What makes you uncomfortable about the answer you got?
Would it make you more comfortable to compare the volume if the height were slightly above or below the height you calculated to the volume you got?
I think the problem means that there are 10 unique pairs of gloves in the drawer and X is the number of pairs of gloves after randomly choosing 6 individual gloves. So one possibility is that you chose 6 gloves and none of them pair up, meaning that you have 0 pairs (X=0). Another possibility...
That's pretty close: the first term (for your second derivative) is not correct but I think if you take a second look you will figure it out (you just missed one detail). Otherwise, you can combine like terms to clean it up a bit, but it's still a long expression.
Try simplifying the expression by dividing both the numerator and denominator by x (so overall you've just multiplied by x/x=1 and haven't changed the expression). You'll be left with something you can find the limit of without having to use L'Hopital's rule.
height is given by f(x) = f(2i/5) so you plug in x=2i/5:
f\left(\frac{2i}{5}\right)=-\left(\frac{2i}{5}\right)^2+5
same goes for the second equation
you have an expression for x as a function of i that you plug in to find f as a function of i.
Your second guess for the outflow rate appears correct to me, making your differential equation:
\frac{dq}{dt}=2-\frac{q}{40+t}
Any reason you are not sure that you would like to discuss?
Don't forget that when you are answering the problem they are asking for concentration of alcohol; your...
No problem.
You've got d(x,y)=2x+2y, which is what you are trying to maximize.
You've also got y = 4-x2/9
Substitute that into the first one so you have d=d(x) (d as a function of x).
Then find the derivative of d with respect to x.
When you set that equal to zero, you will be able to find your...
Your equation is correct so far, but I think you're making things tougher on yourself than they have to be. I would rewrite U as a function of w (my variable for the number of hours worked) and only w. Then you can take the derivative and find the optimum w.
Anyway, the way to solve your...
For the first one, you need to use the product rule to find the derivative:
if f(x) = u(x)*v(x), then f' = u(dv/dx) + v(du/dx)
then solve for critical points. Critical points occur when the derivative = 0. In your work above, you solved for where the denominator = 0 (making the derivative...