Dimension Changes & Scaling - Math 1

1. A scale model of a building is constructed at a ratio of 1:200. If the actual building has a rectangular floor plan with an area of 8,000 square meters, what is the area of the floor plan in the scale model?
   Answer: 0.2 square meters When dealing with scale models, you're working with proportional relationships where areas scale differently than linear dimensions. The key insight is that while linear measurements scale by the given ratio, areas scale by the square of that ratio. Given a 1:200 scale ratio, this means every linear dimension in the model is $\frac{1}{200}$ the size of the actual building. However, since area involves two dimensions (length × width), the area ratio becomes $\left(\frac{1}{200}\right)^2 = \frac{1}{40,000}$. To find the model's floor area, multiply the actual area by this ratio: $8,000 \text{ m}^2 \times \frac{1}{40,000} = \frac{8,000}{40,000} = 0.2 \text{ m}^2$. This confirms answer B is correct. Looking at the wrong answers: A) 0.4 square meters results from incorrectly using $\frac{1}{20,000}$ as the area ratio, possibly from miscalculating $\left(\frac{1}{200}\right)^2$. C) 40 square meters comes from dividing by 200 instead of 40,000, treating area as if it scales linearly like a single dimension. D) 200 square meters results from dividing by 40 instead of 40,000, another linear scaling error. Remember this pattern: linear scale ratios must be squared for area calculations and cubed for volume calculations. When you see scale model problems involving area, always square the linear ratio first—this is one of the most common mistakes on geometry problems involving similar figures.
2. A cone-shaped container has base radius 5 cm and height 12 cm. A similar cone is created where the height is reduced by 40% but the volume must remain exactly the same. What is the radius of the new cone?
   Answer: 6.45 cm When you encounter problems involving similar shapes with changing dimensions and constant volume, you need to understand how scaling affects different measurements. Volume scales with the cube of linear dimensions, which is the key insight here. The original cone has radius 5 cm, height 12 cm, and volume $V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (5^2)(12) = 100\pi$ cubic cm. The new cone has height reduced by 40%, so the new height is $12 \times 0.6 = 7.2$ cm, and we need to find the radius that maintains the same volume. Setting up the volume equation for the new cone: $100\pi = \frac{1}{3}\pi r^2 (7.2)$. Solving for r: $300 = r^2 (7.2)$, so $r^2 = \frac{300}{7.2} = 41.67$, giving us $r = 6.45$ cm. Looking at the wrong answers: A) 7.22 cm likely comes from incorrectly assuming the radius increases by the same percentage that height decreases (40%), giving $5 \times 1.4 ≈ 7$. C) 8.33 cm might result from setting up the proportion incorrectly, perhaps using $\frac{5}{12} = \frac{r}{7.2}$ and solving linearly. D) 5.77 cm could come from a calculation error or misunderstanding the relationship between the dimensions. The correct answer is B) 6.45 cm. Remember: when volume stays constant but one dimension changes, use the volume formula to set up an equation. Don't assume simple proportional relationships—volume involves cubed terms, so the math requires careful algebra, not just ratios.
3. A company manufactures cylindrical cans. When they increase the diameter by 20% while keeping the height the same, by approximately what percentage does the volume increase?
   Answer: The volume increases by approximately 44% When diameter increases by 20%, the radius also increases by 20%, so the new radius is 1.2 times the original. Since volume $V = \pi r^2 h$, the new volume is $(1.2)^2 = 1.44$ times the original. This represents a 44% increase. Choice A incorrectly applies the linear increase. Choice B represents an incorrect calculation. Choice D might result from incorrectly using diameter squared.
4. A conical water tank has its height doubled while the radius remains the same. How does this change affect the volume of the tank?
   Answer: The volume increases to exactly double the original amount When you encounter questions about how changing dimensions affects volume, you need to understand how each dimension contributes to the volume formula. For a cone, the volume formula is $V = \frac{1}{3}\pi r^2 h$. Let's work through what happens when height doubles while radius stays constant. If the original volume is $V_1 = \frac{1}{3}\pi r^2 h$, then the new volume with doubled height is $V_2 = \frac{1}{3}\pi r^2 (2h) = 2 \cdot \frac{1}{3}\pi r^2 h = 2V_1$. The volume becomes exactly twice the original amount, making B correct. Here's why the other answers are wrong: A suggests the volume triples, which would happen if you had a factor of 3 somewhere in the calculation, but doubling height only introduces a factor of 2. C claims the volume quadruples (increases by a factor of 4), which would occur if you doubled the radius instead, since radius is squared in the formula ($2^2 = 4$). D suggests the volume increases eightfold, which would happen if you doubled all three dimensions of a rectangular solid ($2 \times 2 \times 2 = 8$), but that's not the case here. Remember this pattern: when one dimension changes in a volume formula, the volume changes by that same factor. However, when a squared dimension (like radius) changes, the volume changes by the square of that factor. Always identify which variable is changing and how it appears in the formula.
