related rates 5.6. first, a review problem: consider a sphere of radius 10cm. if the radius changes...
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First, a review problem:
Consider a sphere of radius 10cm.
If the radius changes 0.1cm (a very small amount) how much does the volume change?
34
3V r
24dV r dr
24 10cm 0.1cmdV
340 cmdV
The volume would change by approximately .340 cm
Now, suppose that the radius is changing at an instantaneous rate of 0.1 cm/sec.
(Possible if the sphere is a soap bubble or a balloon.)
34
3V r
24dV dr
rdt dt
2 cm4 10cm 0.1
sec
dV
dt
3cm
40sec
dV
dt
The sphere is growing at a rate of .340 cm / sec
Note: This is an exact answer, not an approximation like we got with the differential problems.
Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?
L3
sec
dV
dt
3cm3000
sec
(r is a constant.)
(We need a formula to relate V and h. )
Steps for Related Rates Problems:
1. Draw a picture (sketch).
2. Write down known information.
3. Write down what you are looking for.
4. Write an equation to relate the variables.
5. Differentiate both sides with respect to t.
6. Evaluate – PLUG IN WHAT YOU KNOW AFTER DIFFERENTIATING .
Truck Problem:Truck A travels east at 40 mi/hr.Truck B travels north at 30 mi/hr.
How fast is the distance between the trucks changing 6 minutes later?
Related Rates
Assume that the radius r and height h of a cone are differentiable functions of t and let V be the volume V=(πr2h)/3 of the cone. Find an equation that relates dV/dt, dr/dt, and dh/dt.
A Highway CaseA police cruiser, approaching a right-angled
intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the same instant of measurement, what is the speed of the car?
Filling a Conical Tank
Water runs into a conical tank at the rate of 9 ft3/m. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep.
The sides of a right triangle with legs and and hypotenuse
increase in such a way that / 1 and / 3 / . At the instant
when 4 and 3, what is / ?
(A) 1/3
(B) 1
(C) 2
(D) 5
(E) 5
x y z
dz dt dx dt dy dt
x y dx dt