3.2 RELATED RATES

In a related rates problem, we are given the rate of change of one quantity and wish to find the rate of change of another. Such problems can often be solved by implicit differentiation.

EXAMPLE 1

The point of a fountain pen is placed on an ink blotter, forming a circle of ink whose area increases at the constant rate of 0.03 in. ²/sec. Find the rate at which the radius of the circle is changing when the circle has a radius of ____ inch. We solve the problem in four steps.

 

Step 1  Label all quantities involved and draw a diagram.

                       t = time   A = area     r = radius of circle

       Both A and r are functions of t. The diagram is shown in Figure 3.2.1.

 

 

 

 

 

 

 

Figure 3.2.1

 

Step 1  Write the given information in the form of equations.

                          dA/dt = 0.03.      A= πr².

         The problem is to find dr/dt when r =1/2.

Step 3   Differentiate both sides of the equation A = πr². with respect to t.

    

 

 

Step 4  Set r = ___ and solve for dr/dt.

                      

 

 

EXAMPLE  2 

A 10 foot ladder is propped against a wall. The bottom end is being pulled along the floor away from the wall at the constant rate of 2ft/sec. Find the rate at which the top of the ladder is sliding down the wall when the bottom end is 5ft from the wall.

Warning : although the bottom end of the ladder is being moved at a constant rate, the rate at which the top end moves will vary with time.

 

Step 1  t =time,

       x = distance of the bottom end from the wall,

       y = height of the top end above the floor.

 

The diagram is shown in Figure 3.2.2.

 

 

 

 

 

 

Figure 3.2.2

 

Step 2 dx/dt = 2,  x²+y² =10² =100.

Step 3  we differentiate both sides of x²+y² =100with respect to t.

                       

 

 

Step 4  Set x=5 ft and solve for dy/dt. We first find the value of y when x=5.

             x²+y² =100,   

 

       Then we can solve for dy/dt.

                 

 

          

 

 

 

The sing of dy/dt is negative because y is decreasing.

 

Related rates problems have the following form.

Given:

(a)Two quantities which depend on time, say x and y.

(b)The rate of change of one of them, say dx/dt.

(c)An equation showing the relationship between x and y.

 

(Usually this information is given in the form of a verbal description of a physical situation and part of the problem is to express it in the form of an equation.)

 

The problem: Find the rate of change of y, dy/dt, at a certain time t0. ( the time t0 is sometimes specified by giving the value which x, or y, has at that time.)

 

Related rates problems can frequently be solved in four steps as we did in the examples.

 

Step 1   Label all quantities in the problem and draw a picture. If the labels are x, y,

        and t (time ). The remaining steps are as follows:

Step 2  Write an equation for the given rate of change dx/dt. Write another equation for the   

        given relation between x and y.

Step 3  Differentiate both sides of the equation relating x and y with respect to t.

       We choose the time t as the independent variable.

       The result is a new equation involving x, y, dx/dt, and dy/dt.

 

Step 4   Set t=t0 and solve for dy/dt. It may be necessary to find the values of x, y, and dx/dt at t=t0 first.

 

The hardest step is usually step 2, because one has to start with the given verbal description of the problem and set up as a system of formulas. Sometimes more than two quantities that depend on time are given. Here is an example with three.

 

EXAMPLE 3 

One car moves north at 40mph (miles per hour) and passes a point P at time 1:00. Another car moves east at 60 mph and passes the same point P at time 2:30. How fast is the distance between the two cars changing at the time 2:00?

 

It it not even easy to tell whether the two cars are getting closer or farther away at time 2:00. This is part of the problem.

 

Step 1  t= time,

       y= position of the first car travelling north,

       x= position of the second car travelling east,

       z= distance between the two cars.

 

       In the diagram in Figure 3.2.3, the point P is placed at the origin.

 

 

 

 

 

 

 

Figure 3.2.3

 

Step 2     

 

  

              x² + y² =z².

 

Step 3  

 

 

         

 

 

Step 4  We first find the values of x, yand z at the time t=2 hrs. We are given that when t=1,

        y = 0. In the next hour the car goes 40 miles, so at t=2 , y=40.