5. A circular pizza has a radius of 8 inches. If the pizza parlor creates a larger pizza with 2.25 times the area of the original, what is the radius of the larger pizza?
   Answer: The larger pizza has radius 12.0 inches If the area increases by a factor of 2.25, the radius increases by a factor of $\sqrt{2.25} = 1.5$. The new radius is $8 \times 1.5 = 12$ inches. Choice B incorrectly calculates $8 \times 1.6875$. Choice C incorrectly multiplies the radius by the area factor 2.25. Choice D incorrectly uses $8 \times 2.25$.
6. A cylindrical water tank has its radius increased by a factor of 3 while keeping the height constant. By what factor does the volume of the tank increase?
   Answer: 9 times the original volume The volume of a cylinder is $V = \pi r^2 h$. When the radius increases by a factor of 3, the new volume is $V_{new} = \pi (3r)^2 h = \pi \cdot 9r^2 \cdot h = 9V_{original}$. Choice A incorrectly applies the linear scaling factor. Choice B incorrectly multiplies by 2×3. Choice D incorrectly cubes the scaling factor as if all three dimensions changed.
7. A triangular piece of fabric has an area of 150 square inches. If a similar triangular piece is made with sides that are 2.5 times as long, what will be the area of the larger piece?
   Answer: 937.5 square inches of fabric area When you encounter problems about similar shapes, remember that their areas don't scale the same way as their side lengths. This is a key concept in geometry that often trips students up. Since the triangles are similar, all corresponding sides of the larger triangle are 2.5 times the length of the original triangle's sides. However, area is a two-dimensional measurement, so it scales by the square of the scaling factor. The area scaling factor is $2.5^2 = 6.25$. Therefore, the larger triangle's area is $150 \times 6.25 = 937.5$ square inches. Let's examine why the other answers are wrong: Choice A (375) represents multiplying the original area by 2.5 instead of $2.5^2$. This is the most common error—applying the linear scaling factor directly to area. Choice B (625) comes from incorrectly calculating $150 \times 2.5^2$ but using $2.5^2 = 4.17$ (rounded incorrectly) or making an arithmetic error. Choice C (1,562.5) suggests using $2.5^3 = 15.625$ as the scaling factor, which would be appropriate for volume, not area. This shows confusion between two-dimensional and three-dimensional scaling. Choice D (937.5) correctly applies the area scaling principle: $150 \times 2.5^2 = 150 \times 6.25 = 937.5$. Study tip: Remember the scaling rules: linear measurements scale by the factor, areas scale by the factor squared, and volumes scale by the factor cubed. Always square the scaling factor for area problems involving similar shapes.
8. A wooden block in the shape of a rectangular prism has dimensions 6 cm × 4 cm × 10 cm. If each dimension is increased by the same factor k, and the resulting volume is 1,920 cubic cm, what is the value of k?
   Answer: The scaling factor k equals 2.0 The original volume is $6 \times 4 \times 10 = 240$ cubic cm. The new volume is $k^3 \times 240 = 1920$, so $k^3 = 8$, which means $k = 2$. Choice B would give $k^3 = 13.824$. Choice C would give $k^3 = 64$. Choice D would give $k^3 = 512$.
9. A spherical balloon has radius 6 inches. When heated, its radius increases to 8 inches while maintaining its spherical shape. By what factor does the volume of air inside increase, and what does this suggest about the relationship between linear and volumetric scaling?
   Answer: Factor of 2.37; volume scales with the cube of linear dimensions Volume scales as radius cubed. Original volume: $\frac{4}{3}\pi(6)^3 = 288\pi$. New volume: $\frac{4}{3}\pi(8)^3 = \frac{4}{3}\pi(512) = \frac{2048\pi}{3}$. Ratio: $\frac{512}{216} = \frac{64}{27} \approx 2.37$. This confirms $V \propto r^3$. Choice B incorrectly uses linear scaling. Choice C confuses with area scaling. Choice D invents incorrect power relationship.
