A boat leaves port and follows a course of at 9 knots for 3 hr and 20 min . Then, the boat changes to a new course of at 12 knots for 5 hr.
Part:
Part 1 of 3
(a) How far is the boat from port? Round the answer to one decimal place if necessary.
The boat is approximately

Ask by Young Chan. in the United States

Mar 25,2025

Question

A boat leaves port and follows a course of at 9 knots for 3 hr and 20 min . Then, the boat changes to a new course of at 12 knots for 5 hr.
Part:
Part 1 of 3
(a) How far is the boat from port? Round the answer to one decimal place if necessary.
The boat is approximately

Ask by Young Chan. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

Let the east direction be the positive $$ x $$-axis and the north be the positive $$ y $$-axis.

1. **Leg 1 (Course: $$ \mathrm{N}76^\circ\mathrm{E} $$)**  
   - Speed $$ = 9 $$ knots, time $$ = 3 $$ hr $$ 20 $$ min $$ = 3.\overline{3} $$ hr.  
   - Distance traveled:  
     $$
     d_1 = 9 \times \frac{10}{3} = 30 \text{ nautical miles.}
     $$
   - For a course of $$ \mathrm{N}76^\circ\mathrm{E} $$, the east ( $$ x $$ ) and north ( $$ y $$ ) components are:  
     $$
     x_1 = 30\sin76^\circ,\quad y_1 = 30\cos76^\circ.
     $$
     
2. **Leg 2 (Course: $$ \mathrm{S}24^\circ\mathrm{E} $$)**  
   - Speed $$ = 12 $$ knots, time $$ = 5 $$ hr.  
   - Distance traveled:  
     $$
     d_2 = 12 \times 5 = 60 \text{ nautical miles.}
     $$
   - For a course of $$ \mathrm{S}24^\circ\mathrm{E} $$, the boat is moving eastward and southward. Therefore, the components are:  
     $$
     x_2 = 60\sin24^\circ,\quad y_2 = -60\cos24^\circ.
     $$
     
3. **Calculate Numerical Values**  
   - Using approximations:
     $$
     \sin76^\circ \approx 0.9709,\quad \cos76^\circ \approx 0.2419,
     $$
     $$
     \sin24^\circ \approx 0.4067,\quad \cos24^\circ \approx 0.9135.
     $$
   - Leg 1 components:
     $$
     x_1 \approx 30 \times 0.9709 = 29.127,\quad y_1 \approx 30 \times 0.2419 = 7.257.
     $$
   - Leg 2 components:
     $$
     x_2 \approx 60 \times 0.4067 = 24.402,\quad y_2 \approx -60 \times 0.9135 = -54.81.
     $$

4. **Determine the Total Displacement**  
   - Sum of components:
     $$
     x_{\text{total}} = x_1 + x_2 \approx 29.127 + 24.402 = 53.529,
     $$
     $$
     y_{\text{total}} = y_1 + y_2 \approx 7.257 + (-54.81) = -47.553.
     $$
   - The distance from port is:
     $$
     d = \sqrt{x_{\text{total}}^2 + y_{\text{total}}^2} \approx \sqrt{(53.529)^2 + (-47.553)^2}.
     $$
   - Calculating:
     $$
     (53.529)^2 \approx 2866.0,\quad (-47.553)^2 \approx 2261.8,
     $$
     $$
     d \approx \sqrt{2866.0 + 2261.8} = \sqrt{5127.8} \approx 71.6.
     $$
     
5. **Final Answer**  
   The boat is approximately $$71.6$$ nautical miles from port.
The boat is approximately 71.6 nautical miles from port.

A boat leaves port and follows a course of at 9 knots for 3 hr and 20 min . Then, the boat changes to a new course of at 12 knots for 5 hr.
Part:
Part 1 of 3
(a) How far is the boat from port? Round the answer to one decimal place if necessary.
The boat is approximately

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A boat leaves port and follows a course of at 9 knots for 3 hr and 20 min . Then, the boat changes to a new course of at 12 knots for 5 hr. Part: Part 1 of 3 (a) How far is the boat from port? Round the answer to one decimal place if necessary. The boat is approximately

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A boat leaves port and follows a course of at 9 knots for 3 hr and 20 min . Then, the boat changes to a new course of at 12 knots for 5 hr.
Part:
Part 1 of 3
(a) How far is the boat from port? Round the answer to one decimal place if necessary.
The boat is approximately