A stop sign has the shape of an octagon with each side measuring 0.5 ft. To make a scale model of the sign, every dimension is multiplied by 1/20. How is the ratio of the areas related to the ratio of the corresponding dimensions? What is the ration of the areas?

area varies as the square of length dimension. (note square A = X^2, circle A = pi R^2)

If you multiply every length by 1/20, you multiply every area by 1/400

To find the ratio of the areas, we need to understand how area scales with the dimensions of an object. Area is a two-dimensional measurement, so it scales with the square of the linear dimensions.

In this case, the linear dimensions of the original stop sign are given as each side measuring 0.5 ft. To create a scale model, we multiply every dimension by 1/20.

Let's start by calculating the linear dimensions of the scale model:
Original side length = 0.5 ft
Scale factor = 1/20

To find the corresponding dimensions in the scale model, multiply the original dimensions by the scale factor:

Scale model side length = (0.5 ft) * (1/20)
= 0.025 ft

Next, let's calculate the ratio of the areas. The ratio of the areas will be the square of the ratio of the corresponding dimensions.

Original area = (0.5 ft)²
Scale model area = (0.025 ft)²

Ratio of the areas = (Scale model area) / (Original area)
= [(0.025 ft)²] / [(0.5 ft)²]
= (0.000625 ft²) / (0.25 ft²)
= 0.0025

Therefore, the ratio of the areas is 0.0025.

To summarize, the ratio of the areas is obtained by squaring the ratio of the corresponding dimensions. In this case, the ratio of the areas is 0.0025.