10. A spherical balloon has its radius doubled. If the original balloon used 50 grams of rubber material, approximately how much rubber material will the larger balloon require, assuming the thickness remains constant?
    Answer: 200 grams of rubber material needed When you encounter problems involving scaling of three-dimensional objects, remember that surface area scales with the square of the scaling factor, while volume scales with the cube. Since the balloon's thickness remains constant, the amount of rubber material needed depends on the surface area of the sphere. The surface area of a sphere is $4\pi r^2$, where $r$ is the radius. When the radius doubles from $r$ to $2r$, the new surface area becomes $4\pi(2r)^2 = 4\pi \cdot 4r^2 = 16\pi r^2$. Comparing this to the original surface area of $4\pi r^2$, we see the new balloon has exactly 4 times the surface area. Since the material needed is proportional to surface area, the larger balloon requires $50 \times 4 = 200$ grams of rubber material. Looking at the wrong answers: Choice A (100 grams) represents a common error of thinking the material scales linearly with radius (doubling radius = doubling material). Choice B (150 grams) might come from incorrectly adding 2 times the original amount to the base amount. Choice C (400 grams) results from mistakenly using volume scaling—since volume scales with the cube of the radius, this would give $50 \times 2^3 = 400$ grams, but volume isn't relevant when thickness stays constant. The correct answer is D (200 grams). Study tip: Remember the scaling rules: length scales by the factor, area by the factor squared, and volume by the factor cubed. Always identify which geometric property determines the quantity you're calculating.
11. A scale model of a building uses a ratio of 1:200. If the model has a floor area of 0.8 square meters, and the architect wants to create a new model at 1:150 scale with the same floor area, what must be the floor area of the actual building represented by the new model?
    Answer: 18,000 square meters Original model at 1:200 represents actual area of $0.8 \times 200^2 = 32,000$ m². For new 1:150 model with same 0.8 m² area, the actual area is $0.8 \times 150^2 = 0.8 \times 22,500 = 18,000$ m². Choice A uses incorrect ratio calculation. Choice B gives the original building area. Choice C uses $\sqrt{200 \times 150}$ incorrectly.
12. A cube-shaped storage container is redesigned so that each edge length is reduced to 80% of its original size. What percentage of the original volume does the new container have?
    Answer: 51.2% of the original container volume When you encounter problems involving scaling three-dimensional objects, remember that volume changes differently than linear dimensions because volume involves three dimensions multiplied together. Let's work through this step by step. If the original cube has edge length $s$, its volume is $s^3$. When each edge is reduced to 80% of its original size, the new edge length becomes $0.8s$. The new volume is therefore $(0.8s)^3 = 0.8^3 \times s^3 = 0.512 \times s^3$. This means the new container has 51.2% of the original volume, confirming answer B. Looking at the wrong answers: Answer D (80.0%) represents a common misconception—thinking that reducing each dimension by 20% reduces the volume by 20%. This ignores the fact that volume is three-dimensional. Answer A (64.0%) comes from incorrectly calculating $0.8^2 = 0.64$, treating this as if it were a two-dimensional area problem rather than three-dimensional volume. Answer C (68.8%) doesn't correspond to any straightforward calculation error but might arise from confused arithmetic. The key insight is that when you scale a three-dimensional object, you must raise the scaling factor to the third power to find the volume change. A reduction to 80% of linear dimensions means $0.8^3 = 0.512$ or 51.2% of the original volume. Study tip: For scaling problems, always match the exponent to the dimension—linear measurements use the scaling factor to the first power, areas to the second power, and volumes to the third power.
13. A triangular piece of fabric has area 24 square inches. A designer creates a similar triangle by increasing one side by 50% and decreasing an adjacent side by 20%. If the angle between these sides remains constant, what is the area of the new triangle?
    Answer: 28.8 square inches For a triangle with two sides $a, b$ and included angle $C$, area = $\frac{1}{2}ab\sin C$. New sides: $1.5a$ and $0.8b$. New area = $\frac{1}{2}(1.5a)(0.8b)\sin C = 1.2 \times \frac{1}{2}ab\sin C = 1.2 \times 24 = 28.8$ sq in. Choice B uses $1.5 - 0.2 = 1.3$ incorrectly. Choice C uses $1.5 \times 24$. Choice D uses $0.8 \times 0.75 \times 24$.
14. A cube-shaped container holds 64 cubic feet of liquid. The container is redesigned as a rectangular box with the same volume, where the length is doubled and the width is tripled compared to the original cube's edge. What is the height of the new container?
