Heat is escaping at a constant rate, (dq/dt is constant) through the walls of a long cylindrical pipe. find the temperaturet at a distance r from the axis of the cylinder if the inside wall has radius r=1, and temperaturet=100, and the outside wall has r=2, andt=0.

Step-by-step explanation:

Hello,

To solve this, we can use the famous Fourier's Law of Heat Conduction

What do we know so far?

When solving keep in mind that the values provided are "boundary conditions" that we'll need when solving this differential equation.

keep in mind that the whole term is a constant and will not take part in the integration, so we can just call this whole term

We can simplify now,

This is our solution to the differential equation, lets call it Eq(1)

Putting all the known values, and forming an equation we can find the value of C

We can put this in value in our original equation Eq(1)

This is our general equation, but the question is asking to find the temperature at a distance r, i.e , given that and , this means that the above equation needs to be converted in the form such that is only a function of

We can remove the subscripts if we like

Finally, this is our equation for the outer temperature of the cylinder at any distance r

the smallest possible measurement of the width is 14 feet

step-by-step explanation:

let's assume length of parallelogram is 'l'

width of parallelogram is 'w'

the perimeter of a parallelogram must be no less than 40 feet

so, we get

we are given

length of parallelogram is 6 feet

so,

l=6

we can plug it

now, we can solve for w

subtract both sides by 12

divide both sides by 2

and we get

so,

the smallest possible measurement of the width is 14 feet