a stop sign has the shape of and octagon with each side measuring 0.5ft. to make a scale model to the sign, every dimesion is multiplied by 1/20. how is the ratio of th areas related to the ratio of the corrsesponding dimensions? what is the ratio of the ares?
The ratio of the areas is (1/20)^2 = 1/400 if ANY figure is reduced in its linear dimesnions by 1/20.
It does not matter if it is an octagon or a triangle
To understand the relationship between the areas and dimensions of the stop sign and its scale model, we need to recall that area is determined by the square of the corresponding linear measurement.
Let's start by calculating the area of the stop sign. The formula for the area of an octagon is given by:
Area = (2 + 2√2) × s²
Where s is the length of each side.
In this case, s = 0.5 ft.
Substituting the values, we have:
Area = (2 + 2√2) × (0.5 ft)²
Next, let's calculate the area of the scale model. Since each dimension is multiplied by 1/20, the length of each side in the scale model would be:
New side length = (1/20) × 0.5 ft
Now, we can calculate the area of the scale model using the same formula:
New Area = (2 + 2√2) × (1/20) × (0.5 ft)²
To understand the relationship between the areas, we need to find the ratio of the areas:
Ratio of Areas = New Area / Area
Now, let's substitute the values to find the ratio of the areas:
Ratio of Areas = [(2 + 2√2) × (1/20) × (0.5 ft)²] / [(2 + 2√2) × (0.5 ft)²]
Simplifying, we get:
Ratio of Areas = (1/20)²
Fractional form: Ratio of Areas = 1/400
So, the ratio of the areas of the scale model to the original sign is 1/400.