At noon, ship A is 40 nautical miles due west of ship B. Ship A is sailing west at 18 knots and ship B is sailing north at 23 knots. How fast (in knots) is the distance between the ships changing at 4 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)
To find the rate at which the distance between the two ships is changing at 4 PM, we can use the concept of related rates. Let's break down the problem step by step:
Step 1: Define the variables
Let's define:
- x as the distance ship A has traveled (in nautical miles) from noon to 4 PM
- y as the distance ship B has traveled (in nautical miles) from noon to 4 PM
- d as the distance between the two ships at 4 PM
- t as the time elapsed from noon to 4 PM (in hours).
Step 2: Establish the relationships
At any given time t, the position of ship A can be represented as (40 - x, 0) (given that it is 40 nautical miles due west of ship B), and the position of ship B can be represented as (0, 23t) (given that it is sailing north at 23 knots).
Step 3: Calculate the distance between the ships
Using the distance formula, the distance d between the two ships at time t is given by:
d = sqrt[(40 - x)^2 + (23t)^2]
Step 4: Differentiate both sides of the equation with respect to time
Differentiating both sides of the equation d = sqrt[(40 - x)^2 + (23t)^2] with respect to time t, we get:
dd/dt = (1/2) * ( (40 - x)(-dx/dt) + 2(23t)(23) )
Step 5: Substitute the given values to find the speed at 4 PM
At 4 PM, t = 4 hours. We need to find d at t = 4 and dx/dt and x at t = 4 to substitute into the equation dd/dt.
Since ship A is sailing west at a constant speed of 18 knots, the distance it has traveled from noon to 4 PM is x = 18 * 4 = 72 nautical miles. Since ship B is sailing north at a constant speed of 23 knots, the distance it has traveled from noon to 4 PM is y = 23 * 4 = 92 nautical miles.
Plugging in the values, at t = 4 hours:
d = sqrt[(40 - 72)^2 + (23 * 4)^2]
d = sqrt[(-32)^2 + (92)^2]
d = sqrt[1024 + 8464]
d = sqrt[9488]
d is approximately 97.37 nautical miles.
Substituting the values into the equation dd/dt = (1/2) * ( (40 - x)(-dx/dt) + 2(23t)(23) ):
(dd/dt) | t=4 = (1/2) * ( (40 - 72)(-18) + 2(23 * 4)(23) )
(dd/dt) | t=4 = (1/2) * ( -32(-18) + 2(92)(23) )
(dd/dt) | t=4 = (1/2) * ( 576 + 4232 )
(dd/dt) | t=4 ≈ 2404
Therefore, the speed at which the distance between the ships is changing at 4 PM is approximately 2404 knots.
To find the speed at which the distance between the ships is changing, we can use the concept of related rates from calculus.
Let's start by identifying the relevant information:
- Ship A is moving west at a constant speed of 18 knots.
- Ship B is moving north at a constant speed of 23 knots.
- The initial distance between the ships at noon is 40 nautical miles due west of each other.
Now, let's proceed with finding the speed at which the distance between the ships is changing at 4 PM. To do this, we can follow these steps:
Step 1: Determine the position of each ship at 4 PM.
Since the ships were initially 40 nautical miles west of each other, at 4 PM, Ship A would have traveled for 4 hours at a speed of 18 knots, covering a distance of 4 * 18 = 72 nautical miles west from its initial position.
Since Ship B is moving north at a speed of 23 knots, it would have covered a distance of 4 * 23 = 92 nautical miles north from its initial position.
Step 2: Determine the distance between the ships at 4 PM.
To find the distance between the ships at 4 PM, we can use the Pythagorean theorem. The distance between the ships can be calculated as the square root of the sum of the squares of the east-west and north-south distances.
Distance^2 = (east-west distance)^2 + (north-south distance)^2
Distance^2 = (72)^2 + (92)^2
Distance^2 = 5184 + 8464
Distance^2 = 13648
Distance ≈ 116.84 nautical miles
Step 3: Determine the rate at which the distance is changing.
To find the rate at which the distance between the ships is changing, we need to differentiate the distance equation with respect to time (t).
Differentiating both sides of the equation, we get:
2 * Distance * (d(Distance)/dt) = 2 * (east-west distance) * (d(east-west distance)/dt) + 2 * (north-south distance) * (d(north-south distance)/dt)
Since the east-west distance is constant (Ship A is only moving west), the derivative term (d(east-west distance)/dt) would be zero.
We know that the east-west distance is 72 nautical miles and the north-south distance is 92 nautical miles. Substituting these values into the equation, we get:
2 * 116.84 * (d(Distance)/dt) = 2 * 72 * 0 + 2 * 92 * 23
Simplifying the equation gives:
2 * 116.84 * (d(Distance)/dt) = 2 * 92 * 23
Now, we can solve for (d(Distance)/dt), which represents the rate of change of the distance between the ships.
116.84 * (d(Distance)/dt) = 92 * 23
(d(Distance)/dt) = (92 * 23) / 116.84
Calculating this value gives:
(d(Distance)/dt) ≈ 18.24 knots
Therefore, the speed at which the distance between the ships is changing at 4 PM is approximately 18.24 knots.
check this question and response from October.
Just change the numbers to fit yours.
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