a) If x^2 + y^2 = z^2, find dy/dt when x = 3, y = 4, dx/dt = 4, and dz/dt = 4. (

Question

a) If x^2 + y^2 = z^2, find dy/dt when x = 3, y = 4, dx/dt = 4, and dz/dt = 4. (Enter your answers as a comma-separated list.)

dy/dt= ______

b) Suppose that the daily profit (in dollars) from the production and sale of x units of a product is given by the following. P = 180x - (x^(2))/1000 - 2200 At what rate per day is the profit changing when the number of units produced and sold is 100 and is increasing at a rate of 20 units per day? $______

c) Two cars are approaching an intersection on roads that are perpendicular to each other. Car A is north of the intersection and traveling south at 35 mph. Car B is east of the intersection and traveling west at 30 mph. How fast is the distance between the cars changing when car A is 24 miles from the intersection and car B is 18 miles from the intersection? (Round your answer to two decimal places.) ______ mph

Explanation / Answer

solution:

a)

x^2 + y^2 = z^2

at x=3

y=4

put to get z

z=5

now

x^2 + y^2 = z^2

differentiating both sides wrt t

xdx/dt + ydy/dt=zdz/dt

now putting values

3*4 +4dy/dt=5*4

4dy/dt=8

dy/dt=2

b)given

x=20 , dx/dt=100

now

P=180x -x^2/1000 -2200

differentiating wrt t

dP/dt = 180dx/dt -(x/500)(dx/dt)

dP/dt= 180*20 - (100*20/500)

dP/dt=3600-4 = 3596 $/day

c)

both are prependicular to each other

distance between them D is

D^2= x^2 + y^2

now

dy/dt=35

dx/dt=30

y=24.

x=18

D=sqrt(24^2+18^2)=30

differentiating the equation

D(dD/dt)=xdx/dt + ydy/dt

30.(dD/dt)=18*(30) + 24*(35)

dD/dt=46

and its sign will be negative as both these are appproaching

ans: 46

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