# Problem of the Day

The statue of liberty is approximately 150 feet tall from the base to the tip of the torch. Â A scale model is built that is 3 feet tall. Â If painting the scale model requires 5 cans of paint, approximately how many cans of paint would be required to paint the real Statue of Liberty? (Assume that paint is used at the same rate on the real statue as it is on the model.)

A. 50

B. 250

C. 2,500

D. 10,000

E. 12,500

### Solution

[latexpage]

How much paint is required to paint an object is a function of that object’s surface area. Â This should be obvious. Â If you’re painting the walls in your house and you were able to cover 400 square feet with 1 gallon of paint, and you have 1200 more square feet to paint, that should take about 3 more gallons of paint. Â So, we have to figure out how the surface areas of the model and the real statue are related.

There is a rule in geometry that says if two similar figures have linear measures in the ratio of $\frac{a}{b}$, then their areas will be in the ratio $\frac{a^2}{b^2}$. Â For example, consider two squares, one with side length 4 and one with side length 6. Â The ratio of the linear measures of the square is $\frac{2}{3}$. Â The areas of the squares are 16 and 36 , respectively, andÂ $\frac{16}{36}=\frac{4}{9}=\frac{2^2}{3^2}$. Â Notice that the ratio of the areas was the square of the ratio of the linear measures. Â This rule applies to all shapes.

Here, the given information indicates that the real statue is about 50 times taller than the model, so the ratio of the linear measures is $\frac{50}{1}$. Â This means that the ratio of the surface areas is $\frac{50^2}{1^2}=\frac{2500}{1}$. Â So, 2500 times as much paint is required to paint the real statue, which is $2500 \cdot 5=12,500$ cans of paint.

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