Geometry - Scale Drawings

A map is drawn to a scale of $1:50,000$. If the distance between two villages on the map is $7.5 \text{ cm}$, calculate the actual distance in $\text{km}$.

Step 1: Multiply the map distance by the scale factor. $\text{Actual distance in cm} = 7.5 \times 50,000 = 375,000 \text{ cm}$ Step 2: Convert centimeters to kilometers. Since $1 \text{ km} = 100,000 \text{ cm}$: $\text{Actual distance in km} = \frac{375,000}{100,000} = 3.75 \text{ km}$

To find the real-world distance, we multiply the map distance by the scale ratio. Finally, we divide by $100,000$ to convert from centimeters to kilometers.

A forest has an area of $12 \text{ km}^2$. On a map, the area of the forest is $3 \text{ cm}^2$. Find the scale of the map in the form $1:n$.

Step 1: Express the area ratio. $\text{Area Scale} = 3 \text{ cm}^2 : 12 \text{ km}^2$ Step 2: Simplify the area ratio to $1 \text{ unit}^2$. $1 \text{ cm}^2 : 4 \text{ km}^2$ Step 3: Find the linear scale by taking the square root of both sides. $\sqrt{1 \text{ cm}^2} : \sqrt{4 \text{ km}^2}$ $1 \text{ cm} : 2 \text{ km}$ Step 4: Convert to the same units ($1 \text{ km} = 100,000 \text{ cm}$). $1 : 2 \times 100,000$ $1 : 200,000$

Since map area relates to actual area by the square of the linear scale factor, we must take the square root of the simplified area ratio to find the linear scale, then ensure units are identical for the final ratio.

Two points on a blueprint are $15.8 \text{ mm}$ apart. A third point is $6.3 \text{ mm}$ from the first. Calculate the difference in their drawing lengths and then find the actual difference in meters if the scale is $1:500$.

Calculate the difference in drawing length: $\begin{array}{r} 15.8 \\ - 6.3 \\ \hline 9.5 \end{array} \text{ mm}$ Now, convert to actual distance: $\text{Actual difference in mm} = 9.5 \times 500 = 4750 \text{ mm}$ Convert to meters ($1 \text{ m} = 1000 \text{ mm}$): $\text{Actual difference in m} = \frac{4750}{1000} = 4.75 \text{ m}$

First, subtract the blueprint measurements to find the difference in drawing length. Multiply this by the scale factor ($500$) and convert to meters by dividing by $1000$.

$\text{Scale} = 1:500$ $\text{Drawing Length} = \frac{\text{Actual Length}}{\text{Scale Factor}}$ $\text{Length} = \frac{80 \text{ m}}{500} = 0.16 \text{ m}$ $\text{Width} = \frac{50 \text{ m}}{500} = 0.1 \text{ m}$ Converting to $\text{cm}$: $\text{Length in cm} = 0.16 \times 100 = 16 \text{ cm}$ $\text{Width in cm} = 0.1 \times 100 = 10 \text{ cm}$

Divide the real-world dimensions by the scale factor. Then, multiply by 100 to convert meters to centimeters for the final drawing units.

$\text{Linear Scale Factor } (n) = 20,000$ $\text{Area Scale Factor} = n^2 = (20,000)^2 = 400,000,000$ $\text{Actual Area} = 4.5 \text{ cm}^2 \times 400,000,000 = 1,800,000,000 \text{ cm}^2$ To convert $\text{cm}^2$ to $\text{m}^2$, divide by $100^2 = 10,000$: $\text{Actual Area in } m^2 = \frac{1,800,000,000}{10,000} = 180,000 \text{ m}^2$

First find the area scale factor by squaring the linear scale factor. Multiply the map area by this factor, then convert square centimeters to square meters by dividing by 10,000.