#148

We are told that is inversely proportional to the square of

This tells us that , where is a constant of proportionality.

You need to solve this differential equation, and the best way to do so would be with separation of variables.

Thus, we see that

Now, this is where two conditions come into play:

The first condition: "The initial value of the machine was $500,000"

This is saying that

Applying this condition to the equation we have for V, we see that

Now let us look at the second condition: "Its value decreased by $100,000 in the first year"

This is saying that

Applying this condition to the equation we have for V, we see that

We have to solve this system for and :

I leave it for you to verify that and

Thus, our equation for V is

Now all you have to do is find

# 96This question is similar to the first one here.

Following the same idea, we see that the equation modeling the number of sales per week is , where is the constant of proportionality.

Using separation of variables, we see that

We are given two conditions: , and .

Use a similar process to# 148to get an equation for

# 68

Set up the integral:

Make the substitution .

I leave it for you to verify that:

Then evaluate the integral.

I hope this makes sense!

--Chris