Part 1: 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 travelling to Clara's house. Abigail wants to compare the two distances of her choices.A drawing of the story problem. Abigail lives at point A(-3, -1), Clara lives at point C(2, 3) and Bob's Store is at point B(2, -1). The three points form a right triangle.
a2+b2=c2
(3 points)
How long is the distance traveled if Abigail stops by Bob's Store before going to Clara's house?
units.
Question 2
Part 2: Now determine the distance traveled if Abigail goes directly to Clara's house. What is the difference in measurement between the two routes Abigail can take, and which is faster? You must show all work for full credit. (12 points)
To find the distance traveled if Abigail stops by Bob's Store before going to Clara's house, we can use the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
First, let's find the distance from Abigail's house to Bob's Store:
distance_AB = √((2 - (-3))^2 + (-1 - (-1))^2)
distance_AB = √(5^2 + 0^2)
distance_AB = √(25 + 0)
distance_AB = √25
distance_AB = 5 units
Next, let's find the distance from Bob's Store to Clara's house:
distance_BC = √((2 - 2)^2 + (3 - (-1))^2)
distance_BC = √(0^2 + 4^2)
distance_BC = √(0 + 16)
distance_BC = √16
distance_BC = 4 units
Now, let's find the total distance traveled if Abigail stops by Bob's Store before going to Clara's house:
distance_total = distance_AB + distance_BC
distance_total = 5 + 4
distance_total = 9 units
Therefore, if Abigail stops by Bob's Store before going to Clara's house, she will travel a distance of 9 units.
Now, let's determine the distance traveled if Abigail goes directly to Clara's house. We can use the distance formula again:
distance_AC = √((2 - (-3))^2 + (3 - (-1))^2)
distance_AC = √(5^2 + 4^2)
distance_AC = √(25 + 16)
distance_AC = √41
distance_AC ≈ 6.4 units
The difference in measurement between the two routes Abigail can take is:
differenence = distance_total - distance_AC
difference = 9 - 6.4
difference ≈ 2.6 units
Therefore, the route where Abigail stops by Bob's Store is approximately 2.6 units longer than going directly to Clara's house. The direct route to Clara's house is faster.
To find the distance traveled if Abigail stops by Bob's Store before going to Clara's house, we need to calculate the lengths of the sides of the right triangle formed by the three points.
Using the distance formula, we can calculate the distance between Abigail's house at (-3, -1) and Bob's Store at (2, -1):
d_AB = √((x2 - x1)^2 + (y2 - y1)^2)
= √((2 - (-3))^2 + (-1 - (-1))^2)
= √((5)^2 + (0)^2)
= √(25 + 0)
= √25
= 5 units
Next, we can calculate the distance between Bob's Store at (2, -1) and Clara's house at (2, 3):
d_BC = √((x2 - x1)^2 + (y2 - y1)^2)
= √((2 - 2)^2 + (3 - (-1))^2)
= √((0)^2 + (4)^2)
= √(0 + 16)
= √16
= 4 units
Therefore, the distance traveled if Abigail stops by Bob's Store before going to Clara's house is:
d_AB + d_BC = 5 units + 4 units = 9 units.
To find the distance traveled if Abigail goes directly to Clara's house, we can calculate the distance between Abigail's house at (-3, -1) and Clara's house at (2, 3):
d_AC = √((x2 - x1)^2 + (y2 - y1)^2)
= √((2 - (-3))^2 + (3 - (-1))^2)
= √((5)^2 + (4)^2)
= √(25 + 16)
= √41 units.
The difference in measurement between the two routes Abigail can take is:
d_AC - (d_AB + d_BC) = √41 - 9 = √41 - √9 units.
To determine which route is faster, we need to compare the distances. Since √41 is greater than √9, the direct route to Clara's house is faster than going through Bob's Store.
To find the distance traveled if Abigail stops by Bob's Store before going to Clara's house, we need to find the lengths of the two segments of the triangle formed by Abigail's house, Bob's Store, and Clara's house.
Using the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by the square root of [(x2 - x1)^2 + (y2 - y1)^2], we can calculate the distances.
Distance from Abigail's house to Bob's Store:
AB = sqrt[(2 - (-3))^2 + (-1 - (-1))^2]
= sqrt[(2 + 3)^2 + (-1 + 1)^2]
= sqrt[5^2 + 0^2]
= sqrt[25]
= 5 units
Distance from Bob's Store to Clara's house:
BC = sqrt[(2 - 2)^2 + (-1 - 3)^2]
= sqrt[0^2 + (-4)^2]
= sqrt[0 + 16]
= sqrt[16]
= 4 units
Total distance traveled if Abigail stops by Bob's Store before going to Clara's house:
Total distance = AB + BC
= 5 + 4
= 9 units
To find the distance traveled if Abigail goes directly to Clara's house, we only need to find the length of the hypotenuse of the right triangle formed by Abigail's house and Clara's house.
Distance from Abigail's house to Clara's house (Direct route):
AC = sqrt[(2 - (-3))^2 + (3 - (-1))^2]
= sqrt[(2 + 3)^2 + (3 + 1)^2]
= sqrt[5^2 + 4^2]
= sqrt[25 + 16]
= sqrt[41] units
The difference in measurement between the two routes is:
Difference = Total distance - Direct route distance
= 9 - sqrt[41] units
To determine which route is faster, we compare the distances. If the Total distance is shorter, then stopping by Bob's Store is faster. If the Direct route distance is shorter, then going directly to Clara's house is faster. We can calculate this by comparing the numeric values of the distances.
Therefore, the difference in measurement between the two routes is 9 - sqrt[41] units, and the faster route will depend on the numeric values of this difference.