A model of a planned city is made where ans actual distance 10 miles is represented as 5 meters. If the area of the city is 60 square, and 1 mile = 1600 meters, what is the area of the model in square miles ?

Options

A. `5.9 xx 10^(4)` square miles
B. `5.9 xx 10^(-5)` square miles
C. `5.9 xx 10^(-6)` square miles
D. `5.9 xx 10^(-7)` square miles

Answer

To solve the problem step-by-step, we need to determine the area of the model city in square miles based on the given information.

### Step 1: Understand the Scale
We know that 10 miles in reality is represented as 5 meters in the model. Therefore, we can find the scale of the model:
\[ \text{Scale} = \frac{5 \text{ meters}}{10 \text{ miles}} = \frac{1}{2} \text{ meters per mile} \]
**Hint:** To find the scale, divide the model distance by the actual distance.

### Step 2: Calculate the Area of the Model
The area of the actual city is given as 60 square miles. To convert this area into square meters, we need to convert miles to meters. Since 1 mile = 1600 meters, we can express the area in square meters:
\[ \text{Area in square meters} = 60 \text{ miles}^2 \times (1600 \text{ meters/mile})^2 \]
Calculating this gives:
\[ = 60 \times 1600^2 \text{ square meters} \]
**Hint:** When converting area, remember to square the conversion factor.

### Step 3: Calculate the Area in the Model
Now, using the scale we found earlier, we can find the area of the model. Since 1 mile is represented as 0.5 meters, we can express the area in the model as:
\[ \text{Area of the model} = \text{Area of the city} \times \left(\frac{1 \text{ meter}}{0.5 \text{ meters}}\right)^2 \]
Calculating this gives:
\[ = 60 \times \left(\frac{1}{0.5}\right)^2 = 60 \times 4 = 240 \text{ square meters} \]
**Hint:** When converting area, remember to square the scale factor.

### Step 4: Convert Area from Square Meters to Square Miles
Now we need to convert the area from square meters back to square miles. We know that:
\[ 1 \text{ mile} = 1600 \text{ meters} \implies 1 \text{ square mile} = (1600 \text{ meters})^2 = 2560000 \text{ square meters} \]
To convert the area of the model from square meters to square miles:
\[ \text{Area in square miles} = \frac{240 \text{ square meters}}{2560000 \text{ square meters/square mile}} \]
Calculating this gives:
\[ = \frac{240}{2560000} \approx 0.00009375 \text{ square miles} \]
**Hint:** To convert from square meters to square miles, divide by the number of square meters in one square mile.

### Final Result
Thus, the area of the model in square miles is approximately:
\[ 0.00009375 \text{ square miles} \]