Answer :
Let's solve the problem step-by-step for each part given the scale of the map which is [tex]\(1 \text{ cm} : 8 \text{ km}\)[/tex].
### Understanding the Scale
The scale [tex]\(1 \text{ cm} : 8 \text{ km}\)[/tex] means that 1 cm on the map corresponds to 8 km in reality. To find the area representation on the map, we'll follow these steps:
1. Calculate the side length [tex]\( \text{in km}^2 \)[/tex] of the given area.
2. Convert that side length from kilometers to centimeters.
3. Calculate the area in square centimeters.
### Given Areas
We are asked to find the area on the map for each of the following real-world areas:
[tex]\[ (a) \ 64 \text{ km}^2, \quad (b) \ 128 \text{ km}^2, \quad (c) \ 320 \text{ km}^2, \quad (d) \ 1600 \text{ km}^2 \][/tex]
### Step-by-Step Solution
#### (a) [tex]\(64 \text{ km}^2\)[/tex]
1. Find the side length (km):
[tex]\[ \sqrt{64 \text{ km}^2} = 8 \text{ km} \][/tex]
2. Convert side length to cm:
[tex]\[ \frac{8 \text{ km}}{8 \text{ km/cm}} = 1 \text{ cm} \][/tex]
3. Calculate the area on the map (cm[tex]\(^2\)[/tex]):
[tex]\[ (1 \text{ cm})^2 = 1 \text{ cm}^2 \][/tex]
So, the area on the map that represents [tex]\(64 \text{ km}^2\)[/tex] is [tex]\( 1 \text{ cm}^2 \)[/tex].
#### (b) [tex]\(128 \text{ km}^2\)[/tex]
1. Find the side length (km):
[tex]\[ \sqrt{128 \text{ km}^2} = 11.313708498984761 \text{ km} \][/tex]
2. Convert side length to cm:
[tex]\[ \frac{11.313708498984761 \text{ km}}{8 \text{ km/cm}} = 1.4142135623730951 \text{ cm} \][/tex]
3. Calculate the area on the map (cm[tex]\(^2\)[/tex]):
[tex]\[ (1.4142135623730951 \text{ cm})^2 = 2.0000000000000004 \text{ cm}^2 \][/tex]
So, the area on the map that represents [tex]\(128 \text{ km}^2\)[/tex] is approximately [tex]\( 2.000 \text{ cm}^2 \)[/tex].
#### (c) [tex]\(320 \text{ km}^2\)[/tex]
1. Find the side length (km):
[tex]\[ \sqrt{320 \text{ km}^2} = 17.88854381999832 \text{ km} \][/tex]
2. Convert side length to cm:
[tex]\[ \frac{17.88854381999832 \text{ km}}{8 \text{ km/cm}} = 2.23606797749979 \text{ cm} \][/tex]
3. Calculate the area on the map (cm[tex]\(^2\)[/tex]):
[tex]\[ (2.23606797749979 \text{ cm})^2 = 5.000000000000001 \text{ cm}^2 \][/tex]
So, the area on the map that represents [tex]\(320 \text{ km}^2\)[/tex] is approximately [tex]\( 5.000 \text{ cm}^2 \)[/tex].
#### (d) [tex]\(1600 \text{ km}^2\)[/tex]
1. Find the side length (km):
[tex]\[ \sqrt{1600 \text{ km}^2} = 40 \text{ km} \][/tex]
2. Convert side length to cm:
[tex]\[ \frac{40 \text{ km}}{8 \text{ km/cm}} = 5 \text{ cm} \][/tex]
3. Calculate the area on the map (cm[tex]\(^2\)[/tex]):
[tex]\[ (5 \text{ cm})^2 = 25 \text{ cm}^2 \][/tex]
So, the area on the map that represents [tex]\(1600 \text{ km}^2\)[/tex] is [tex]\( 25 \text{ cm}^2 \)[/tex].
