Scaling Relationships in Models - Math 3

1. A scale model of a wind turbine is built at 1:50 scale. In a wind tunnel test, the model generates 0.8 watts of power. If wind speed and air density remain constant, and power output scales with the swept area of the turbine blades, what power would the full-size turbine generate under similar conditions?
   Answer: 2,000 watts Power scales with swept area (the circular area covered by rotating blades). The linear scale factor is 50, so the area scale factor is 50² = 2,500. The full-size turbine would generate 0.8 × 2,500 = 2,000 watts. Choice A uses the linear scale factor. Choice C uses the volume scale factor. Choice D appears to use an incorrect scaling relationship.
2. If Park A generates $8,000 in annual revenue and costs $3,600 to maintain, what is the net profit for Park B using the same proportional relationships?
   Answer: $9,422 Revenue scales with area (radius²) and costs scale with perimeter (radius). Scale factor for radius is 200/150 = 4/3. Revenue for Park B: $8,000 × (4/3)² = $8,000 × 16/9 = $14,222. Costs for Park B: $3,600 × (4/3) = $4,800. Net profit: $14,222 - $4,800 = $9,422. Choice A and D reflect calculation errors. Choice C incorrectly applies scaling relationships.
3. Two similar cylindrical tanks have a height ratio of 3:5. If the smaller tank holds 810 liters when full, how many liters does the larger tank hold when full?
   Answer: 3,750 liters For similar solids, volume scales as the cube of the linear scale factor. The height ratio is 3:5, so the volume ratio is 3³:5³ = 27:125. If the smaller tank holds 810 liters, the larger tank holds 810 × (125/27) = 810 × 125/27 = 3,750 liters. Choice A incorrectly uses the linear ratio (810 × 5/3). Choice B uses the square of the ratio. Choice D reflects a computational error.
4. A cube-shaped container is scaled up by a factor of 1.5 in each dimension. If the original container's surface area was 96 square units, what is the ratio of the volume of the new container to the volume of the original container?
   Answer: 3.375:1 When linear dimensions are scaled by a factor of 1.5, volume is scaled by (1.5)³ = 3.375. The surface area information is given as a distractor but is not needed to solve the problem. The ratio of new volume to original volume is 3.375:1. Choice A uses the linear scale factor. Choice B uses the area scale factor (1.5)². Choice D incorrectly squares the volume scale factor.
5. A company produces two sizes of similar cone-shaped containers. The larger container has a height that is 2.5 times the height of the smaller container. If the smaller container uses 36 square inches of material to manufacture, how much material does the larger container require?
   Answer: 225 square inches When you encounter problems involving similar geometric shapes with different sizes, you're working with scaling relationships. The key insight is that when a 3D object is scaled up, its surface area scales with the square of the scaling factor. Since the containers are similar cones and the larger container has a height 2.5 times that of the smaller one, all linear dimensions are scaled by a factor of 2.5. For surface area (which determines the material needed), you must square this scaling factor: $2.5^2 = 6.25$. The smaller container uses 36 square inches of material, so the larger container requires: $36 \times 6.25 = 225$ square inches. Looking at the wrong answers: B) 90 square inches represents multiplying by 2.5 instead of squaring it—this would be correct for a linear measurement but not for area. C) 144 square inches comes from multiplying by 4 (perhaps confusing this with $2^2$), but our scaling factor is 2.5, not 2. D) 180 square inches results from multiplying by 5, which might come from adding 2.5 + 2.5, but scaling doesn't work through addition. The correct answer is A) 225 square inches. Remember this pattern: when similar 3D objects are scaled by a linear factor, their surface areas scale by the square of that factor, and their volumes scale by the cube. Always identify what type of measurement you're scaling—length, area, or volume—to apply the correct power of the scaling factor.
6. A company produces two similar cone-shaped containers. The larger container has a base diameter that is 1.5 times that of the smaller container. If it costs $2.40 to fill the smaller container with product, what would be the cost to fill the larger container with the same product at the same price per unit volume?
