A box with a square base and open top must have a volume the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms pase. Hint: use the volume formula to express the height of the E Simplify your formula as much as possible. A(2) = %3D Next, find the derivative, A'(x).
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![A box with a square base and open top must have a volume of 108000 cm. We wish to find the dimensions of
the box that minimize the amount of material used.
ст
First, find a formula for the surface area of the box in terms of only x, the length of one side of the square
base.
[Hint: use the volume formula to express the height of the box in terms of x.]
Simplify your formula as much as possible.
A(x) =
Next, find the derivative, A'(x).
A'(x) =
Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x.]
A'(x) = 0 when x =
We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the
second derivative test. Find A"(x).
A"(x) =
Evaluate A"(x) at the x-value you gave above.
NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that value,
so the zero of A'(x) must indicate a local minimum for A(x). (Your boss is happy now.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1a2d46c-7922-4557-987b-de9ad08df9f1%2Ffd75a9c5-f2c7-4182-84c6-76431008e9fb%2Fbr67c0m_processed.jpeg&w=3840&q=75)
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