How do you find the volume of a box with an open top?

The formula for volume of the box is V=l×l×h . You can determine the maximum value of this function using graphing calculator.

How do you solve a calculus optimization problem?

To solve an optimization problem, begin by drawing a picture and introducing variables. Find an equation relating the variables. Find a function of one variable to describe the quantity that is to be minimized or maximized. Look for critical points to locate local extrema.

When is the volume of the open-top box maximized?

Because the derivative is increasing ( 1 4 > 0 14>0 1 4 > 0) to the left of the critical point, and decreasing ( − 1 3 < 0 -13<0 − 1 3 < 0) to the right of it, the function has a maximum at x = 0. 9 6 x=0.96 x = 0. 9 6, and we can say that the volume of the open-top box is maximized when x = 0. 9 6 x=0.96 x = 0. 9 6.

How can I maximize the enclosed volume of a box?

Determine the dimensions of the box that will maximize the enclosed volume. Solution We want to build a box whose base length is 6 times the base width and the box will enclose 20 in 3. The cost of the material of the sides is $3/in 2 and the cost of the top and bottom is $15/in 2.

Can we use optimization in the real world?

This same optimization process can be used in the real world. When the function we start with models some real-world scenario, then finding the function’s highest and lowest values means that we’re actually finding the maximum and minimum values in that situation.

What are the critical points of the volume function?

So the critical points of the volume function are x ≈ 3. 0 4 x\\approx3.04 x ≈ 3. 0 4 and x ≈ 0. 9 6 x\\approx0.96 x ≈ 0. 9 6. But before we start testing critical points, we should always consider which of the critical points is even plausible.