# Algebra 2

posted by .

Find the dimensions and area of the largest rectangle that can be inscribed in a right triangle whose sides are 9cm, 12cm, 15cm.

• Algebra 2 -

Put the points of the triangle at (0,0), (0,9) and (12,0)
Let the insribed rectangle have corners at:
(0,0), (x,0), (x,y'), and (0,y')
y' is on the hypotenuse of the triangle, so that
y = 12 - (4/3)x
So y' = 12 - (4/3)x
The rectangle area is
A(x) = x*y' = 12 x - (4/3)x^2
There is a maximum when dA/dx = 0
12 = (8/3) x
x = (3/8)*12 = 4.5
y' = 12 - (4/3)(9/2) = 5
The maximum rectangle area is xy'= 22 and the side lengths are 4.5 and 5.

• Algebra 2 -

This is normally a challenging calculus problem, however it can be worked with similar triangles, or algebra.

First, note the triangle is a right triangle. Lay it out so the 12 dimension is on the x axis, and the 9 is on the y axis.

Note the equation of the hypotenuse is
y=-9/12 x + 9
let one side of the rectangle going upward as h, and the side along the x axis as W. So the intersection with the hypotenuse is w,-9w/12 + 9

So the drill is to find the max area.
h= -9/12 w + 9
Area= hw= -9/12 w^2+9w
so when is Area max?

• Algebra 2 (correction) -

I calculated the point on the hypotenuse incorrectly. It is
y' = 12 - (4/3)(9/2) = 6
The rectangle side lengths are 4.5 and 6, and the area is 27. That equals half the area of the triangle, (1/2)*9*12 = 54. BobPursley tells me that Euclid proved this in many different ways, without calculus.

Thanks to Bob for pointing this out

• Algebra 2 -

A package with square ends has a combined length and girth (girth is the perimeter of a cross section) of 120 in. The surface area of the entire package is 3600 sq. in.
Determine the dimensions of thr package. s in.*s in.*l in.
one solution is: 11.08in*11.08in*75.68in.
find the other solution
hint: if 4S + length=120, then the length=120-4S
Enter the 3 dimensions separated by comas

## Similar Questions

1. ### CALCULUS

Find the area A of the largest rectangle that can be inscribed in a right triangle withh legs of length 3cm and 4cm if two sides of the rectangle lie along the two legs of the triangle.
2. ### calculus

A rectangle is to be inscribed in a right triangle having sides of length 36 in, 48 in, and 60 in. Find the dimensions of the rectangle with greatest area assuming the rectangle is positioned as in the accompanying figure. right triangle: …

For each right triangle, draw the rectangle made by drawing sides opposite the two shorter sides in the triangle. Find the area of each rectangle. 9cm height 4cm width How does the area of each rectangle relate to the area of either …
4. ### arithmetic

The base of a solid right prism is a triangle whose sides are 9cm,12cm and 15 cm.the total surface area of the prism.

6. ### maths

the base of a solid right prism is a triangle whose sides are 9cm, 12cm and 15cm. The height of the prism is 5cm. Then the total surface area of the prism is.
7. ### math

the base of a solid right prism is a triangle whose sides are 9 cm,12cm,15cm. the height of the prism is 5 cm. then find the total surface area of the prism is?
8. ### Calculus AB

A rectangle is inscribed in a right triangle with legs of length 5 and 12. The sides of the rectangle are parallel to the legs of he triangle. Find the dimensions of the rectangle that has the largest area?
9. ### math

The area of a right triangle whose hypotenuse is 15cm and whose height is 12cm is 54sq. cm.What is its perimeter?
10. ### Caclulus

Let x and y be to positive numbers whose product is 500: (a) Find the maximum sum of x and y (b) Find the minimum sum of x and y: 2. A cylindrical container can hold a volume of 1 liter. Find the dimensions of the container that minimizes …

More Similar Questions