Understand the map scale.

Here are the solutions for Exercise 49: a) What distance on land will be represented by a length of $5 \text{ cm}$ on a map of scale $1:50\,000$? Step 1: Understand the map scale. A scale of $1:50\,000$ means that $1 \text{ unit on the map represents } 50\,000 \text{ units on the land}$. So, $1 \text{ cm on the map represents } 50\,000 \text{ cm on the land}$. Step 2: Calculate the land distance for $5 \text{ cm}$ on the map. $$ \text{Land distance} = \text{Map length} \times \text{Scale denominator} $$ $$ \text{Land distance} = 5 \text{ cm} \times 50\,000 $$ $$ \text{Land distance} = 250\,000 \text{ cm} $$ Step 3: Convert the land distance to a more practical unit (kilometers). Since $1 \text{ m} = 100 \text{ cm}$ and $1 \text{ km} = 1\,000 \text{ m}$, then $1 \text{ km} = 100\,000 \text{ cm}$. $$ \text{Land distance in km} = \frac{250\,000 \text{ cm}}{100\,000 \text{ cm/km}} $$ $$ \text{Land distance in km} = 2.5 \text{ km} $$ The distance on land will be represented by a length of $5 \text{ cm}$ on the map is $\boxed{\text{2.5 km}}$. b) What length will a distance of $3 \text{ km}$ represent on a map of scale $1:10\,000$? Step 1: Understand the map scale. A scale of $1:10\,000$ means that $1 \text{ unit on the map represents } 10\,000 \text{ units on the land}$. So, $1 \text{ cm on the map represents } 10\,000 \text{ cm on the land}$. Step 2: Convert the land distance to centimeters. $$ 3 \text{ km} = 3 \times 1\,000 \text{ m} = 3\,000 \text{ m} $$ $$ 3\,000 \text{ m} = 3\,000 \times 100 \text{ cm} = 300\,000 \text{ cm} $$ Step 3: Calculate the map length. $$ \text{Map length} = \frac{\text{Land distance}}{\text{Scale denominator}} $$ $$ \text{Map length} = \frac{300\,000 \text{ cm}}{10\,000} $$ $$ \text{Map length} = 30 \text{ cm} $$ A distance of $3 \text{ km}$ will represent a length of $\boxed{\text{30 cm}}$ on the map. Last free one today — make it count tomorrow, or type /upgrade for unlimited.