   Answer: $8.10 Since the containers are similar and the diameter ratio is 1.5:1, the linear scale factor is 1.5. The volume scale factor is (1.5)³ = 3.375. Therefore, the cost to fill the larger container is $2.40 × 3.375 = $8.10. Choice A uses the linear scale factor ($2.40 × 1.5). Choice B uses the area scale factor ($2.40 × 2.25). Choice D appears to use an incorrect calculation mixing ratios.
7. An engineer creates a 1:25 scale model of a bridge support beam. In the model, the beam can support a maximum load of 3.2 pounds before breaking. Assuming the materials have the same properties, what is the maximum load the actual beam can support?
   Answer: 2,000 pounds Load-bearing capacity scales with cross-sectional area, not volume. The linear scale factor from model to actual is 25, so the area scale factor is 25² = 625. The actual beam can support 3.2 × 625 = 2,000 pounds. Choice A uses only the linear scale factor (3.2 × 25). Choice C uses the volume scale factor (3.2 × 25³). Choice D uses an incorrect area calculation (3.2 × 500).
8. Two similar rectangular swimming pools have depths in the ratio 4:7. The shallower pool has a volume of 1,200 cubic meters and a surface area of 400 square meters. What is the surface area of the deeper pool?
   Answer: 1,225 square meters When you encounter problems about similar three-dimensional figures, remember that corresponding linear dimensions scale by the same ratio, areas scale by the square of that ratio, and volumes scale by the cube of that ratio. Since the pools are similar with depths in ratio 4:7, all linear dimensions (length, width, depth) follow this same 4:7 ratio. This means surface areas will be in the ratio $4^2:7^2 = 16:49$. Let's find the ratio multiplier: $\frac{49}{16} = 3.0625$ The deeper pool's surface area = $400 \times 3.0625 = 1,225$ square meters. We can verify this makes sense by checking volumes. Volume ratios should be $4^3:7^3 = 64:343$. The deeper pool's volume should be $1,200 \times \frac{343}{64} = 6,431.25$ cubic meters. With surface area 1,225 and the same length-to-width ratio as the shallow pool, this checks out. Looking at the wrong answers: B) 700 represents using the linear ratio (4:7) instead of the area ratio - a common error where students forget to square the ratio for areas. C) 980 might come from incorrectly applying $\frac{7}{4} = 1.75$ and calculating $400 \times 1.75 \times 1.4 = 980$. D) 1,575 could result from using $\frac{7^3}{4^2} = \frac{343}{16}$ - mixing up volume and area scaling rules. Key strategy: For similar figures, always remember the scaling rules: linear dimensions use the ratio as-is, areas use the ratio squared, and volumes use the ratio cubed. Write these relationships down first to avoid mixing them up.
9. A scale model of a car is built at 1:18 scale. If the model's windshield has an area of 2.5 square inches, what is the area of the actual car's windshield in square feet?
   Answer: 5.625 square feet When you encounter scale model problems, remember that scale factors affect linear dimensions differently than area or volume. A 1:18 scale means every linear dimension on the model is 1/18 the size of the actual object. For areas, you must square the scale factor. Since the model is 1/18 the linear size, its areas are $(1/18)^2 = 1/324$ the size of the actual areas. This means the actual windshield area is $324 \times 2.5 = 810$ square inches. Converting to square feet: since 1 foot = 12 inches, then 1 square foot = 144 square inches. Therefore: $810 ÷ 144 = 5.625$ square feet. Looking at the wrong answers: Choice A (4.5 square feet) results from incorrectly using the linear scale factor instead of squaring it, giving $18 \times 2.5 = 45$ square inches, then converting incorrectly. Choice B (6.25 square feet) comes from squaring 2.5 to get 6.25 but forgetting the scale conversion entirely. Choice C (3.125 square feet) appears to use half the correct scale factor, perhaps from confusion about the 1:18 ratio. The key study tip: always square the linear scale factor when working with areas, and cube it for volumes. Don't forget unit conversions at the end—square units require you to square the conversion factor (12 inches per foot becomes 144 square inches per square foot).
