A person starts walking from home and walks: 6 miles East 6 miles Southeast 3 miles South 1 miles Southwest 4 miles East Round your answers to 2 decimal places. This person has walked a total of Find the total displacement vector for this walk: If this person walked straight home, they'd have to walk Question Help:

Sure, let's tackle each part of the problem step by step. --- ### **1. Total Distance Walked** To find the **total distance** walked, simply add up all the individual distances: - **6 miles East** - **6 miles Southeast** - **3 miles South** - **1 mile Southwest** - **4 miles East** **Calculation:** \[ \text{Total Distance} = 6 + 6 + 3 + 1 + 4 = 20 \text{ miles} \] **Rounded to two decimal places:** \[ \boxed{20.00 \text{ miles}} \] --- ### **2. Total Displacement Vector** Displacement is a vector quantity that depends on both **magnitude** and **direction**. To find the total displacement vector, we'll break each movement into its **east (x-axis)** and **north (y-axis)** components, sum them up, and then determine the resultant vector. **Assumptions:** - **East** is positive along the **x-axis**. - **North** is positive along the **y-axis**. Let's break down each movement: 1. **6 miles East** - **x-component:** \( +6 \) miles - **y-component:** \( 0 \) miles 2. **6 miles Southeast** Southeast is **45° South of East**. - **x-component:** \( 6 \times \cos(45^\circ) \approx 6 \times 0.7071 \approx +4.24 \) miles - **y-component:** \( 6 \times \sin(45^\circ) \approx 6 \times (-0.7071) \approx -4.24 \) miles *(negative because it's South)* 3. **3 miles South** - **x-component:** \( 0 \) miles - **y-component:** \( -3 \) miles 4. **1 mile Southwest** Southwest is **45° South of West**. - **x-component:** \( 1 \times \cos(45^\circ) \approx 1 \times (-0.7071) \approx -0.71 \) miles - **y-component:** \( 1 \times \sin(45^\circ) \approx 1 \times (-0.7071) \approx -0.71 \) miles 5. **4 miles East** - **x-component:** \( +4 \) miles - **y-component:** \( 0 \) miles **Summing Components:** - **Total x-component:** \[ 6 + 4.24 + 0 + (-0.71) + 4 = 13.53 \text{ miles East} \] - **Total y-component:** \[ 0 + (-4.24) + (-3) + (-0.71) + 0 = -7.95 \text{ miles South} \] **Total Displacement Vector:** \[ \boxed{13.54 \text{ miles East and } 7.95 \text{ miles South}} \] *(Rounded to two decimal places)* --- ### **3. Straight-Line Distance to Walk Home** To find the **straight-line distance** back home, calculate the **magnitude** of the displacement vector using the Pythagorean theorem. **Given:** - **Eastward displacement (x):** \( 13.54 \) miles - **Southward displacement (y):** \( 7.95 \) miles **Calculation:** \[ \text{Distance} = \sqrt{(13.54)^2 + (7.95)^2} \] \[ \text{Distance} = \sqrt{183.07 + 63.20} = \sqrt{246.27} \approx 15.70 \text{ miles} \] **Rounded to two decimal places:** \[ \boxed{15.70 \text{ miles}} \] --- ### **Summary of Answers** 1. **Total Distance Walked:** \(\boxed{20.00 \text{ miles}}\) 2. **Total Displacement Vector:** \(\boxed{13.54 \text{ miles East and } 7.95 \text{ miles South}}\) 3. **Straight-Line Distance to Walk Home:** \(\boxed{15.70 \text{ miles}}\) --- If you have any further questions or need more clarification on any of the steps, feel free to ask!