### Summary
The areas on the map representing the given real-world areas are:
[tex]\[ (a) \ 64 \text{ km}^2 \rightarrow 1 \text{ cm}^2 \][/tex]
[tex]\[ (b) \ 128 \text{ km}^2 \rightarrow 2 \text{ cm}^2 \][/tex]
[tex]\[ (c) \ 320 \text{ km}^2 \rightarrow 5 \text{ cm}^2 \][/tex]
[tex]\[ (d) \ 1600 \text{ km}^2 \rightarrow 25 \text{ cm}^2 \][/tex]
### Understanding the Scale
The scale [tex]\(1 \text{ cm} : 8 \text{ km}\)[/tex] means that 1 cm on the map corresponds to 8 km in reality. To find the area representation on the map, we'll follow these steps:
1. Calculate the side length [tex]\( \text{in km}^2 \)[/tex] of the given area.
2. Convert that side length from kilometers to centimeters.
3. Calculate the area in square centimeters.
### Given Areas
We are asked to find the area on the map for each of the following real-world areas:
[tex]\[ (a) \ 64 \text{ km}^2, \quad (b) \ 128 \text{ km}^2, \quad (c) \ 320 \text{ km}^2, \quad (d) \ 1600 \text{ km}^2 \][/tex]
### Step-by-Step Solution
#### (a) [tex]\(64 \text{ km}^2\)[/tex]
1. Find the side length (km):
[tex]\[ \sqrt{64 \text{ km}^2} = 8 \text{ km} \][/tex]
2. Convert side length to cm:
[tex]\[ \frac{8 \text{ km}}{8 \text{ km/cm}} = 1 \text{ cm} \][/tex]
3. Calculate the area on the map (cm[tex]\(^2\)[/tex]):
[tex]\[ (1 \text{ cm})^2 = 1 \text{ cm}^2 \][/tex]
So, the area on the map that represents [tex]\(64 \text{ km}^2\)[/tex] is [tex]\( 1 \text{ cm}^2 \)[/tex].
#### (b) [tex]\(128 \text{ km}^2\)[/tex]
1. Find the side length (km):
[tex]\[ \sqrt{128 \text{ km}^2} = 11.313708498984761 \text{ km} \][/tex]
2. Convert side length to cm:
[tex]\[ \frac{11.313708498984761 \text{ km}}{8 \text{ km/cm}} = 1.4142135623730951 \text{ cm} \][/tex]
3. Calculate the area on the map (cm[tex]\(^2\)[/tex]):
[tex]\[ (1.4142135623730951 \text{ cm})^2 = 2.0000000000000004 \text{ cm}^2 \][/tex]
So, the area on the map that represents [tex]\(128 \text{ km}^2\)[/tex] is approximately [tex]\( 2.000 \text{ cm}^2 \)[/tex].
#### (c) [tex]\(320 \text{ km}^2\)[/tex]
1. Find the side length (km):
[tex]\[ \sqrt{320 \text{ km}^2} = 17.88854381999832 \text{ km} \][/tex]
2. Convert side length to cm:
[tex]\[ \frac{17.88854381999832 \text{ km}}{8 \text{ km/cm}} = 2.23606797749979 \text{ cm} \][/tex]
3. Calculate the area on the map (cm[tex]\(^2\)[/tex]):
[tex]\[ (2.23606797749979 \text{ cm})^2 = 5.000000000000001 \text{ cm}^2 \][/tex]
So, the area on the map that represents [tex]\(320 \text{ km}^2\)[/tex] is approximately [tex]\( 5.000 \text{ cm}^2 \)[/tex].
#### (d) [tex]\(1600 \text{ km}^2\)[/tex]
1. Find the side length (km):
[tex]\[ \sqrt{1600 \text{ km}^2} = 40 \text{ km} \][/tex]
2. Convert side length to cm:
[tex]\[ \frac{40 \text{ km}}{8 \text{ km/cm}} = 5 \text{ cm} \][/tex]
3. Calculate the area on the map (cm[tex]\(^2\)[/tex]):
[tex]\[ (5 \text{ cm})^2 = 25 \text{ cm}^2 \][/tex]
So, the area on the map that represents [tex]\(1600 \text{ km}^2\)[/tex] is [tex]\( 25 \text{ cm}^2 \)[/tex].
### Summary
The areas on the map representing the given real-world areas are:
[tex]\[ (a) \ 64 \text{ km}^2 \rightarrow 1 \text{ cm}^2 \][/tex]
[tex]\[ (b) \ 128 \text{ km}^2 \rightarrow 2 \text{ cm}^2 \][/tex]
[tex]\[ (c) \ 320 \text{ km}^2 \rightarrow 5 \text{ cm}^2 \][/tex]
[tex]\[ (d) \ 1600 \text{ km}^2 \rightarrow 25 \text{ cm}^2 \][/tex]
