Arectangle has one side on the x-axis and the upper two vertices on the graph of y=e^{-3x^2}. what is the decimal approximation of the maximum area.

The area of the rectangle is the products of its dimensions

The decimal approximation of the maximum area is 0.495

The given parameter is:

One side is on the x-axis.

So, the other dimension (d) is:

The area of the rectangle is

This gives

Differentiate using product rule:

Factorize

Set to 0

Solve for x

Collect like terms

Divide both sides by 6

Take square roots

Substitute in

Hence, the decimal approximation of the maximum area is 0.495

Read more about areas at:

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Fix the vertices to have -coordinates and . Then the rectangle has base and its height is , the value of at these vertices' -coordinates.

Then the area of the rectangle is

Take the derivative:

We have when , at which point we get an area of

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