Question

1. to get to the top of the mountain, heather drove 10 east and 4 miles south. then she hiked the rest of the way to the top which was about 1 mile high. assuming heather started at (0, 0, 0), what is the total distance she traveled to the top of the mountain? 7 miles 10.8 miles 15 miles 11.8 miles

Explanation:

Step1: Calculate driving distance

$\text{Driving distance} = 10 + 4 = 14$ miles

Step2: Calculate hiking distance

Use 3D distance formula: $\sqrt{10^2 + 4^2 + 1^2} = \sqrt{100 + 16 + 1} = \sqrt{117} \approx 10.8167$ miles

Step3: Sum total travel distance

$\text{Total distance} = 14 + 10.8167 \approx 24.8167$
*Note: Correction: Reinterpret question: total distance is the straight-line path from start to top? No, re-read: total distance traveled is driving + hiking. Wait, no—wait, the hiking is the straight line from end of drive to top. Drive is 10+4=14, hike is $\sqrt{10^2+4^2+1^2}\approx10.8$? No, no—wait, no: the question says "total distance she traveled": driving (10+4) plus hiking (the straight line from (10,-4,0) to (0,0,1)? No, no: start at (0,0,0), drive 10 east to (10,0,0), then 4 south to (10,-4,0), then hike to top which is 1 mile high from start? No, the top is at (10,-4,1). So hiking distance is distance from (10,-4,0) to (10,-4,1)? No, that can't be. Wait, no, the question says "the rest of the way to the top which was about 1 mile high"—so the top is 1 mile above the starting elevation, so the 3D displacement from start to top is (10, -4, 1). The total distance traveled is driving (10+4) plus hiking (the straight line from end of drive to top). Wait, no, maybe the question means the total straight-line distance from start to top? No, the options don't have 10.8 is the straight line from start to top. Wait, 10.8 is $\sqrt{10^2+4^2+1^2}=\sqrt{117}\approx10.8$, which is an option. Oh! Maybe the question means the total straight-line distance from start to top, not the sum of driving and hiking. That must be the intended question (likely wording error). So recalculate:

Step1: Apply 3D distance formula

Total straight-line distance from $(0,0,0)$ to $(10,-4,1)$:
$$d = \sqrt{(10-0)^2 + (-4-0)^2 + (1-0)^2}$$

Step2: Compute inside the square root

$10^2 + (-4)^2 + 1^2 = 100 + 16 + 1 = 117$

Step3: Calculate square root

$d = \sqrt{117} \approx 10.8$ miles

Answer:

10.8 miles