Abigail lives at the point (-3,-1) and wants to travel to her friend Clara's house at (2,3) on the map. She has two choices to get there: either travel directly to Clara's house, or Abigail can stop by Bob's store at (2,-1) to pick up snacks before traveling to Clara's house. Abigail wants to compare the two distances of her choices.
How long is the distance traveled if Abigail stops by bobs store before going to Clara's house?
Determine the distance traveled if Abigail goes directly to Clara's house. What is the difference in measurement between the two routes, and which one is faster?
To find the distance traveled if Abigail stops by Bob's store before going to Clara's house, we can use the distance formula.
The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's denote Abigail's house as A (-3, -1), Bob's store as B (2, -1), and Clara's house as C (2, 3).
To find the distance from A to B, we use the formula:
d_AB = sqrt((2 - (-3))^2 + (-1 - (-1))^2)
= sqrt(5^2 + 0^2)
= sqrt(25)
= 5
To find the distance from B to C, we use the formula:
d_BC = sqrt((2 - 2)^2 + (3 - (-1))^2)
= sqrt(0^2 + 4^2)
= sqrt(16)
= 4
Therefore, the distance traveled if Abigail stops by Bob's store before going to Clara's house is d_AB + d_BC = 5 + 4 = 9 units.
To find the distance traveled if Abigail goes directly to Clara's house, we use the formula:
d_AC = sqrt((2 - (-3))^2 + (3 - (-1))^2)
= sqrt(5^2 + 4^2)
= sqrt(25 + 16)
= sqrt(41)
Therefore, the distance traveled if Abigail goes directly to Clara's house is sqrt(41) units.
The difference in measurement between the two routes is sqrt(41) - 9 units.
To determine which one is faster, we compare the distances. Since sqrt(41) < 9, it means that going directly to Clara's house is faster than stopping by Bob's store before going to Clara's house.
To find the distance traveled if Abigail stops by Bob's store before going to Clara's house, we need to calculate the sum of the distances between Abigail and Bob's store, and between Bob's store and Clara's house.
First, we need to find the distance between two points on a coordinate plane using the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Let's calculate the distance from Abigail's house to Bob's store:
Distance_AB_to_B = √[(2 - (-3))^2 + (-1 - (-1))^2]
= √[(5)^2 + (0)^2]
= √[25]
= 5 units
Next, let's calculate the distance from Bob's store to Clara's house:
Distance_B_to_C = √[(2 - 2)^2 + (3 - (-1))^2]
= √[(0)^2 + (4)^2]
= √[16]
= 4 units
To find the total distance traveled, we add the distances together:
Total Distance = Distance_AB_to_B + Distance_B_to_C
= 5 + 4
= 9 units
Now, let's find the distance if Abigail goes directly to Clara's house:
Distance_A_to_C = √[(2 - (-3))^2 + (3 - (-1))^2]
= √[(5)^2 + (4)^2]
= √[25 + 16]
= √[41]
≈ 6.40 units (rounded to two decimal places)
The difference in measurement between the two routes is:
Difference = Distance_A_to_C - Total Distance
= 6.40 - 9
≈ -2.60 units (rounded to two decimal places)
Since the difference is negative, it means that going directly to Clara's house is shorter than stopping by Bob's store. Therefore, going directly to Clara's house is faster.
To determine the distance traveled if Abigail stops by Bob's store before going to Clara's house, we need to calculate the sum of the distances from Abigail's house to Bob's store, and then from Bob's store to Clara's house.
Step 1: Calculate the distance from Abigail's house to Bob's store.
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
= √((2 - (-3))^2 + (-1 - (-1))^2)
= √((2 + 3)^2 + (0)^2)
= √(5^2 + 0)
= √(25)
= 5
Step 2: Calculate the distance from Bob's store to Clara's house.
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
= √((2 - 2)^2 + (3 - (-1))^2)
= √((0)^2 + (4)^2)
= √(0 + 16)
= √(16)
= 4
Step 3: Calculate the total distance traveled.
Total distance = Distance from Abigail's house to Bob's store + Distance from Bob's store to Clara's house
= 5 + 4
= 9
Therefore, the distance traveled if Abigail stops by Bob's store before going to Clara's house is 9 units.
To determine the distance traveled if Abigail goes directly to Clara's house, we need to calculate the distance between Abigail's house and Clara's house.
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
= √((2 - (-3))^2 + (3 - (-1))^2)
= √((2 + 3)^2 + (3 + 1)^2)
= √((5)^2 + (4)^2)
= √(25 + 16)
= √(41)
Therefore, the distance traveled if Abigail goes directly to Clara's house is √(41) units.
The difference in measurement between the two routes is:
Difference = Distance traveled with Bob's store - Distance traveled directly
= 9 - √(41)
≈ 2.24 units.
The route without stopping at Bob's store is faster as it has a shorter distance.