10. A spherical balloon's radius increases from 6 inches to 9 inches. If the original balloon contained 2.4 cubic feet of air, how much air does the expanded balloon contain?
    Answer: 8.1 cubic feet The radius scale factor is 9/6 = 1.5. Volume scales as the cube of the linear scale factor: (1.5)³ = 3.375. The expanded balloon contains 2.4 × 3.375 = 8.1 cubic feet. Choice A uses the linear scale factor (2.4 × 1.5). Choice B uses the area scale factor (2.4 × 2.25). Choice D uses an incorrect intermediate calculation.
11. A architect creates a scale model of a building where 1 inch represents 8 feet. If the model building has a volume of 125 cubic inches and a surface area of 150 square inches, what is the surface area of the actual building in square feet?
    Answer: 9,600 square feet Since the linear scale factor is 8, the area scale factor is 8² = 64. The actual building's surface area is 150 × 64 = 9,600 square feet. Choice B uses the linear scale factor incorrectly (150 × 8). Choice C incorrectly uses the volume scale factor (150 × 512). Choice D uses an incorrect calculation (8³ × 125 instead of working with surface area).
12. A scale model of a building is constructed with a scale factor of 1:200. If the actual building has a total floor area of 48,000 square feet, what is the total floor area of the scale model?
    Answer: 1.2 square feet When scaling areas, the scale factor must be squared. The linear scale factor is 1:200, so the area scale factor is (1/200)² = 1/40,000. The model's floor area is 48,000 ÷ 40,000 = 1.2 square feet. Choice A incorrectly uses the linear scale factor (48,000 ÷ 200). Choice C uses an incorrect calculation. Choice D applies the scale factor incorrectly.
13. A manufacturer produces similar spherical ball bearings in two sizes. The cost to produce the smaller bearing is $0.45, and its diameter is 8 mm. If the larger bearing has a diameter of 12 mm and the cost is proportional to the amount of material used, what is the cost to produce the larger bearing?
    Answer: $1.52 The material cost is proportional to volume. The diameter ratio is 12:8 = 1.5, so the volume ratio is (1.5)³ = 3.375. The cost for the larger bearing is $0.45 × 3.375 = $1.52. Choice A uses the linear scale factor ($0.45 × 1.5). Choice B uses the area scale factor ($0.45 × 2.25). Choice D appears to use an incorrect calculation.
14. Based on the information above, how much grass seed will be needed to cover Park B completely?
    Answer: 337.5 kg When you encounter problems involving similar geometric figures, remember that linear dimensions scale proportionally, but areas scale with the square of that proportion. This is a fundamental principle in geometry that applies to any similar shapes. Since these triangular parks are similar, you need to find the scale factor between them. The perimeter ratio gives you this: $\frac{360}{240} = 1.5$, meaning Park B is 1.5 times larger in all linear dimensions than Park A. However, grass seed coverage depends on area, not perimeter. When linear dimensions scale by a factor of 1.5, the area scales by $(1.5)^2 = 2.25$. Therefore, Park B requires $150 \times 2.25 = 337.5$ kg of grass seed. Looking at the wrong answers: Choice A (225 kg) comes from multiplying by 1.5 instead of 2.25 – this treats grass coverage as if it were a linear measurement like perimeter. Choice B (300 kg) doubles the original amount, which would only be correct if the area doubled. Choice C (506.25 kg) appears to use an incorrect scale factor calculation, possibly confusing the relationship between the parks. The key study tip here is to always remember the scaling rules: linear measurements scale by the factor, areas scale by the factor squared, and volumes scale by the factor cubed. When you see "coverage," "paint needed," or "material required," you're almost always dealing with area, so square that scale factor.
