problem n solution

Problem solving

1. Thermometer reading 75^oF is taken out where temperature is 20^oF. The reading is 30^oF 4 minutes later. Find:

a.)Thermometer reading 7 minutes after the thermometer was brought outside. b.)The time taken for the reading to drop from 75^oF to within a half degree of the air temperature

2. Proved the identity of trigonometry

3. Determine the differential equation in behind

4. If y=x²-4x and x=

, find when t=√2

5. Find the solution

Solution:

1. Known :

75^oF 30^oF

u is the thermometer temperature (^oF)

t is time (minutes)

when t=0 temperature is 75^o (u=75^o)

when t=4 temperature is 30^o (u=30^o)

-k = decrease of temperature in thermometer

(u-20)=difference of temperature

Differential equation

Substitution t=0 and u=75 in formula where we get it.

u=20+ce^-kt

75=20+ce^-k.0

75-20=ce⁰

55=c.1

c=55

the equation become u=20+55e^-kt

to find k, substitution t=4 and u=30^o

30=20+55e^-kt

10=55 e^-kt

e^4t=5,5

ln e^4k=ln5,5

4k lne=ln 5,5

k=

so u=20+55exp(-0,43t)

a) If t=7

And you input t in the formula so

u=20+55exp(-0,43.7)

=20+55.0,5

=22,7

So the thermometer reading 22,7^oF

b) If u=75/2=37,5

And you input u in the formula so

u=20+55exp(-0,43t)

37,5=20+55exp(-0,43t)

17,5=55exp(-0,43t)

17,5:55=exp(-0,43t)

So the time taken for the reading to drop from 75^oF whiting a half degree of the air temperature is 2,66 minutes

2.

Answer:

We choose left segment to prove it

So the equation proven

3.

we can write the differential equation

From that:

Integration factor :

The general solution differential equation is:

So the solution is

4. If

and

find






when t=√2

Answer:

If t=√2, x=√5 and

So the solution dy/dx is

5. Find the solution

Answer: Choose u=secx, dv=sec²xdx, so dv=secxtanxdx, v=tanx

so

So the solution