Solve each problem. See Example 5. Height of a Carving of Lincoln Assume that Lincoln was 6 1/3 ft tall and his head was 3/4 ft long. Knowing that the carved head of Lincoln at Mt. Rushmore is 60 ft tall, find how tall his entire body would be if it were carved into the mountain.

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First, convert Lincoln's actual height into an improper fraction or decimal for easier calculation. Since he is 6 1/3 feet tall, express this as \(6 \frac{1}{3} = \frac{19}{3}\) feet.

Next, note the length of Lincoln's actual head, which is \(\frac{3}{4}\) feet.

Set up a proportion comparing the actual head length to the actual height, and the carved head length to the unknown carved height. The proportion is: \(\frac{\text{Actual Head Length}}{\text{Actual Height}} = \frac{\text{Carved Head Length}}{\text{Carved Height}}\).

Substitute the known values into the proportion: \(\frac{\frac{3}{4}}{\frac{19}{3}} = \frac{60}{x}\), where \(x\) is the height of the entire carved body.

Solve the proportion for \(x\) by cross-multiplying and isolating \(x\): \(x = \frac{60 \times \frac{19}{3}}{\frac{3}{4}}\). Simplify this expression step-by-step to find the carved height.

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Ratio and Proportion

Ratio and proportion involve comparing two quantities and determining how one quantity relates to another. In this problem, the ratio of Lincoln's actual head height to his total height is used to find the corresponding total height from the scaled-up head measurement on the carving.

Scaling and Similar Figures

Scaling refers to enlarging or reducing objects while maintaining their shape and proportions. The carved head and the actual head are similar figures, so their dimensions are proportional. This allows us to use the scale factor from the head to find the total height of the carved body.

Unit Conversion and Fraction Operations

Understanding how to work with mixed numbers and fractions is essential here, as Lincoln's height is given as 6 1/3 ft and his head as 3/4 ft. Converting these to improper fractions or decimals simplifies calculations and ensures accuracy in solving the proportion.