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The following table shows distances in miles between some cities in the united states. The distances have been written in scientific notation.

The table is too big, so just know the cities that I'm traveling to are Los Angeles to Chicago, Chicago to Boston, Boston to Baltimore, Baltimore to Los Angeles

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. 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.

11 answers

To calculate the total distance for your trip from Los Angeles to Chicago, then to Boston, then to Baltimore, and back to Los Angeles, we need to find the distances between each of these cities.

Assuming the distances between the cities in miles are as follows (expressed in scientific notation):

  • Los Angeles to Chicago: \( 2.0 \times 10^{3} \) miles
  • Chicago to Boston: \( 1.0 \times 10^{2} \) miles
  • Boston to Baltimore: \( 2.0 \times 10^{1} \) miles
  • Baltimore to Los Angeles: \( 3.0 \times 10^{3} \) miles

Now, let's break down the route step by step and find the total distance traveled:

  1. Distance from Los Angeles to Chicago:
    \[ 2.0 \times 10^3 \text{ miles} = 2000 \text{ miles} \]

  2. Distance from Chicago to Boston:
    \[ 1.0 \times 10^2 \text{ miles} = 100 \text{ miles} \]

  3. Distance from Boston to Baltimore:
    \[ 2.0 \times 10^1 \text{ miles} = 20 \text{ miles} \]

  4. Distance from Baltimore to Los Angeles:
    \[ 3.0 \times 10^3 \text{ miles} = 3000 \text{ miles} \]

Now, we can find the total distance traveled by adding these distances together:

\[ \text{Total Distance} = (2.0 \times 10^3) + (1.0 \times 10^2) + (2.0 \times 10^1) + (3.0 \times 10^3) \]

Converting all distances to decimal notation for easier addition:

  1. \( 2.0 \times 10^3 = 2000 \)
  2. \( 1.0 \times 10^2 = 100 \)
  3. \( 2.0 \times 10^1 = 20 \)
  4. \( 3.0 \times 10^3 = 3000 \)

Now summing them up:

\[ 2000 + 100 + 20 + 3000 = 4120 \text{ miles} \]

Now, we present the total distance traveled in both decimal notation and scientific notation.

  • Decimal Notation: \( 4120 \) miles
  • Scientific Notation: \( 4.12 \times 10^3 \) miles

So, the total distance traveled for your trip is:

  • 4120 miles (Decimal Notation)
  • \( 4.12 \times 10^3 \) miles (Scientific Notation)
forgot to mention that the miles between the cities that i have chosen are:

Los Angeles to Chicago: 2.79980 x 10^3
Chicago to Boston: 1.36663 x 10^3
Boston to Baltimore: 5.7785 x 10^2
Baltimore to Log Angeles: 4.16643 x 10^3

Thank you for providing the distances in scientific notation. Let's recalculate the total distance for your trip from Los Angeles to Chicago, then to Boston, then to Baltimore, and back to Los Angeles using the specific distances you provided.

Distances:

  1. Los Angeles to Chicago: \[ 2.79980 \times 10^3 \text{ miles} = 2799.80 \text{ miles} \]

  2. Chicago to Boston: \[ 1.36663 \times 10^3 \text{ miles} = 1366.63 \text{ miles} \]

  3. Boston to Baltimore: \[ 5.7785 \times 10^2 \text{ miles} = 577.85 \text{ miles} \]

  4. Baltimore to Los Angeles: \[ 4.16643 \times 10^3 \text{ miles} = 4166.43 \text{ miles} \]

Calculating Total Distance

Now add these distances together:

\[ \text{Total Distance} = (2.79980 \times 10^3) + (1.36663 \times 10^3) + (5.7785 \times 10^2) + (4.16643 \times 10^3) \]

Let's convert each distance into decimal for easier addition:

  1. Los Angeles to Chicago: \[ 2.79980 \times 10^3 = 2799.80 \text{ miles} \]

  2. Chicago to Boston: \[ 1.36663 \times 10^3 = 1366.63 \text{ miles} \]

  3. Boston to Baltimore: \[ 5.7785 \times 10^2 = 577.85 \text{ miles} \]

  4. Baltimore to Los Angeles: \[ 4.16643 \times 10^3 = 4166.43 \text{ miles} \]

Now sum them up:

\[ 2799.80 + 1366.63 + 577.85 + 4166.43 = 10010.71 \text{ miles} \]

Total Distance

  • Total Distance in Decimal Notation: \[ \boxed{10010.71} \text{ miles} \]

Now, let's convert this total distance into scientific notation.