        We are given that at time t=__, x=0. One -half hour before that the car was 30 miles to

        the left of P, so at t =2, x = -30. To sum up,

                         at  t =2,   y=40  and x= -30.

        We can now find the value of z at t = 2,

 

 

         Finally, we solve for dz/dt at t =2

 

 

      The negative sign shows that z is decreasing. Therefore at 2:00 the cars are getting closer to each other at the rate of 4 mph.

 

EXAMPLE  4 

The population of a country is growing at the rate of one million people per year, while gasoline consumption is decreasing by one billion gallons per year. Find the rate of change of the per capita gasoline consumption when the population is 30 million and total gasoline consumption is 15 billion gallons per year.

 

By the per capita gasoline consumption we mean the total consumption divided by the population.

 

Step 1  t = time

       x = population

       y = gasoline consumption

       z = per capita gasoline consumption.

 

Step 2   At t= t0,

 

                    dx/dt = 1million =106

                    dy/dt = -1 billion = -109

                        z =y/x.

 

 

Step 3  

 

 

 

 

 

Step 4   at t= t0, we are given

                         x = 30 million =30×106

                         y = 15 billion = 15×109

    

           Thus          

                        

 

 

 

The per capita gasoline consumption is decreasing at the annual rate of 50 gallons per person.

 

We conclude with another example from economics. In this example the independent variable is the quantity x of a commodity. The quantity x which can be sold at price p is called the demand function D(p).

                       x=D(p).

 

When a quantity x is sold at price p, the revenue is the product

                            R = px.

 

The additional revenue from the sale of an additional unit of the commodity is called the marginal revenue and is given by the derivative

                             marginal revenue = dR/dx.

 

EXAMPLE  5 Suppose the demand for a product is equal to the inverse of the square of the

        price. Find the marginal revenue when the price is $10 per unit.

 

Step 1   p= price, x= demand, R= revenue.

Step 2   x=1/p²,   R=px

Step 3  

 

 

 

         

so by the Inverse Function Rule,

       

 

 

 

     Substituting,   

 

 

 

Step 4  We are given p=$10. Therefore the marginal revenue is

                                dR/dx = $5.

An additional unit sold would bring in an additional revenue of $5.

 

Here is a list of formulas from plane and solid geometry which will be useful in related rates problems. We always let A= area and V= volume.

 

Rectangle with sides a and b:  A = ab,   perimeter = 2a + 2b

Triangle with base b and height h:   A =____ bh

Circle of radius r :    A= πr ² ,  circumference = 2 πr

Sector (pie slice) of a circle of radius r and central angle θ (measured in radians ) : A = ____

Rectangular solid with sides a, b, c:   V = abc

 

Sphere of radius r :

 

Right circular cylinder, base of radius r, height of h:  V= πr ² h,   A = 2πr h

Prism with base of area B and height h:  V = Bh

Right circular cone, base of radius r, height h: V=πr ² h/3,

 

 

PROBLEMS  FOR  SECTION 3.2

 

1  Each side of a square is expanding at the rate of 5 cm/sec. How fast is the area changing when the

   length of each side is 10cm?

2  The area of a square is decreasing at the constant rate of 2 sq cm/sec. How fast is the length of each side

   decreasing when the area is 1 sq cm?

3  The vertical side of a rectangle is expanding at the rate of 1 in. /sec, while the horizontal side is

   contracting at the rate of 1 in ./sec. At time t =1 sec the rectangle is a square whose sides are 2 in.long.  

   How fast is the area of the rectangle changing at time t =2sec?

4  Each edge of a cube is expanding at the rate of 1 in./sec. How fast is the volume of the cube changing

   when the volume is 27 cu in.?

5  Two cars pass point P at approximately the same time, one travelling north at 50 mph, the other

   travelling west at 60 mph. Find the rate of change of the distance between the two cars one hour after

   they pass the point P.

6  A cup in the form of a right circular cone with radius r and height h is being filled with water at the

   rate of 5cu in./sec. How fast is the level of the water rising when the volume of the water is equal to one

   half the volume of the cup?

7  A spherical balloon is being inflated at the rate of 10 cu in. /sec. Find the rate of change of the area 

   when the ballon has radius 6 in.

8  A snowball melts at the rate equal to twice its surface area, with area in square inches and melting

   measured in cubic inches per hour. How fast is the radius shrinking?

