# Time Rates

## 38 - Rate of rotation of search light pointing to a ship

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## 37 - A ladder sliding downward

**Problem 37**

A ladder 15 ft long leans against a vertical wall. If the top slides down at 2 ft/sec, how fast is the angle of elevation of the ladder decreasing, when the lower end is 12 ft from the wall?

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## Problems in Caculus Involving Inverse Trigonometric Functions

The following are problems involving inverse trigonometric functions.

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## 37-38 How fast a ship leaving from its starting point

**Problem 37**

A ship sails east 20 miles and then turns N 30° W. If the ship's speed is 10 mi/hr, find how fast it will be leaving the starting point 6 hr after the start.

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## 35-36 Time Rates: Lengthening of shadow and movement of its tip in 3D space

**Problem 35**

An arc light hangs at the height of 30 ft above the center of a street 60 ft wide. A man 6 ft tall walks along the sidewalk at the rate of 4 ft/sec. How fast is his shadow lengthening when he is 40 ft up the street?

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## 33-34 Time Rates: A car traveling east and airplane traveling north

**Problem 33**

From a car traveling east at 40 miles per hour, an airplane traveling horizontally north at 100 miles per hour is visible 1 mile east, 2 miles south, and 2 miles up. Find when this two will be nearest together.

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## 31-32 Train in an elevated track and car in perpendicular road

**Problem 31**

An elevated train on a track 30 ft above the ground crosses a street at the rate of 20 ft/sec at the instant that a car, approaching at the rate of 30 ft/sec, is 40 ft up the street. Find how fast the train and the car separating 1 second later.

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## 30 - Two trains in perpendicular railroad tracks

**Problem 30**

Two railroad tracks intersect at right angles, at noon there is a train on each track approaching the crossing at 40 mi/hr, one being 100 mi, the other 200 mi distant. Find (a) when they will be nearest together, and (b) what will be their minimum distance apart.

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## 28-29 Time Rates: Two cars driving on roads that intersects at 60 degree

**Problem 28**

At noon a car drives from A (Fig. 48) toward C at 60 miles per hour. Another car starting from B at the same time drives toward A at 30 miles per hour. If AB = 42 miles, find when the cars will be nearest each other.

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## 26-27 Time Rates: Kite moving horizontally

**Problem 26**

A kite is 40 ft high with 50 ft cord out. If the kite moves horizontally at 5 miles per hour directly away from the boy flying it, how fast is the cord being paid out?

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