Optimization with maxima and minima

In summary, we are discussing the dimensions of a rectangle inscribed in a right triangle with sides measuring 6 inches, 8 inches, and 10 inches. The optimal dimensions for this rectangle can be found by using derivatives of the area or by considering the area of strips added and subtracted as the rectangle is moved. The result does not depend on the specific measurements of the right triangle and can be generalized for any right triangle.
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electritron
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A rectangle is to be inscribed in a right triangle having sides 6 inches, 8 inches, and 10 inches. Determine the dimensions of the rectangle with greatest area.

I recently tried doing it and the answer was found by finding the slope and then using the first and second derivatives of the area.
Now I would like to know how to solve this by using proportions.
So if there are any suggestions whatsoever, Thanks.
 
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  • #2
I confess I was not sure what is meant by a rectangle inscribed in...
Didi it mean a rectangle with base on the hypotenuse, or one with the same right angle as the right triangle?

After working out an answer for one I realized the answer would be the same for the other!

What is your answer?

You are given a classical 3, 4, 5 right triangle - but does your result really depend on that? As a fraction of the total area of the triangle can you state maximum area of inscribed rectangle in the most general fashion, i.e. for any right triangle?

I cannot think of how to do it by proportions, but if you consider the area of strip added on one side and subtracted on the other side as you move a corner of the rectangle I think you can get the result without formal differential calculus, indeed without having to consider the strips to be narrow.
 
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1. What is optimization with maxima and minima?

Optimization with maxima and minima is a mathematical process used to find the maximum or minimum value of a function. This is done by finding the critical points of the function, where the slope is equal to zero, and then determining whether these points correspond to a maximum or minimum value.

2. How is optimization with maxima and minima used in real life?

Optimization with maxima and minima is used in various fields, including economics, engineering, and science. For example, it can be used to determine the most efficient production level for a company, the optimal design for a bridge, or the best conditions for a chemical reaction.

3. What is the difference between a local maximum and a global maximum?

A local maximum is a point on a function where the value is greater than all the points around it, but it may not be the highest point on the entire function. A global maximum, on the other hand, is the highest point on the entire function and is also a local maximum.

4. How do you find the critical points of a function for optimization?

To find the critical points, you need to take the derivative of the function and set it equal to zero. Then, solve for the variable to find the x-values of the critical points. These points can then be plugged back into the original function to determine if they correspond to a maximum or minimum value.

5. Can optimization with maxima and minima be used for functions with multiple variables?

Yes, optimization with maxima and minima can be used for functions with multiple variables. In this case, the critical points are found by taking the partial derivatives of the function with respect to each variable and setting them equal to zero. The resulting system of equations can then be solved to find the critical points.

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