15. Two similar triangular plots of land have perimeters in the ratio 5:8. If the smaller plot requires 200 pounds of fertilizer for complete coverage, how many pounds of fertilizer does the larger plot require?
    Answer: 512 pounds When you encounter problems involving similar figures, remember that corresponding measurements scale at different rates. Linear measurements (like perimeter) scale directly with the ratio, while area measurements scale with the square of that ratio. Since these triangular plots are similar with perimeters in the ratio 5:8, all corresponding linear dimensions are in this same 5:8 ratio. However, fertilizer coverage depends on area, not perimeter. For similar figures, areas are in the ratio of the square of the linear ratio. The area ratio is therefore $\left(\frac{5}{8}\right)^2 = \frac{25}{64}$. This means if the smaller plot has area 25 units, the larger plot has area 64 units. Since the smaller plot requires 200 pounds of fertilizer, you can set up a proportion: $\frac{200 \text{ pounds}}{25 \text{ units}} = \frac{x \text{ pounds}}{64 \text{ units}}$ Solving: $x = \frac{200 \times 64}{25} = \frac{12,800}{25} = 512$ pounds. Looking at the wrong answers: A) 320 pounds incorrectly applies the linear ratio 5:8 directly to fertilizer ($200 \times \frac{8}{5} = 320$), forgetting that area scales quadratically. B) 800 pounds appears to double the linear scaling error. C) 128 pounds seems to apply an inverse relationship incorrectly. The key strategy here is recognizing that similar figures have area ratios equal to the square of their linear ratios. Always square the given ratio when dealing with area-related quantities like paint, fertilizer, or carpeting coverage.
16. A photographer enlarges a rectangular photo from 4 inches by 6 inches to 10 inches by 15 inches. If the original photo used 48 square millimeters of ink coverage, how much ink coverage will the enlarged photo require, assuming the same ink density?
    Answer: 300 square millimeters The linear scale factor is 10/4 = 2.5 (or 15/6 = 2.5). For area-based quantities like ink coverage, the scale factor is (2.5)² = 6.25. The enlarged photo requires 48 × 6.25 = 300 square millimeters. Choice A incorrectly uses the linear scale factor (48 × 2.5). Choice C uses an intermediate incorrect calculation. Choice D reflects a computational error in scaling.
17. Two similar rectangular swimming pools have a depth ratio of 2:3. The shallower pool has a volume of 1,800 cubic feet and loses 45 gallons per hour due to evaporation from its surface. How many gallons per hour does the deeper pool lose to evaporation?
    Answer: 101.25 gallons per hour When you encounter problems involving similar geometric figures, remember that corresponding linear dimensions scale by the same ratio, areas scale by the ratio squared, and volumes scale by the ratio cubed. Since these pools have a depth ratio of 2:3, and they're similar rectangles, all linear dimensions follow this ratio. The volume ratio is therefore $(2:3)^3 = 8:27$. If the shallower pool has 1,800 cubic feet, the deeper pool has $1,800 \times \frac{27}{8} = 6,075$ cubic feet. For evaporation, the key insight is that water loss depends on surface area, not volume. Since the pools are similar with a linear ratio of 2:3, their surface areas have a ratio of $(2:3)^2 = 4:9$. The deeper pool's surface area is $\frac{9}{4}$ times larger than the shallower pool's surface area. Therefore, the deeper pool loses $45 \times \frac{9}{4} = 101.25$ gallons per hour to evaporation. Choice A (67.5) incorrectly uses the linear ratio: $45 \times \frac{3}{2} = 67.5$. Choice B (90) doubles the original rate without considering the area relationship. Choice C (152) appears to use an incorrect ratio calculation, possibly confusing the volume and area scaling relationships. The correct answer is D) 101.25 gallons per hour. Remember: for similar figures, always identify whether the question asks about linear measurements (scale by ratio), areas (scale by ratio squared), or volumes (scale by ratio cubed). Evaporation involves surface area, so use the squared ratio.
