The following table shows distances in miles between some cities in the United States. The distances have been written in scientific notation. Atlanta Baltimore Boston Chicago Los Angeles Seattle Tampa Atlanta 0 9.2735×102 1.50511×103 9.444×102 3.10801×103 3.50307×103 6.7037×102 Baltimore 9.2735×102 0 5.7785×102 9.7323×102 3.72245×103 3.7441×103 1.37072×103 Boston 1.50511×103 5.7785×102 0 1.36663×103 4.16643×103 3.99708×103 1.90293×103 Chicago 9.444×102 9.7323×102 1.36663×103 0 2.79980×103 2.78657×103 1.61477×103 Los Angeles 3.10801×103 3.72245×103 4.16643×103 2.79980×103 0 1.5442×103 3.45696×103 Seattle 3.50307×103 3.7441×103 3.99708×103 2.78657×103 1.5442×103 0 4.06249×103 Tampa 6.7037×102 1.37072×103 1.90293×103 1.61477×103 3.45696×103 4.06249×103 0

Question

The following table shows distances in miles between some cities in the United States. The distances have been written in scientific notation. Atlanta Baltimore Boston Chicago Los Angeles Seattle Tampa Atlanta 0 9.2735×102 1.50511×103 9.444×102 3.10801×103 3.50307×103 6.7037×102 Baltimore 9.2735×102 0 5.7785×102 9.7323×102 3.72245×103 3.7441×103 1.37072×103 Boston 1.50511×103 5.7785×102 0 1.36663×103 4.16643×103 3.99708×103 1.90293×103 Chicago 9.444×102 9.7323×102 1.36663×103 0 2.79980×103 2.78657×103 1.61477×103 Los Angeles 3.10801×103 3.72245×103 4.16643×103 2.79980×103 0 1.5442×103 3.45696×103 Seattle 3.50307×103 3.7441×103 3.99708×103 2.78657×103 1.5442×103 0 4.06249×103 Tampa 6.7037×102 1.37072×103 1.90293×103 1.61477×103 3.45696×103 4.06249×103 0
Plan a trip where you will start at one city, travel to a second city, travel to a third city, and then return to the original city. Assuming you could travel the direct straight-line distance between each pair of cities, how far will you travel all together? Write your total distance traveled in both decimal notation and scientific notation. Show all your work so it is clear what cities you are going to and in what order.

GPT-4o mini · Answer

Let's plan a trip starting from Atlanta, then visiting Chicago, Seattle, and finally returning to Atlanta. We can summarize the distance between the cities using the table provided and perform the calculations.

Step 1: Identify the Distances

From the table:

Atlanta to Chicago: \( 9.444 \times 10^2 \) miles
Chicago to Seattle: \( 2.78657 \times 10^3 \) miles
Seattle to Atlanta: \( 3.50307 \times 10^3 \) miles

Step 2: Calculate the Total Distance

Now we will add all the distances together.

Distance from Atlanta to Chicago:
\[ 9.444 \times 10^2 = 944.4 \text{ miles} \]
Distance from Chicago to Seattle:
\[ 2.78657 \times 10^3 = 2786.57 \text{ miles} \]
Distance from Seattle to Atlanta:
\[ 3.50307 \times 10^3 = 3503.07 \text{ miles} \]

Total Distance Calculation

Now we add the distances: \[ 944.4 + 2786.57 + 3503.07 = 7234.04 \text{ miles} \]

Step 3: Convert the Total Distance to Scientific Notation

To express \( 7234.04 \) in scientific notation: \[ 7234.04 = 7.23404 \times 10^3 \]

Final Result

So the total distance traveled in decimal notation and scientific notation is:

Total Distance Traveled (Decimal Notation): \( 7234.04 \) miles
Total Distance Traveled (Scientific Notation): \( 7.23404 \times 10^3 \) miles

Recap of Trip Route

Start at Atlanta
Travel to Chicago
Travel to Seattle
Return to Atlanta

Total distance traveled:

Decimal Notation: \( 7234.04 \) miles
Scientific Notation: \( 7.23404 \times 10^3 \) miles