Sita lives in Los Angeles, and her grandmother lives in Seattle. Sita plans to visit her grandmother three times in the next year. Assuming she travels the direct straight-line distance between the cities on each trip, 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.
To calculate the total distance Sita will travel, we need to multiply the distance between Los Angeles and Seattle by the number of trips she plans to make. Let's assume the distance between Los Angeles and Seattle is 1,000 miles.
The expression to calculate the total distance traveled is: 1,000 miles/trip * 3 trips.
In decimal notation, the total distance traveled is 1,000 miles/trip * 3 trips = 3,000 miles.
In scientific notation, the total distance traveled is 3 x 10^3 miles.
To calculate the total distance Sita will travel, we need to multiply the distance between Los Angeles and Seattle by the number of trips.
The direct straight-line distance between Los Angeles and Seattle is approximately 963 miles.
Let's say the number of trips Sita plans to make is denoted by "n", which in this case is 3.
Therefore, the expression to calculate the total distance traveled is:
Expression: 963 * 3
Decimal Notation Calculation: 963 * 3 = 2,889 miles
Scientific Notation Calculation: 963 * 3 = 8.667 x 10^2 miles
To calculate the total distance Sita will travel to make these three trips, we need to determine the direct straight-line distance between Los Angeles and Seattle, and multiply it by the number of trips.
Step 1: Find the direct straight-line distance between Los Angeles and Seattle.
The direct straight-line distance between two points can be calculated using the distance formula, which is derived from the Pythagorean theorem. In this case, we can use the coordinates of Los Angeles (34.0522° N, 118.2437° W) and Seattle (47.6062° N, 122.3321° W) to calculate the distance.
Using the Haversine formula, the distance between two points on the Earth's surface given their latitudes and longitudes can be calculated as follows:
a = sin²(Δφ/2) + cos(φ1) * cos(φ2) * sin²(Δλ/2)
c = 2 * atan2(√a, √(1−a))
distance = R * c
where:
φ1 and φ2 are the latitudes of the two points
Δφ = φ2 - φ1
Δλ = λ2 - λ1
R is the radius of the Earth (approximately 6371 km)
Converting the latitude and longitude values to radians:
φ1 = 34.0522° * π/180 ≈ 0.5934 rad
φ2 = 47.6062° * π/180 ≈ 0.8293 rad
Δφ = 0.8293 rad - 0.5934 rad ≈ 0.2359 rad
Δλ = 122.3321° * π/180 - 118.2437° * π/180 ≈ 0.0692 rad
Using the Haversine formula:
a = sin²(0.2359/2) + cos(0.5934) * cos(0.8293) * sin²(0.0692/2)
c = 2 * atan2(√a, √(1−a))
distance = 6371 * c
Evaluating the expression:
a = sin²(0.2359/2) + cos(0.5934) * cos(0.8293) * sin²(0.0692/2) ≈ 0.1205
c = 2 * atan2(√0.1205, √(1−0.1205)) ≈ 0.4221
distance = 6371 * 0.4221 ≈ 2697.84 km
Step 2: Multiply the direct straight-line distance by the number of trips.
Since Sita plans to visit her grandmother three times in the next year, we multiply the distance calculated in step 1 by 3 to get the total distance traveled.
total distance = 2697.84 km * 3 ≈ 8093.52 km
Answer in decimal notation: The total distance Sita will travel to make these three trips is approximately 8093.52 kilometers.
Answer in scientific notation: The total distance Sita will travel to make these three trips is approximately 8.09352 x 10^3 kilometers.