To express \( 10010.71 \) in scientific notation:

\[ 10010.71 = 1.001071 \times 10^4 \]

Thus, in scientific notation:

  • Total Distance in Scientific Notation: \[ \boxed{1.001071 \times 10^4} \text{ miles} \]

Summary of Total Distance:

  • In Decimal Notation: \( 10010.71 \text{ miles} \)
  • In Scientific Notation: \( 1.001071 \times 10^4 \text{ miles} \)
Sita lives in Los Angeles, and her grandmother lives in Seattle. Sita plans to visit her grandmother three times in the next year. What is the total distance she will travel to make these three trips? Write and evaluate an expression that uses multiplication to calculate the total distance traveled. Give your answer in both decimal notation and scientific notation.

The miles between Los Angeles and Seattle are 1.5442 x 10^3, so i guess its just that 3 times.

You're correct! To find the total distance Sita will travel for her visits from Los Angeles to Seattle and back, we can use multiplication.

Distance Calculation

  1. Distance from Los Angeles to Seattle: \[ 1.5442 \times 10^3 \text{ miles} = 1544.2 \text{ miles} \]

  2. Total Distance for One Round Trip: Since Sita will travel to Seattle and back, we need to double the one-way distance: \[ \text{Round Trip Distance} = 2 \times (1.5442 \times 10^3) \text{ miles} \] \[ = 2 \times 1.5442 \times 10^3 \text{ miles} = 3.0884 \times 10^3 \text{ miles} \]

  3. Total Distance for Three Round Trips: Since she plans to make three round trips, we multiply the round trip distance by 3: \[ \text{Total Distance} = 3 \times (3.0884 \times 10^3) \text{ miles} \] \[ = 9.2652 \times 10^3 \text{ miles} \]

Evaluating the Expression

To summarize, the expression for her total travel distance can be written as: \[ \text{Total Distance} = 3 \times 2 \times (1.5442 \times 10^3) \]

  1. Convert to Decimal Notation:
    • Calculating the total distance in decimal notation: \[ 9.2652 \times 10^3 \text{ miles} = 9265.2 \text{ miles} \]

Final Answer

  • Total Distance in Decimal Notation: \[ \boxed{9265.2} \text{ miles} \]

  • Total Distance in Scientific Notation: \[ \boxed{9.2652 \times 10^3} \text{ miles} \]

These results represent the total distance Sita will travel to visit her grandmother in Seattle three times in the next year.

Using the tables of distances, (the ones i listed earlier) make up your own word problem that would require someone to use division to solve. Write the problem, and provide the solution.

the table is too large for me to type, so like i said use the ones i listed earlier.

Sure! Based on the distances you provided earlier, here's a word problem that would require the use of division to solve:

Word Problem:

Sita is planning a charity run that takes place on a circular route. Each lap around the route is equal to the distance from Los Angeles to Chicago, which is \( 2.79980 \times 10^3 \) miles. She intends to run a total of 5,598.60 miles over the course of the campaign.

  1. How many laps will Sita complete if she runs her charity route to raise money?

Given:

  • Distance from Los Angeles to Chicago: \( 2.79980 \times 10^3 \) miles (or 2799.80 miles)
  • Total distance to be run: 5,598.60 miles

Solution:

To find out how many laps Sita will complete, we need to divide the total distance she plans to run by the distance of one lap:

  1. Calculate the number of laps using division:

\[ \text{Number of Laps} = \frac{\text{Total Distance}}{\text{Distance per Lap}} = \frac{5598.60 \text{ miles}}{2799.80 \text{ miles}} \]

  1. Perform the calculation:

\[ \text{Number of Laps} = \frac{5598.60}{2799.80} \approx 2 \]

So, Sita will complete approximately 2 laps around her charity run route.

