Question

1. a map of colorado says that the scale is 1 inch to 20 miles or 1 to 1,267,200. are these two ways of reporting the scale the same? explain your reasoning.

(from unit 1, lesson 11)

1. here is a polygon on a grid.

image of a grid with a polygon

a. draw a scaled copy of the polygon using a scale factor 3. label the copy a.

b. draw a scaled copy of the polygon with a scale factor \\(\frac{1}{2}\\). label it b.

c. is polygon a a scaled copy of polygon b? if so, what is the scale factor that takes b to a?

Explanation:

Question 4

Step 1: Convert miles to inches

We know that 1 mile = 63360 inches. So, 20 miles in inches is calculated as:
\(20\times63360 = 1267200\) inches.

Step 2: Compare the scales

The first scale is 1 inch (map) to 20 miles (actual), which is equivalent to 1 inch (map) to 1267200 inches (actual). The second scale is 1 to 1,267,200, which means 1 unit on the map to 1,267,200 units in actual (where the unit for map and actual should be consistent, and here when we convert 20 miles to inches we get 1267200 inches, so the scales are consistent in terms of the ratio of map distance to actual distance).

Step 1: Analyze the original polygon's dimensions

First, we need to determine the lengths of the sides of the original polygon. Let's assume each grid square has a side length of 1 unit. For the rectangular part: let's say the length (horizontal) is, for example, if we look at the horizontal sides, suppose the rectangle has a width (vertical) of 2 units and a length (horizontal) of 2 units, and the triangular part: the base of the triangle (horizontal) is 2 units and the height (from the base to the tip) is 2 units (this depends on the actual grid, but generally, to scale with a factor of 3, we multiply each side length by 3).

Step 2: Draw the scaled polygon

For each side of the original polygon, multiply its length by 3. So, if a side was 2 units, it becomes \(2\times3 = 6\) units. Then, using these new side lengths, draw the polygon (the rectangle part will have width 6 and length 6, and the triangle part will have base 6 and height 6, maintaining the same shape and proportions) and label it A.

Step 1: Analyze the original polygon's dimensions

Again, using the side lengths of the original polygon (as determined in part a, e.g., if a side was 2 units).

Step 2: Draw the scaled polygon

Multiply each side length of the original polygon by \(\frac{1}{2}\). So, if a side was 2 units, it becomes \(2\times\frac{1}{2}=1\) unit. Then, draw the new polygon with these scaled - side lengths, keeping the same shape and proportions, and label it B.

Answer:

Yes, these two ways of reporting the scale are the same. Because 20 miles is equal to \(20\times63360 = 1267200\) inches, so a scale of 1 inch to 20 miles is equivalent to a scale of 1 to 1,267,200 (when both are in the same unit, inches for the actual distance corresponding to 1 inch on the map).

Question 5a