Question

A city is planning to place a 22 -foot-high statue in a local park. The artist who earned the commission is working from a model that is 18 inches high. A feature on the model measure 3 inches across. How large will the feature on the completed statue be? A. 2.45 feet B. 3.67 feet C. 4.00 feet D. 4.33 feet

Short answer

The feature on the completed statue will be 3.67 feet. Answer: B. 3.67 feet.

Step-by-step solution

- Convert Model Height to Feet

First, convert the model height from inches to feet. Since 1 foot = 12 inches, we have: \ \[ \text{Model height in feet} = \frac{18}{12} = 1.5 \text{ feet} \]

- Calculate Scale Factor

Calculate the scale factor by dividing the actual statue height by the model height: \ \[ \text{Scale factor} = \frac{22 \text{ feet}}{1.5 \text{ feet}} = 14.67 \]

- Compute Feature Size on Actual Statue

Multiply the feature size on the model by the scale factor to get the feature size on the actual statue: \ \[ \text{Feature size on statue} = 3 \text{ inches} \times 14.67 \] Convert inches to feet: \(3 \text{ inches} = \frac{3}{12} \text{ feet}\) which simplifies to \(0.25 \text{ feet}\). Therefore: \ \[ \text{Feature size on statue} = 0.25 \text{ feet} \times 14.67 = 3.67 \text{ feet} \]

Key concepts

Scale Factor

Understanding the scale factor is crucial when dealing with models and their real-life counterparts. The scale factor helps you determine how much larger or smaller the real object is compared to the model. To find the scale factor, you divide the actual size by the model size.

For example, in the exercise given, the actual statue's height is 22 feet, and the model's height is 1.5 feet. Thus, the scale factor is calculated as:
\[ \text{Scale factor} = \frac{22 \text{ feet}}{1.5 \text{ feet}} = 14.67 \]

This means the statue is 14.67 times taller than the model. Whenever you need to find unknown dimensions on the actual object, you can multiply the dimensions of the model by the scale factor.

This skill is essential in various fields such as architecture, engineering, and even art. It helps to create accurate representations and models from which larger structures or sculptures are built.

Unit Conversion

Unit conversion is another fundamental concept in mathematics. It allows you to switch between different units of measurement, ensuring you compare quantities accurately. In the example provided, converting the model's height from inches to feet is a crucial step:

\[ \text{Model height in feet} = \frac{18 \text{ inches}}{12 \text{ inches/foot}} = 1.5 \text{ feet} \]

Knowing that 1 foot equals 12 inches helps you make this conversion. Likewise, to convert inches to feet for the feature size on the model: \[ 3 \text{ inches} = \frac{3 \text{ inches}}{12 \text{ inches/foot}} = 0.25 \text{ feet} \]

Unit conversion is useful in daily life too. Whether you are cooking and following a recipe, traveling and needing to convert kilometers to miles, or even shopping and converting currency, understanding how to switch units can save you a lot of trouble and ensure precision in measurements.

Proportion

The concept of proportion is closely related to both scale factor and unit conversion. Proportion establishes a relationship between two quantities, showing how one value changes in relation to another.

In the exercise, the feature size on the actual statue is proportional to its size on the model, based on the scale factor: \[ \text{Feature size on statue} = 0.25 \text{ feet} \times 14.67 = 3.67 \text{ feet} \]

Here, 0.25 feet (the feature size on the model) is multiplied by the scale factor (14.67) to find the feature size on the actual statue. This concept of proportion helps in solving various real-world problems—such as determining quantities in recipes when scaling up or down, resizing images, or creating accurate drawings.

Proportion is fundamental in everyday activities and many professional fields, enabling you to understand and maintain the correct relationship between different quantities.