Certainly! Let's solve the problem step-by-step:
Step 1: Understanding the Scale
The scale of the map is given as 1:100. This means that 1 cm on the map represents 100 cm in actual size.
Step 2: Calculating the Actual Width
The width of the field on the map is given as 8 cm. To find the actual width of the field, we use the scale:
[tex]\[ \text{Actual width} = \text{Map width} \times \text{Scale factor} \][/tex]
[tex]\[ \text{Actual width} = 8 \, \text{cm} \times 100 \][/tex]
[tex]\[ \text{Actual width} = 800 \, \text{cm} \][/tex]
Step 3: Calculating the Area Conversion Factor
The linear scale is 1:100. However, when we are dealing with areas, the scale factor must be squared. Therefore, the area scale factor is:
[tex]\[ \text{Area scale factor} = (\text{Scale factor})^2 \][/tex]
[tex]\[ \text{Area scale factor} = (100)^2 \][/tex]
[tex]\[ \text{Area scale factor} = 10000 \][/tex]
Step 4: Calculating the Actual Area in [tex]$cm^2$[/tex]
The area of the field on the map is given as [tex]\(88 \, cm^2\)[/tex]. To find the actual area, we multiply by the area scale factor:
[tex]\[ \text{Actual area in } cm^2 = \text{Map area} \times \text{Area scale factor} \][/tex]
[tex]\[ \text{Actual area in } cm^2 = 88 \, cm^2 \times 10000 \][/tex]
[tex]\[ \text{Actual area in } cm^2 = 880000 \, cm^2 \][/tex]
Step 5: Converting [tex]$cm^2$[/tex] to [tex]$m^2$[/tex]
Since 1 [tex]$m^2$[/tex] is equal to 10,000 [tex]$cm^2$[/tex], we need to convert the actual area from [tex]$cm^2$[/tex] to [tex]$m^2$[/tex]:
[tex]\[ \text{Actual area in } m^2 = \frac{\text{Actual area in } cm^2}{10000} \][/tex]
[tex]\[ \text{Actual area in } m^2 = \frac{880000 \, cm^2}{10000} \][/tex]
[tex]\[ \text{Actual area in } m^2 = 88 \, m^2 \][/tex]
Thus, the area of the actual field is [tex]\(88 \, m^2\)[/tex].
