Answer :
The distance between the ships changing at 6 PM is 31.667 knots
Let x equal the distance covered by ship A,
y equal the distance covered by ship B, and
z represent the separation between ships A and B.
At noon, ship A is 40 nautical miles west of ship B.
While ship B is moving north at 21 knots,
ship A is moving west at 24 knots.
So, x = miles at noon + knots of ship A
x = 40 + 24t .....ep. 1
y = 21t .....ep. 2
Ships changing at 6 PM, So t = 6
Substituting values in eq 1 and 2
x = 40 + 24 (6) = 40 + 144 = 184
y = 21(6) = 126
We will use Pythagoras' theorem
Pythagoras theorem formula is z² = x² + y², where x is the perpendicular side, y is the base, and z is the hypotenuse side. The Pythagoras equation is applied to any triangle that has one of its angles equal to 90°.
z² = x² + y² .....ep. 3
z² = (184)² + (126)²
z² = 33856 + 15876
z² = 49732
z = [tex]\sqrt{49732}[/tex]
z = 223.0067
Substitute the x and y in equation 3
z² = (40 + 24t)² + (21t)² .....ep. 4
Using differentiation
The geometrical meaning of the derivative of y = f(x) is the slope of the tangent to the curve y = f(x) at ( x, f(x)).
Power Rule: [tex]\frac{d}{dx} (x^{n}) = nx^{n-1}[/tex]
Differentiating both sides of equation 4,
[tex]2z\frac{dz}{dt} = 2(40 + 24t)(24)\frac{dt}{dt} + 2(21t)(21) \frac{dt}{dt} \\\\2z\frac{dz}{dt} = 2(40 + 24t)(24) + 2(21t)(21) \\\\z\frac{dz}{dt} = (40 + 24t)(24)\frac{dt}{dt} + (21t)(21) \frac{dt}{dt} \\\\\frac{dz}{dt} = \frac{(40 + 24t)(24)+ (21t)(21) }{z} \\ \\\frac{dz}{dt} = \frac{(40 + 24(6))(24)+ 21(6)(21) }{223.0067} \\ \\\frac{dz}{dt} = \frac{4416\;+\; 2646}{223.0067} \\\\\frac{dz}{dt} = \frac{4416\;+\; 2646}{223.0067} \\\\\frac{dz}{dt} = 31.667[/tex]
Hence, The distance between the ships changing at 6 PM is 31.667 knots
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