A = πr 2. , the area changes in m2

Transcription

1 1 Related Rates So,what sarelatedratewordproblem? Inthese type of word problems two quantities are related somehow, like area and radius of a circle. The area, A, and radius, r, ofacirclearerelatehroughthe formula A = πr 2 As the radius changes in say m s, the area changes in m2 s. That is, the rate at which the radius changes effects the rate at which the area changes. The rates are related, hence, the title. The derivative comes into play because we are talking about rates of change and we can think of the tangent slope as the instantaneous rate of change of a function. Now, if we differentiate the function, A = πr 2, with respect to r we get da dr =2πr but there is no reference to time here. Where s the per second part? What we have is a formula that relates the change in radius to the change in area but doesn t relate the rates. To introduce a time component we differentiate the equation, A = πr 2, with respect to time. In a sense we are considering radius and area to be functions of time. To illustrate this, consider the function rewritten

2 as A(t) =π [r(t)] 2 and differentiate it with respect to time using the chain rule. That results in da(t) =2π [r(t)] dr(t) Normally we would skip the formality of rewriting the equation and our end result would be da =2πrdr This is the formula that relates the rate of change of the area, da, with the rate of change of the radius, dr da m2. Notice that the units for are sec because the da part is area and is measured in metres 2 and the is time measured in seconds in this case. The units for dr m would therefore be sec. Let s do two examples to illustrate the related rate idea in use. Example Q? A spherical balloon is loosing air at a rate of 2 cm3 min. How fast is the radius of the balloon shrinking when the radius is 3 m? A. The first sentence tells us how the volume is changing, that is dv = 2. Notice that we make it negative because the balloon s volume is shrinking, the change is negative. The question wants us to

3 find dr. First we must decide how the volume and the radius are related. We use the volume of a sphere formula. V = 4 3 πr3 Now differentiate that formula with respect to time to get dv = 4 3 π3r2dr =4πr 2dr Now substitute dv = 2,r = 3into the equation and solve for dr. Therefore dr = 1 18π 2=4π(3) 2dr dr = 1 18π cm min. Example Q? The water in a cylindrical glass is rising at arateof1 cm sec. Theradiusoftheglassis2cm. What is the rate at which water is being poured into the glass? A. The formula for the volume of a cylinder is V = πr 2 h where h is the height and r is the radius. This is

4 the formula that relates volume and height. We want the rate of change of volume, dv. Differentiating the above formula with respect to time involves the product rule because both radius and height can be thought of as functions of time. This yields dv =2πrdr h + πr2dh Note that we know that the radius is constant so we could have treated it as such but the math will take care of it. We now set dh dr =1, =0and r =2to get dv =2π(2)(0)h + π2 2 (1) =4π Therefore the volume is increasing at 4π cm3 sec. So, when I see a word problem, how do I know it s a related rate problem? The word rate is usually there, but also the presence of units like m cm3 s or min could signal a related rate question. Then you write the function that relates the variables in question, differentiate it with respect to time, substitute for the known quantities and solve for the one left over unknown. Remember to include units in your answer. Also look at the examples in section 4.1 of the text.

5 Section 4.1, #1-3, 5-12, 15, 18, 21, 27 Submit # 6, 12, 18 on Monday. Use the odd number questions to verify that you know what you are doing.

6 Example 1 Oil spills from an Exxon tanker in a huge circular slick. The radius of the slick is increasing 10 m min.how fast is the area of the slick increasing 1 hour after the spill started? Example 2 A canoe is being pulled into a dock by a rope attached to the bow and passing thru a pulley on the dock that is 1 metre higher than the bow of the canoe. If the rope is being pulled in at a rate of 1 m s, how fast is the canoe approaching the dock when it is 8 m from the dock? Can we create a function that relates speed of the canoe with the distance from the dock? Example 3 I was in lock at the canal near Pittsford has a trapezoidal cross section that is 30 feet at the bottom and 40 feet at the top. The total height is 40 feet. The lock is 60 feet long. If my boat is dropping at 1 ft min at the halfway point, how fast is the water being drained from the lock?

7 Example 4 Sound intensity various inversely with the square of the distance from the speaker stacks. Decibels are measured withe the equation I D =log What is the rate of change of the decibel levels as I walk away from the speaker stack at 3 m s when I am 10m away. I 0