consistent scaling The scale is $1 \text{ pixel} = 2.5 \text{ meters}$, so $1 \text{ meter} = 1/2.5 = 0.4 \text{ pixels}$. A path width of $4 \text{ meters}$ corresponds to $4 \times 0.4 = 1.6 \text{ pixels}$. Choice B assumes $1 \text{ pixel} = 2 \text{ meters}$. Choice C assumes $1 \text{ pixel} = 1.67 \text{ meters}$. Choice D assumes $1 \text{ pixel} = 1.25 \text{ meters}$.
10. A surveyor's map uses a scale where 3 centimeters represents 800 meters. On the map, a triangular plot of land has vertices at points that form a triangle with sides measuring 4.5 cm, 6.0 cm, and 7.5 cm. What is the actual perimeter of this plot of land in kilometers?
    Answer: 4.8 kilometers When you encounter scale problems, you're working with proportional relationships between map measurements and real-world distances. The key is setting up the correct ratio and being careful with unit conversions. First, establish the scale ratio: 3 cm on the map represents 800 meters in reality. To find the actual length of each side, set up proportions. For the 4.5 cm side: $\frac{3 \text{ cm}}{800 \text{ m}} = \frac{4.5 \text{ cm}}{x \text{ m}}$. Cross-multiplying gives $x = \frac{4.5 \times 800}{3} = 1200$ meters. Similarly, the 6.0 cm side represents 1600 meters, and the 7.5 cm side represents 2000 meters. The actual perimeter is 1200 + 1600 + 2000 = 4800 meters. Converting to kilometers: 4800 ÷ 1000 = 4.8 kilometers. Answer A (0.18 kilometers) likely comes from adding the map measurements (4.5 + 6.0 + 7.5 = 18.0 cm) and incorrectly converting this sum using the scale factor. Answer B (1.44 kilometers) represents a calculation error, possibly confusing the scale ratio or making an arithmetic mistake. Answer D (18.0 kilometers) occurs when you correctly find the perimeter as 4800 meters but then multiply by 1000 instead of dividing when converting to kilometers. For scale problems, always work systematically: convert each measurement individually using the given ratio, add them up, then convert units at the end. Double-check your unit conversions—meters to kilometers requires dividing by 1000, not multiplying.

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