this is an optimation question. Help!!!! and i just dont want the answer must explain cos i need 2 understand.

Draw the triangle on an x-y plot with the right angle at 0,0 and the other two points at (12,0) and (0,5). First write an expression for the hypotenuse line. The y-intercept is B = 5 and the slope is:

m = rise/run

m = -5/12

the equation of the line would then be:

y = mx + B

y = (-5/12)x + 5

now calculate the area of a rectangle:

A = x*y

A = x*((-5/12)x + 5)

A = (-5/12)x^2 + 5x

to get the maximum you need to take the derivative and set it to 0:

dA/dx = 2*(-5/12)x + 5

0 = (-5/6)x + 5

(5/6)x = 5

x = 6

y = (-5/12)x + 5

y = (-30/12) + 5 = 2.5

You should be able to do part b on your own now.

For part c just replace the numbers with variables:

a will be the length and b will be the height

The y-intercept is B = b and the slope is:

m = rise/run

m = -b/a

y = (-b/a)x + b

now calculate the area of a rectangle:

A = x*y

A = x*((-b/a)x + b)

A = (-b/a)x^2 + bx

to get the maximum you need to take the derivative and set it to 0. Treat b and a just as if they were actual numbers:

dA/dx = 2*(-b/a)x + b

0 = (-2b/a)x + b

(2b/a)x = b

x = a/2

y = (-b/a)x + b

y = (-b/a)(a/2) + b

y = (-b/2) + b

y = b/2

Ok I am going to try to explain this without a board or graph paper maybe this will help you:

First thing if you do not have any, you are going to need some graph paper:

If you do not have graph paper: Get you a sheet of paper out. At the center of the page draw you a line all the way across it. Once you get that done: at the center of the paper draw you a in the center from the top to the bottom of the page:

When you finish you should have two lines on the page meeting in roughly the center of the page.forming four boxes:

The line going up and down (from the top to the bottom of your page is called the Y axis:

Beside that line label it Y

The line going across your page is called the X axis.

Beside that line label it X:

When the two lines meet in the center of the page is known as the X,Y intercept:

Label it X,Y

Now the box made by the two lines that is on the upper right hand side of your paper is known at the first quadrant: It is symbolized with the Roman number one or I:

The box on the upper right hand side of the paper is known at the second quadrant it is symbolized with the Roman number II

The box on the lower left hand side is known as the thrid quadrant it is symbolized by the Roman number three III

The box on the lower right hand side of the paper is the four quadrant it is symbolized by the Roman number four IV:

We will be working in the first quadrant which is the upper right hand box: The reason for this is the first quadrant is the only quadrant where both X and Y are positive numbers:

Now right under where you labeled the line going up and down the page Y do this like this.

Y

(B) (b)

(height)

(altitude)

(opposite)

(rise) used in construction mainly concerning roofs

(reactive power which is in reactance either Yc or Yl depending on of the circuit is reactive or capacitive)

Now where you have labeled the ling going across the page X do this:

X

(A) (a)

(base)

(Adjacent)

(run) used in construction mainly concerning roofs.

(real power which is in watts) W

Now we are ready to get started: Note this first. Always start drawing your triangle at the X,Y intercept until you get to knowing and feel comfortable otherwise. it is a lot easier working this way when you are starting out as you will see:

when writing numbers on an graph the X axis number is always written first: and the Y axis is always written second when dealing with trig.

X,Y

so your first set of numbers is 5cm and 12 cm correct:

What that means is X = 5cm: and Y = 12 cm.

Look at the two numbers in the set: You see that X is roughly a little less than half what Y is: This gives us a starting point to draw out out triangle on your paper:

This does not have to be exact when drawing your triangle right now:

I want you to draw a right triangle in the upper left hand box on your paper (first quadrant) starting where the two lines cross each other or intercept each other.

Starting at the X,Y intercept draw a line along the X axis roughly about half the length between the X,Y lines cross each other and the edge of the page.

When you get that done make a right angle and now draw a lind going up roughly 2 times longer than the line you just drew on the the X axis line:

When you have that done draw a line going from the top of the line you drew going up and down back to you X,Y intercept:

That line is going to be your hypotenuse. It is the line opposite the right angle on the triangle you just drew: the hypote

(a) To optimize the space inside of the triangle, I divided each leg by 2, leaving me a rectangle that is 2.5 cm by 6 cm. Multiplying the two together to find the area gives you 15 cm².

(b) Same thing here. Dividing each by 2 gives us a rectangle 4 cm by 7.5 cm. Multiplying the two together to find the area gives you 30 cm².

(c) For right triangle ABC, with C = 90°, a and b are the legs, and c is the hypotenuse: The area of the largest rectangle that can be inscribed into this right triangle is the area of the triangle [(1/2)(b)(h)] divided by two is correct.