A ship is sailing away from a lighthouse at a constant speed of 20 knots. At a certain point, the ship is 10 nautical miles from the lighthouse, and the lighthouse is 6 nautical miles inland. Find the rate at which the distance between the ship and the lighthouse is increasing.
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To solve this problem, let's draw a diagram:
```
*
\ |
d2 \ | d1
\ |
* S
\
d2 \ |
\|
L
```
Let:
- S be the ship
- L be the lighthouse
- d1 be the distance between the ship and the lighthouse
- d2 be the distance between the ship and the coast
We are given that the ship is sailing away from the lighthouse at a constant speed of 20 knots. This tells us that the distance between the ship and the coast, d2, is increasing at a rate of 20 knots.
We want to find the rate at which the distance between the ship and the lighthouse, d1, is increasing. To do this, we use the Pythagorean theorem:
d1² = d2² + 6²
Differentiating both sides of the equation with respect to time t, we get:
2d1 * d1' = 2d2 * d2'
Plugging in d1 = 10 and d2 = 20, we can solve for d1':
2 * 10 * d1' = 2 * 20 * 20'
Simplifying, we get:
10 * d1' = 20 * 20'
We are given that the ship is moving at a constant speed of 20 knots, so 20' = 20. Plugging this in, we get:
10 * d1' = 20 * 20
d1' = 400 / 10
d1' = 40
Therefore, the rate at which the distance between the ship and the lighthouse is increasing is 40 knots.
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