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A right triangle is formed in the first quadrant by a line passing through the point (3,5) and the coordinate axes. Find the coordinates of its vertices if its area is to be minimized.

$$(0,0),(6,0),(0,10)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

Campbell University

Harvey Mudd College

Baylor University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

21:23

10. A right triangle is fo…

03:06

A right triangle is formed…

01:54

02:04

A triangle is to be formed…

03:42

05:18

A line with a slope of $-5…

01:22

Find the area of a triangl…

so as to maximize the area of a right triangle With the line that has to go through the .35. So let's see here. So what I did is I said, okay we can we know at this point and so we know we're gonna need the X. And the Y intercepts here. And so we're going to find this line and basically we just don't know the slope, right, it could be this line or it could be this, you know, basically this line or any other line in here that goes through this point and so they all have different areas. So this is the line and so we know the why the X intercept is when Y equals zero, so that is actually cause three m minus five over em and when X equals zero, that's the Y intercept and that is why equals five minus three M. So the area is 1/2 the base times height. So this is the base and this is the height and length of the base length of the height. We can take the derivative of this. Um And after some simplifications, we get minus nine halves plus 25/25 halves, 25 divided by two m squared, setting Amica antoine and setting the record is zero, You get are optimal value. We get to values um plus or -5/3. Well we can clearly see that if we have a plus five thirds, we're going to have something like like this, is it going to go through the origin? Yeah, it goes through the origin here. So basically we have zero area there. Um It's kind of a weird problem um because as soon as you get a slope that is you know negative is non negative zero or positive. Um We really never have a triangle here. So we know we need we need to take the value with a negative um slope. And so the coordinates of that Of the triangle then our 60 out here, 00 and 010. So that would maximize the area And that area would be, what would that be? 30? Um Yeah, 30. Um And then again if we if we plug in The positive one, we get just 000. So um everything that goes through this, so we don't really have a triangle and again we don't really have a triangle with any time. And this is um the slope is greater than Greater than or equal to zero.

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