Differentiation, related rates?

In summary, to determine the dimensions of the rectangle of largest area inscribed in the given right triangle, we first set up a coordinate system and use the equation of the hypotenuse to relate the base and height of the rectangle. Then, we maximize the area of the rectangle using the constraint and complete the square to find the vertex of the parabola.
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Homework Statement


Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown?


Homework Equations


AB/AD = BC/DE


The Attempt at a Solution


I've been trying to do this problem. I looked online and saw an explanation without directly using the area of the triangle, found http://answers.yahoo.com/question/index?qid=20101128161728AA21pl7".

Can someone remind me how to differentiate A = x H - (H/X)x^2 , in the last part of the problem? Is this from related rates?
 
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  • #2
Picture added:

lCBl = X

lABl = H

H/h = X/x
 

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  • #3
I'm not sure exactly what you want here. You have not given any dimensions for the triangle.

Assuming a right triangle of base b and height h, we can set up a coordinate system having the two legs as axes. Then the hypotenuse is a line from (0, h) to (b, 0). The equation of that line is y= h- (h/b)x. Let a be the length of the rectangle along the x-axis, b, the length of the rectangle along the y-axis. Since the fourth vertex must lie on the hypotenuse, a and b must satisfy b= h- (h/b)x. Maximize the area, ab, with the constraing b= h- (h/b)a. Yes, that gives A= a(h-(h/b)x)= ah- (h/b)x2, just what you have with a slightly different notation.

As far as differentiating Hx- (H/X)x2, one of the very first derivative rules you learned is that the derivative of [itex]x^n[/itex] is [itex]nx^{n-1}[/itex]. However, you don't really need to use the derivative to maximize this. Complete the square to find the vertex of the parabola.
 

1. What is differentiation?

Differentiation is a mathematical process that involves finding the rate of change of a function. It is used to calculate how a variable changes with respect to another variable.

2. What is the purpose of differentiation?

The purpose of differentiation is to help us understand how a function changes and how it is related to other variables. It is also used to find maximum and minimum values of functions, which has many practical applications in fields such as engineering, physics, and economics.

3. What is a related rate?

A related rate is the rate of change of one variable with respect to another variable. In other words, it is the rate at which one variable changes while another variable is changing.

4. How is differentiation used to solve related rate problems?

Differentiation is used to solve related rate problems by finding the derivatives of the given functions and using them to set up equations that relate the rates of change of the variables involved. These equations can then be solved to find the desired rate of change.

5. What are some real-life applications of differentiation and related rates?

Differentiation and related rates have many real-life applications, such as calculating the speed and acceleration of moving objects, determining optimal production rates in manufacturing, and predicting changes in stock prices in economics. They are also used in fields like medicine, engineering, and biology to model and analyze various phenomena.

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