9  A ball is dropped from a height of 100ft, at which time its shadow is 500ft from the ball. How fast is the

   shadow moving when the ball hits the ground? The ball falls with velocity 32 ft/sec, and the shadow is

   cast by the sun.

10 A 6 foot man walks away from a 10 foot hight lamp at the rate of 3 ft/sec. How fast is the tip of his

   shadow moving?

 

11 A car is moving along a road at 60 mph. To the right of the road is a bush 10 ft away and a parallel wall

  30 ft away. Find the rate of motion of the shadow of the bush on the wall cast by the car headlights.

 

 

 

 

 

 

 

 

 

12 A car moves along a road at 60 mph. There is a bush 10 ft to the right of the road, and a wall 30 ft

  behind the bush is perpendicular to the road. Find the rate of motion of the shadow of the bush on the

  wall when the car is 26 ft from the bush.

 

 

 

 

 

 

 

 

 

 

13 An airplane passes directly above a train at an altitude of 6 miles. If the airplane moves north at 500

  mph and the train moves north at 100mph, find the rate at which the distance between them is

  increasing two hours after the airplane passes over the train.

 

14 A rectangle has constant area, but its length is growing at the rate of 10 ft/sec. Find the rate at which the

  width is decreasing when the rectangle is 3 ft long and 1 ft wide.

 

15 A cylinder has constant volume, but its radius is growing at the rate of 1ft/sec. Find the rate of change

  of its height when the radius and height are both 1 ft.

 

16 A country has constant national income, but its population is growing at the rate of one million people

   per year. Find the rate of change of the per capita income (national income divided by population)

   when the population is 20 million and the national income is 20 billion dollars.

 

17 If at time t a country has a birth rate of 1,000,000t births per year and a death rate of 300,000____

   deaths per year, how fast is the population growing?

 

18 The population of a country is 10 million and is increasing at the rate of 5000,000 people per year. The

   national income is $10 billion and is increasing at the rate of $100 million per year . Find the rate of

   change of the per capita income.

 

19 Work Problem 18 assuming that the population is decreasing by 500,000 per year.

 

20  Sand is poured at the rate of 4 cu in./sec and forms a conical pile whose height is equal to the radius of

    its base. Find the rate of increase of the height when the pile is 12 in. high.

 

21 A circular clock has radius 5 in. At time t minutes past noon, how fast is the area of the sector of the

   circle between the hour and minute hand increasing? (t ≤ 60).

 

22 The demand x for a commodity at price p is    

 Find the marginal revenue, that is , the change in revenue per unit change in x, when the price is $100

per unit.

 

23 x units of a commodity can be produced at a total cost of y= 100+5x. The average cost is defined as the

  total cost divided by x. Find the change in average cost per unit change in x (the marginal average cost )

  when x=100.

 

24 The demand for a commodity at price p is x =1 /(p+p3).

   Find the change of the price per unit change in x, dp/dx, when the price is 3 dollars per unit.

 

25 In one day a company can produce x items at a total cost of 200 + 3x dollars and can sell x items at a

   price of 5 - x/1000 dollars per item. Profit is defined as revenue minus cost. Find the change in profit

   per unit change in the number of items x (marginal profit).

 

26 In one day a company can produce x items at a total cost of 200+ 3x dollars and can sell 

items at a price of y dollars per item.

(a) Find the change in profit per dollar change in the price y (the marginal profit with respect to price).

(b) Find the change in profit per unit change in x (the marginal profit).

 

27 An airplane P flies at 400 mph one mile above a line L on the surface.

  An observer is at the point O and L. Find the rate of change (in radians per hour) of the angle θ between

  the line L and the line OP from the observer to the airplane when θ = π/6.

 

28 A train 20 ft wide is approaching an observer standing in the middle of the track at 100ft/sec.

  Find the rate of increase of the angle subtended by the train (in radians per second )

  when the train is 20 tf from the observer.

 

29 Find the rate of increase of e 2x+3y when x = 0, y=0 dx/dt = 5, and dy/dt =4.

 

30 Find the rate of change of 1n A where A is the area of a rectangle of sides x and y

   when x =1, y=2 , dx/dt = 3, dy/dt = -2.