18. A landscape designer creates a scale drawing where 1 inch represents 4 feet. A circular garden bed in the drawing has a circumference of 3.14 inches. If mulch costs $8 per square foot, what is the cost to cover the actual garden bed with mulch?
    Answer: $401.92 Scale drawing problems require you to work through three key steps: find the actual dimensions, calculate the required measurement (often area), then apply the unit cost. First, you need the actual circumference. Since 1 inch represents 4 feet, the actual circumference is $3.14 \times 4 = 12.56$ feet. Next, find the radius using $C = 2\pi r$, so $r = \frac{12.56}{2\pi} = \frac{12.56}{6.28} = 2$ feet. Then calculate the actual area: $A = \pi r^2 = 3.14 \times 2^2 = 12.56$ square feet. Finally, multiply by the cost per square foot: $12.56 \times 8 = \$100.48$. Wait - let me recalculate this more carefully. The drawing circumference is 3.14 inches, so the actual circumference is $3.14 \times 4 = 12.56$ feet. The radius is $r = \frac{12.56}{2\pi} = 2$ feet. The area is $A = \pi r^2 = 3.14 \times 4 = 12.56$ square feet. Cost is $12.56 \times 8 = \$100.48$. Actually, I need to be more precise with the scaling. The area scales by the square of the linear scale factor. The drawing area is $\pi \times 1^2 = 3.14$ square inches. Since 1 inch = 4 feet, 1 square inch = 16 square feet. So the actual area is $3.14 \times 16 = 50.24$ square feet, and the cost is $50.24 \times 8 = \$401.92$. Answer A is correct. Answer B ($100.48) uses the wrong area scaling. Answers C and D represent other calculation errors with the scale conversion. Remember: when scaling areas, square the linear scale factor - if length scales by 4, area scales by 16.
19. An architect creates a scale model where 2 centimeters represents 3 meters. If a room in the model has a floor area of 8 square centimeters, what is the actual floor area of the room in square meters?
    Answer: 18 square meters The scale is 2 cm : 3 m, so the linear scale factor from model to reality is 3/2 = 1.5 meters per cm. For areas, we square this factor: (1.5)² = 2.25. The actual area is 8 × 2.25 = 18 square meters. Choice A incorrectly uses the linear scale factor (8 × 1.5). Choice C uses the cube of the scale factor. Choice D uses an incorrect ratio interpretation.
20. If the regular size contains 350 mL of product, approximately how much product does the family size contain?
    Answer: 980 mL When you encounter problems about similar shapes with different sizes, you're dealing with scaling relationships. Since the family size can maintains the same proportional shape as the regular size, this is a volume scaling problem where you need to find the scale factor. First, find the scale factor by comparing the diameters: $\frac{9.1 \text{ cm}}{6.5 \text{ cm}} = 1.4$. Since volume scales with the cube of the linear scale factor, the volume ratio is $1.4^3 = 2.744$. Therefore, the family size volume is $350 \text{ mL} \times 2.744 = 960.4 \text{ mL}$, which rounds to approximately 980 mL. Looking at the wrong answers: Choice A (490 mL) represents using just the scale factor itself ($350 \times 1.4 = 490$), forgetting that volume requires cubing the linear scale factor. Choice B (686 mL) comes from squaring the scale factor ($350 \times 1.4^2 = 686$), which would be correct for area scaling but not volume. Choice C (1,225 mL) appears to use an incorrect scale factor calculation, possibly confusing diameter relationships. The correct answer is D) 980 mL. Remember this key principle: when similar 3D shapes are scaled, volume changes by the cube of the linear scale factor. Linear measurements scale by the factor itself, areas scale by the factor squared, and volumes scale by the factor cubed. Always identify what type of measurement you're working with before applying the scale factor.

Start this quiz