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If a particle is moving along a straight line according to the equation of motion , since the velocity may be interpreted as a rate of change of distance with respect to time, thus we have shown that the velocity of the particle at time “t” is the derivative of “s” with respect to “t”. There are many problems in which we are concerned with the rate of change of two or more related variables with respect to time, in which it is not necessary to express each of these variables directly as function of time. For example, we are given an equation involving the variables x and y, and that both x and y are functions of the third variable t, where t denotes time.
Since the rate of change of x and y with respect to t is given by and , respectively, we differentiate both sides of the given equation with respect to t by applying the chain rule. When two or more variables, all functions of t, are related by an equation, the relation between their rates of change may be obtained by differentiating the equation with respect to t.
Example 1 A 17 ft ladder is leaning against a wall. If the bottom of the ladder is pulled along the ground away from the wall at the constant rate of 5 ft/sec, how fast will the top of the ladder be moving down the wall when it is 8 ft above the ground? 17 ft. y x
Note: • Values which changes as time changes are denoted by variable. • The rate is positive if the variable increases as time increases and • is negative if the variable decreases as time increases.
Example 2 A balloon leaving the ground 60 feet from an observer, rises vertically at the rate 10 ft/sec . How fast is the balloon receding from the observer after 8 seconds? L h Viewer 60 feet
Example 3 As a man walks across a bridge at the rate of 5 ft/sec , a boat passes directly beneath him at 10 ft/sec. If the bridge is 30 feet above the water, how fast are the man and the boat separating 3 seconds later?
30’ S 5t L 10t 30’
Example 4 A man on a wharf of 20 feet above the water pulls in a rope, to which a boat is attached, at the rate of 4 ft/sec. At what rate is the boat approaching the wharf when there is 25 feet of rope out? R 20ft x
Example 5 Water is flowing into a conical reservoir 20 feet deep and 10 feet across the top, at the rate of 15 ft3/min . Find how fast the surface is rising when the water is 8 feet deep? 10 feet 5 feet r 20 feet h
Example 6 Water is flowing into a vertical tank at the rate of 24 ft3/min . If the radius of the tank is 4 feet, how fast is the surface rising? 4 feet h
Example 7 A triangular trough is 10 feet long, 6 feet across the top, and 3 feet deep. If water flows in at the rate of 12 ft3/min, find how fast the surface is rising when the water is 6 inches deep? 6 feet 3 feet 10 feet x h
Example 8 A train, starting at noon, travels at 40 mph going north. Another train, starting from the same point at 2:00 pm travels east at 50 mph . Find how fast the two trains are separating at 3:00 pm. C 3pm y B 2pm L 80 miles 3pm A 12pm D 2pm x
Example 9 A billboard 10 feet high is located on the edge of a building 45 feet tall. A girl 5 feet in height approaches the building at the rate of 3.4 ft/sec . How fast is the angle subtended at her eye by the billboard changing when she is 30 feet from the billboard? 10’ 45’ 5’ x
Example 10 A picture 40 cm high is placed on a wall with its base 30 cm above the level of the eye of an observer. If the observer is approaching the wall at the rate of 40 cm/sec, how fast is the measure of the angle subtended at the observer’s eye by the picture changing when the observer is 1 m from the wall? x
Example 11 A statue 10ft. high is standing on a base 13ft. high. If an observer’s eye is 5ft. above the ground, how far should he stand from the base in order that the angle between his lines of sight to the top and bottom of the statue be a maximum? 10’ 13’ x 5’
Therefore, the observer must be 12 ft from the base of the statue so that his line of sight from top to bottom of the statue is maximum.
EXERCISE A: What number exceeds its square by the maximum amount? The sum of two numbers is “K”. find the minimum value of the sum of their squares. A rectangular field of given area is to be fenced off along the bank of a river. If no fence is needed along the river, what are the dimensions of the rectangle that will require the least amount of fencing? A Norman window consists of a rectangle surmounted by a semicircle. What shape gives the most light for a given perimeter? A cylindrical glass jar has a plastic top. If the plastic is half as expensive as the glass per unit area, find the most economical proportions for the glass. Find the proportions of the circular cone of maximum volume inscribed in a sphere. A wall 8 feet high and 24.5 feet from a house. Find the shortest ladder which will reach from the ground to the house when leaning over the wall
EXERCISE B: 1. A sign 3 ft high is placed on a wall with its base 2 ft above the eye level of a woman attempting to read it. Find how far from the wall the woman should stand to get the “best view” of the sign; that is, so that the angle subtended at her eye by the sign is maximum. 2. A man on dock is pulling in at the rate of 2ft/sec a rowboat by means of a rope. The man’s hands are 20ft. above the level of the point where the rope is attached to the boat. How fast is the measure of the angle of depression of the rope changing when there are 52 ft. of rope out?
3. Find the equations of the normal line and tangent lines to the graph of the equation at the point . 4. A picture 5 ft high is placed on a wall with its base 7ft above the level of the eye of an observer is approaching the wall at the rate of 3ft/sec. How fast is the measure of the angle subtended at her eye by the picture changing when the observer is 10ft. from the wall? 5. An airplane is flying at a speed of 300mi/hr at an altitude of 4 mi. If an observer is on the ground, find the time rate of change of the measure of the observer’s angle of elevation of the airplane when the airplane is over a point on the ground 2 mi. from the observer.