    Answer: 0.67 feet Original cube edge: $\sqrt[3]{64} = 4$ feet. New dimensions: length = $2 \times 4 = 8$ ft, width = $3 \times 4 = 12$ ft. Volume constraint: $8 \times 12 \times h = 64$, so $96h = 64$, giving $h = \frac{64}{96} = \frac{2}{3} \approx 0.67$ ft. Choice A uses $\frac{4}{3}$. Choice B uses $\frac{8}{3}$. Choice D uses $\frac{16}{3}$.
15. A rectangular garden is enlarged so that each dimension is multiplied by 1.5. If the original garden required 12 bags of fertilizer to cover completely, how many bags will the enlarged garden require?
    Answer: 27 bags of fertilizer for complete coverage When both dimensions are multiplied by 1.5, the area is multiplied by $1.5^2 = 2.25$. Therefore, the enlarged garden needs $12 \times 2.25 = 27$ bags. Choice A incorrectly multiplies by the linear scale factor 1.5. Choice B uses an incorrect factor of 1.75. Choice D incorrectly cubes the scale factor.
16. A square garden plot has an area of 144 square meters. If the gardener wants to create a similar square plot with an area of 400 square meters, by what factor must each side length be increased?
    Answer: Each side length must increase by factor 1.67 The ratio of areas is $\frac{400}{144} = \frac{25}{9} \approx 2.78$. Since area ratios equal the square of length ratios, the length ratio is $\sqrt{\frac{25}{9}} = \frac{5}{3} \approx 1.67$. Choice B incorrectly uses the area ratio as the length ratio. Choice C uses an incorrect calculation. Choice D incorrectly uses $\frac{400}{144} \times \frac{400}{144}$.
17. A manufacturing company produces rectangular metal sheets. When they increase both length and width by the same percentage $p$, the area increases by 44%. However, due to material constraints, they can only increase the length by $p$ while keeping width constant. What percentage increase in area will this produce?
    Answer: 20% This problem tests your understanding of how percentage changes affect area calculations when dimensions change independently versus together. Let's work with the given information systematically. If both length and width increase by the same percentage $p$, the new area becomes $(1 + \frac{p}{100})^2$ times the original area. Since the area increases by 44%, we have: $(1 + \frac{p}{100})^2 = 1.44$ Taking the square root of both sides: $1 + \frac{p}{100} = 1.2$ This gives us $p = 20\%$. Now, when only the length increases by 20% while width stays constant, the new area is simply $1.2$ times the original area, representing a 20% increase. Looking at the wrong answers: Choice A (15%) might tempt you if you incorrectly assumed the relationship would be less than proportional. Choice C (25%) could result from mistakenly thinking that since 44% came from squaring the effect, you need to add something extra. Choice D (12%) might come from incorrectly trying to take half of 44% ÷ 2, thinking that changing one dimension instead of two would halve the effect. The key insight is recognizing that when both dimensions change by percentage $p$, the area changes by $(1 + \frac{p}{100})^2 - 1$, but when only one dimension changes by $p$, the area simply changes by $p$. Always identify what's changing and work backwards from compound effects to find the individual percentage increase.
18. A rectangular swimming pool cover has length 24 feet and width 16 feet. If both dimensions are reduced by the same percentage, and the resulting area is 75% of the original area, by what percentage was each dimension reduced?
    Answer: Each dimension was reduced by exactly 13.4% If the area becomes 75% of the original, the area ratio is 0.75. The linear scale factor is $\sqrt{0.75} \approx 0.866$. This means each dimension is reduced to 86.6% of its original size, representing a reduction of $100\% - 86.6\% = 13.4\%$. Choice B incorrectly uses the area reduction percentage. Choice C incorrectly calculates $100\% - 75\% + 12.5\%$. Choice D incorrectly assumes linear relationship between area and dimension reductions.
19. A scale model airplane has a wingspan of 15 inches and requires 0.8 square feet of material to cover its wings. If a larger model is built with a wingspan of 25 inches, approximately how much material will be needed for its wings?
    Answer: Approximately 2.2 square feet of wing material The scale factor is $\frac{25}{15} = \frac{5}{3} \approx 1.667$. The wing area scales as the square of this factor: $\left(\frac{5}{3}\right)^2 = \frac{25}{9} \approx 2.78$. The new material needed is $0.8 \times 2.78 \approx 2.2$ square feet. Choice A incorrectly uses approximately the linear scale factor. Choice B uses an incorrect intermediate calculation. Choice D incorrectly calculates the area scaling.

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