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A map has a scale of 4 cm to 1 km. What is the actual area of a lake on ground which is represented as an area of \(60cm^2\) on the map?

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Question:

A
\(3.75km^2\)

explanation

The problem gives the scale of 4cm:1km, 4 cm on the map represents an actual distance of 1 km.

As the question asks for the relationship between the areas, it is first necessary to rewrite the scale ratio in terms of units squared:

\(4^2cm\) square: \(1^2km\) square, which gives \(16cm^2 \div 1km^2\)

Using the ratio of the areas and the map area given in the problem, a proportion can be written that enables calculation of the actual area: \(\frac{\mathrm{16cm^2} }{\mathrm{1km^2}}=\frac{\mathrm{60cm^2} }{\mathrm{xkm^2}}\) , where x is the unknown actual area.
Note that every proportion you write should maintain consistency in the ratios described. In this case, \(cm^2\) both occupy the numerator, and \(km^2\) both occupy the denominator. The proportion will not work if one \(cm^2\) was written in the numerator and the other written in the denominator. Cross-multiply and isolate to solve for the unknown area x: \(x \times \frac{\mathrm{16cm^2} }{\mathrm{1km^2}}= 60cm^2\), which simplifies to \(x=60cm^2 \times \frac{\mathrm{1km^2} }{\mathrm{16cm^2}}\), which simplifies to \(x=3.75km^2\).

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