Question

Instructions: read the problem. Then answer the questions. 1. Imagine you are looking at a map of the United States, but it does not have a scale that indicates the number of miles per inch. You use a ruler to measure the distance between New York City and Washington D.C. in the map. the distance is 2 1/2 inches. You consult a website and find out that the real distance between New York City and Washington DC is. it is 200 miles. A. What is the scale of the map in miles per inch (unit rate)?___________________________ B. Write the equation of the proportional relationship whose constant of proportionality is the scale of the map. It states that the Y represents the distance in miles and the X represents this distance on the map in inches. __________________________________C. Using a ruler, you determine that the distance between New York City and Atlanta on the map is 8.75 inches. What is the actual distance between New York City and Atlanta in miles?______________________________

Answer 1

The scale of the map is 80 miles per inch, determined by dividing the actual distance between New York City and Washington D.C. by the map distance. The proportional relationship is Y = 80X, where Y represents the distance in miles and X represents the distance on the map in inches. Using this scale, the actual distance between New York City and Atlanta is found to be 700 miles.

Step-by-step explanation:

To solve this question, we will first determine the scale of the map by dividing the actual distance by the map distance. Then, we will write the equation for the proportional relationship with the scale as the constant of proportionality. Finally, we'll apply this scale to find the actual distance between New York City and Atlanta.

Part A: Scale of the Map:

We know the actual distance between New York City and Washington D.C. is 200 miles, and on the map, it's 2.5 inches. To find the scale in miles per inch, we divide the actual distance by the map distance:

Scale = 200 miles / 2.5 inches = 80 miles per inch

Part B: Proportional Relationship Equation:

The proportional relationship can be represented as:

Y = 80X
Where Y represents the distance in miles and X represents the distance on the map in inches.

Part C: Actual Distance between New York City and Atlanta:

The distance on the map between New York City and Atlanta is 8.75 inches, so to find the real distance, we multiply by the scale:

Actual distance = 8.75 inches * 80 miles per inch = 700 miles

Answer 2

Given:

Distance between New york and Washington DC in ruler is, d(r) = 2 1/2 inches=2.5 inches.

Actual distance between New york and Washington DC is, d(a) = 200 miles.

The objective is,

a) To find the scale ratio of map

b) To write the equation of relationship

c) To find the actual distance if distance between New york and Atlanta in map is 8.75 inches.

a)

The scale ratio can be calculated as,

`\begin{gathered} r=(d(a))/(d(r)) \\ r=(200)/(2.5) \\ r=80\text{ miles per inch.} \end{gathered}`

Hence, the scale ratio of map is 80 miles per inch.

b)

Consider the actual distance as Y (miles) and the distance in map as X (inches).

The equation can be represented as,

`\begin{gathered} Y\propto X \\ Y=rX \end{gathered}`

Here, r represents the proportionality contant which is the scale ratio of the map.

Hence, the required equation is Y = rX.

c)

The actual distance between New York City and Atlanta can be calculated using the equation obtained in part (b).

It is known that r = 80 miles per inch and the value of X is given as 8.75 inches.

Substitute the obtained values in the equation.

`\begin{gathered} Y=80\cdot8.75 \\ Y=700\text{ miles} \end{gathered}`

Hence, the actual distance between New York City and Atlanta is 700 miles.