Optimization | folding box | maximize volume of box calculus

In this example problem, a piece of cardboard is formed into an open-top box by cutting squares with side length x from each corner and folding up the sides. Find a formula for the volume of the box in terms of x. Then find the value of x that will maximize the volume of the box.

We work an optimization problem by setting up an objective function based on removing the squares in the corners. After expanding the volume function, we take the derivate. Setting the derivative equal to zero and solving down gives two values for x. Only one works based on the dimensions given to us for the cardboard. We then take the second derivative of the volume function and use the second derivative test to check that the function is concave down at our value of x. This ensures that we will have a maximum volume at the value that we found.

[nope-------]

Thank you for your clear, step by step breakdown! Saved me multiple headaches

[violetterra]

Thank you!!!! This was very insightful! 😁

[optimalmiguelmanaig]

What happens if the corners or folded parts are binomials?

[movementmasterypt]

It should be illegal to post videos of maths problems using imperial measurements.