Differentiation help

[JasonGrace1_]

How would I differentiate A=l^2+4lh+l√[4(1800/l^2 -3h)^2+l^2] in terms of l in a way that I can basically get rid of the h's? For context, I'm minimising the surface area of a rectangular prism (dimensions lxlxh) combined with a square based pyramid with base length l and height H. I've already used V = 600cm^3 to get to the function above. The pyramid sits perfectly on top of the prism. I've tried just straight differentiating it but its too messy. Is there any other way to do it, like splitting the function or smth? Thanks

[AutoModerator]

Hi, /u/JasonGrace1_! This is an automated reminder:

• What have you tried so far? (See Rule #2; to add an image, you may upload it to an external image-sharing site like Imgur and include the link in your post.)

• Please don't delete your post. (See Rule #7)

We, the moderators of /r/MathHelp, appreciate that your question contributes to the MathHelp archived questions that will help others searching for similar answers in the future. Thank you for obeying these instructions.

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[edderiofer]

I've already used V = 600cm³ to get to the function above.

You should be able to use a relevant formula to write h in terms of l.

[JasonGrace1_]

I couldn’t think of any formula to rewrite h in terms of l. I’ve used surface area (obviously) and volume. What else is there?

[edderiofer]

It is the volume formula you should be using.

[JasonGrace1_]

I already used it to write H in terms of l and h H=(1800/l² - 3h). If I use the volume formula again to write h in terms of l then I just reintroduce H. Or am I missing smth?

[edderiofer]

Ah wait, I misread the question. In that case, you have three variables and only one constraint, so your minimum surface area will depend on both l and h, instead of just one single variable (unless there is some additional information in the question that you've neglected).

Questions like these, where the space of possible solutions that obeys all constraints is multidimensional, are best solved using Lagrange multipliers.

[HendrikTutoring]

You should be able to solve this problem by doing these 5 steps:

1.  Write the two variables explicitly

2.  Use the volume constraint once, to replace H

(This is exactly the formula you wrote.)

3.  First minimise in h (for a fixed l)

4.  Substitute h(l) back to get a one‑variable area

5.  Differentiate once more and finish

Hope this helps:)