Question

Shahriyar is at a winter park with his family they go sledding first then ice skating then skiing what is the total distance traveled between these location ice skating (2, -4) skiing (-4, -8) sledding (2, 12) to miles

Answer 1

To find the total distance traveled between the locations of sledding, ice skating, and skiing, we can treat the locations as points in a coordinate system and use the distance formula.

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Locations:
- Sledding: \((2, 12)\)
- Ice Skating: \((2, -4)\)
- Skiing: \((-4, -8)\)

### Step 1: Calculate distance from sledding to ice skating
Let \(A = (2, 12)\) and \(B = (2, -4)\).

\[
d_{AB} = \sqrt{(2 - 2)^2 + (-4 - 12)^2} = \sqrt{0 + (-16)^2} = \sqrt{256} = 16
\]

### Step 2: Calculate distance from ice skating to skiing
Let \(B = (2, -4)\) and \(C = (-4, -8)\).

\[
d_{BC} = \sqrt{(-4 - 2)^2 + (-8 - (-4))^2} = \sqrt{(-6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \approx 7.21
\]

### Step 3: Calculate distance from sledding to skiing
Let \(A = (2, 12)\) and \(C = (-4, -8)\).

\[
d_{AC} = \sqrt{(-4 - 2)^2 + (-8 - 12)^2} = \sqrt{(-6)^2 + (-20)^2} = \sqrt{36 + 400} = \sqrt{436} = 2\sqrt{109} \approx 20.88
\]

### Step 4: Total distance traveled
The total distance traveled is:

\[
\text{Total Distance} = d_{AB} + d_{BC} + d_{AC} = 16 + 2\sqrt{13} + 2\sqrt{109}
\]

Approximating the square roots:

- \(2\sqrt{13} \approx 2 \times 3.60555 \approx 7.21\)
- \(2\sqrt{109} \approx 2 \times 10.4403 \approx 20.88\)

Now adding these values:

\[
\text{Total Distance} \approx 16 + 7.21 + 20.88 = 44.09
\]

So, the total distance traveled is approximately **44.09 miles**.