related rates

How fast is the depth of the water changing when the depth of the water is 15 cm?

25.8 cm

19.6

30.4

h

When full, the volume of the truncated cone is 12, 379. 6 cm3.

It took 2 mins and 58 seconds to fill the container so the container was being filled at a rate of

i.e

Related Rates

Finding the relationships between different variables and the rates at which these variables are changing.

Related Rate Problem Strategy

1. Draw a picture, naming all variables and constants. Use t for time and assume all variables are differentiable functions of t.

2. Write down all numerical information, in terms of your variables, stated in the problem.

3. Write down, in terms of your variables, what you are asked to find.

4. Write an equation that relates the variables.

5. Differentiate your equation with respect to t.

6. Evaluate the unknown rate using the known values.

The public observation platform from which to watch the shuttle launch is three miles from the launch pad. Assume that the shuttle rises at an estimated speed of 583 feet per second. How quickly is the angle of elevation changingthree seconds after the launch?

see next page

LPPOP

h

3 miles

Notice: Miss - match of units with ft/sec and distance measured in miles. Shuttle travels 3(583) = 1749 ft in 3 seconds.

GIVEN :

FIND : when t = 3

h = 15840tanθdh = dh dθdt dθ dtdh = 15840sec2 θdθdt dt

We are interested in dθ/dt when t = 3. Since the shuttle is moving at a speed of 583 ft/sec, it will be 583(3) = 1749 ft above the ground after 3 seconds and the situation will be as shown below.

θ

h = 1749 ft

15840 ft

√(158402 + 17492) cosθ = 15840

√(158402 + 17492)

what we want to find

583 = 15840sec2θ dθ dt

dθ = 583 cos2 θdt 15840dθ = 583 158402

dt 15840 √158402 + 17492 )

dθ = 0.0364 rad/secdt

The radius of a sphere is increasing at a constant rate of 0.5 inch/second.

a)When the radius of the sphere is 15 inches, at what rate is the volume of the sphere changing?

b) When the volume and the radius of the sphere are changing at the same rate, what is the radius of the sphere?

The edges of a cube are increasing at a rate of 2 cm/sec.

a) How fast is the volume of the cube increasing when each edge is 5 cm long?

b) How fast is the surface area changing when each edge is 5 cm long?

A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the lift-off point. At the moment the range finder's elevation angle is /4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment?

A police cruiser, approaching a right-angledintersection from the north, is chasing a speeding car that has turned the corner and is now moving east. When the cruiser is 0.6 miles north of the intersection and the car is 0.8 miles to the east, the police determine that the distance between them and the car is increasing at a rate of 20mph. If the cruiser is moving at 60mph at the instant of measurement, what is the speed of the car ?

image by lemoncat1

when x = .8, y = .8 and L = √((.8)2 + (.6)2 ) = 1

1(20) = .8 dx + .6( -60) dt 20 = .8 dx - 36 dt

56 = .8 dx dt

56 = dx = 70 mph. .8 dt

Water runs into a conical tank at a rate of 9 ft3/min. 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 ?

HOMEWORK - 1st day Foerster P.177

DV

HS

s

v

s = distance between Hans Solo and the originv = distance between Darth Vader and the originL = distance between the spaceships

Given :

Find : when v = 1200 and s = 500

L2 = v2 + s2

when v = 1200 and s= 500

So distance between them is decreasing at a rate of approximately 15.4 km/min