Question

A cruise ship heads due west from a port 5 miles directly south of San Francisco. If the ship is travelling at a constant rate of 18 mph, how fast is the distance between the ship and San Francisco changing 1 hour after leaving port? Round your answer to the nearest tenth.

Answer 1

Given that the Ship traveling at 18 mph is 5 miles south of San Francisco, the rate of change of the distance from San Francisco is approximately 17.57 mph

How can the rate of change of the distance be found?

Given;

• Vertical distance from the path of the ship to San Francisco = 5 miles

• Rate at which the ship is traveling = 18 mph

Required;

The rate at which the distance from the ship to San Francisco is changing 1 hour after leaving the port.

Solution;

Let x represent the horizontal distance from the ship to San Francisco, and let y represent the vertical distance, we have;

Using Pythagorean theorem, we have;

The distance from the ship to San Francisco, d is; d² = x² + 4²

Which gives;

$2\cdot d\cdot \dfrac{dd}{dt} = 2\cdot x \cdot \dfrac{dx}{dt}$

After 1 hour, we have;

x = 18 miles

Which gives;

d² = 18² + 4² = 340

d = 2•√(85)

$\dfrac{dd}{dt} = \dfrac{2\cdot x \cdot \dfrac{dx}{dt}}{2\cdot d}$

Therefore;

$\dfrac{dd}{dt} = \dfrac{36 \times 18}{2\times 2•√(85)} ≈ 17.57$

The rate at which the distance between San Francisco is changing after 1 hour is, d ≈ 17.57 m/s

Learn more about Pythagorean theorem here:

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