heya, I absolutely horrible with Calculus Word Problems and was wonder if anyone could help me with these.
#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
# 96
This 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 # 148 to 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