Therefore, the answer is:

  • Sita will complete approximately \(\boxed{2}\) laps.

This problem not only illustrates the use of division but also ties in the context of distances between cities!

instead of that, go off of my starting sentence to complete this word problem: I am planning to visit my best friend who lives in Chicago, i live in Los Angeles.

Sure! Here’s a word problem based on your starting sentence:

Word Problem:

I am planning to visit my best friend who lives in Chicago, and I live in Los Angeles. The one-way distance from Los Angeles to Chicago is \( 2.79980 \times 10^3 \) miles. If I were to drive to Chicago and back home to Los Angeles, I would be making a round trip.

If I want to see how many gallons of gas I will need for the trip, I know my car gets about 25 miles per gallon.

  1. How many gallons of gas will I need for the entire round trip from Los Angeles to Chicago and back to Los Angeles?

Given:

  • Distance from Los Angeles to Chicago: \( 2.79980 \times 10^3 \text{ miles} \) (or 2799.80 miles)
  • Total distance for the round trip: \( 2 \times 2.79980 \times 10^3 \text{ miles} \)
  • Car's fuel efficiency: 25 miles per gallon

Solution:

  1. First, we need to calculate the total distance for the round trip:

\[ \text{Total Round Trip Distance} = 2 \times 2799.80 \text{ miles} = 5599.60 \text{ miles} \]

  1. Next, we will determine how many gallons of gas are needed for the entire trip by dividing the total round trip distance by the car's fuel efficiency:

\[ \text{Gallons of Gas Needed} = \frac{\text{Total Round Trip Distance}}{\text{Fuel Efficiency}} = \frac{5599.60 \text{ miles}}{25 \text{ miles per gallon}} \]

  1. Perform the calculation:

\[ \text{Gallons of Gas Needed} = \frac{5599.60}{25} = 223.984 \text{ gallons} \]

  1. Rounding to two decimal places, I will need approximately \( 224.00 \) gallons of gas for the round trip.

Therefore, the answer is:

  • I will need approximately \(\boxed{224}\) gallons of gas for the entire trip.

This word problem incorporates the distance provided and requires the use of division to solve for the amount of gas needed!

What is the total round trip distance? like what does it mean

The total round trip distance refers to the complete distance traveled when going from a starting point to a destination and then returning back to the starting point.

In Your Context:

  1. Starting Point: Los Angeles (where you live).
  2. Destination: Chicago (where your best friend lives).

How to Calculate Total Round Trip Distance:

  1. One-Way Distance: The distance from Los Angeles to Chicago is given as \( 2.79980 \times 10^3 \) miles:

    • In decimal form, this is approximately 2799.80 miles.

  2. Round Trip Distance: Since you need to travel to Chicago and then return to Los Angeles, you double the one-way distance: \[ \text{Total Round Trip Distance} = \text{Distance to Chicago} + \text{Distance back to Los Angeles} = 2799.80 \text{ miles} + 2799.80 \text{ miles} = 5599.60 \text{ miles} \]

    • Alternatively, you can calculate it as: \[ \text{Total Round Trip Distance} = 2 \times \text{One-Way Distance} = 2 \times 2799.80 \text{ miles} = 5599.60 \text{ miles} \]

Summary:

  • Total Round Trip Distance: The total distance you would travel to go from Los Angeles to Chicago and back to Los Angeles is 5599.60 miles. This means, for your trip, you would cover this distance in total by making the round trip.

  • In simpler terms: If you think of going to a friend's house (Chicago) and then coming back home (Los Angeles), the total round trip distance is how far you drive in total for both the trip there and the trip back.