To solve this problem, we can break it down step-by-step.
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Determine the time it takes for the runner to reach the finish line:
The runner is 12 km from the finish line and is jogging at a speed of \(v_r = 4 \text{ km/hr}\). The time \(t_r\) it takes for the runner to reach the finish line is given by:
\[ t_r = \frac{L}{v_r} = \frac{12 \text{ km}}{4 \text{ km/hr}} = 3 \text{ hours} \]
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Determine the distance the bird travels to the finish line:
The bird starts flying when the runner is 12 km away from the finish line and flies directly to the finish line at a speed of \(v_b = 16 \text{ km/hr}\). The time \(t_b\) it takes for the bird to reach the finish line is given by:
\[ t_b = \frac{L}{v_b} = \frac{12 \text{ km}}{16 \text{ km/hr}} = 0.75 \text{ hours} \]
In this time, the bird travels a distance of 12 km to reach the finish line.
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Determine how far the runner travels during that time:
During the 0.75 hours that it takes for the bird to reach the finish line, the runner will also be moving. The distance the runner covers while the bird is flying to the finish line is:
\[ d_r = v_r \cdot t_b = 4 \text{ km/hr} \times 0.75 \text{ hr} = 3 \text{ km} \]
So, when the bird reaches the finish line, the runner has traveled 3 km and is now 9 km away from the finish line.
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Determine the time it takes the bird to return to the runner:
After reaching the finish line, the bird immediately turns around and flies back to the runner, who is now 9 km away from the finish line. At this point, the runner is 3 km away from the original start point. The distance the bird needs to cover to get back to the runner is also 9 km.
The time \(t_r' \) for the bird to fly back to the runner is:
\[ t_r' = \frac{9 \text{ km}}{16 \text{ km/hr}} = 0.5625 \text{ hours} \]
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Calculate the total distance traveled by the bird:
When the bird first flies to the finish line, it travels 12 km. Then it flies back to the runner, traveling an additional 9 km. Therefore, the total distance traveled by the bird is:
\[ \text{Total Distance} = 12 \text{ km} + 9 \text{ km} = 21 \text{ km} \]
Thus, the cumulative distance the bird travels is \( \boxed{21} \) km.