Need math help :( Illustration included, please give me a hand!

[phatj]

This is the first time I've posted for help with math... I've always helped others, but now I am in need of help

I've done at least 30 of these problems last year, but I just cannot remember how to get the answer...

Here's the illustration

"A rectangle is to be inscribed in a right triangle having sides of length 6 in, 8 in, and 10 in. Find the dimensions of the rectangle with the greatest area asuuming the rectangle is positioned as pictured (in link)"

 

[amdskip]

I'm sure this is simple but I've never come across a problem like this.

 

[phatj]

I know I have to find the first derivative of the area function, but I need the length and width to be in common terms... That's the problem I'm having.

 

[her209]

Assume rectangle has sides of length x and y.

Assume that length y lies on the hypotenuse.

Now we know that the area of the triangle is 0.5*b*h = 0.5*6*8 = 24

You know that:

24 = x*y + (sum of areas of the three triangles)

Since the triangle that is resting on top of the rectangle is an equilatrel triangle to the larger triangle, we can use ratios to find the length of the sides of that triangle.

Therefore:

y / 10 = a / 6

Conversely,

y / 10 = b / 8

We also know that the two remaining triangles are equilatral triangles to teh larger triangle due to complementary angles.

I think you can take it from there.

 

[her209]

Were you able to solve it?

 

[ClueLis]

Originally posted by: her209
Assume rectangle has sides of length x and y.

Assume that length y lies on the hypotenuse.

Now we know that the area of the triangle is 0.5*b*h = 0.5*6*8 = 24

You know that:

24 = x*y + (sum of areas of the three triangles)

Since the triangle that is resting on top of the rectangle is an equilatrel triangle to the larger triangle, we can use ratios to find the length of the sides of that triangle.

Therefore:

y / 10 = a / 6

Conversely,

y / 10 = b / 8

We also know that the two remaining triangles are equilatral triangles to teh larger triangle due to complementary angles.

I think you can take it from there.

To make things clearer, you should change "equilateral" to "congruent".

 

[phatj]

Yea I got it, thanks. I got so deep into the similar triangles (on the first attempt b4 I posted), it just screwed me up (i had one side being radical((6-(3/5)y)^2 + x^2) etc etc.. I just needed to keep things more simple.

 

[RossGr]

Originally posted by: ClueLis

Originally posted by: her209
Assume rectangle has sides of length x and y.

Assume that length y lies on the hypotenuse.

Now we know that the area of the triangle is 0.5*b*h = 0.5*6*8 = 24

You know that:

24 = x*y + (sum of areas of the three triangles)

Since the triangle that is resting on top of the rectangle is an equilatrel triangle to the larger triangle, we can use ratios to find the length of the sides of that triangle.

Therefore:

y / 10 = a / 6

Conversely,

y / 10 = b / 8

We also know that the two remaining triangles are equilatral triangles to teh larger triangle due to complementary angles.

I think you can take it from there.

To make things clearer, you should change "equilateral" to "congruent".

To make it correct change congruent to similar.

All of the triangles have the same angles, the 2 on the sides are congruent, the top one is similar. The final solution may show that all are congruent but you cannot use that assumption in solving the problem.

 

[her209]

Originally posted by: RossGr

Originally posted by: ClueLis

Originally posted by: her209
Assume rectangle has sides of length x and y.

Assume that length y lies on the hypotenuse.

Now we know that the area of the triangle is 0.5*b*h = 0.5*6*8 = 24

You know that:

24 = x*y + (sum of areas of the three triangles)

Since the triangle that is resting on top of the rectangle is an equilatrel triangle to the larger triangle, we can use ratios to find the length of the sides of that triangle.

Therefore:

y / 10 = a / 6

Conversely,

y / 10 = b / 8

We also know that the two remaining triangles are equilatral triangles to teh larger triangle due to complementary angles.

I think you can take it from there.

To make things clearer, you should change "equilateral" to "congruent".

To make it correct change congruent to similar.

All of the triangles have the same angles, the 2 on the sides are congruent, the top one is similar. The final solution may show that all are congruent but you cannot use that assumption in solving the problem.

Whoops, I mean similar. You can say all the triangles are similar to the large triangle (and each other) due to their complentary angles.

 

[silverpig]

lagrange multipliers d00d...