# Solid Mensuration

A triangle has a base of 16 inches and an altitude of 8 inches. Find the dimensions of the largest rectangle that can be inscribed in the triangle if the base of the rectangle coincides with the base of the triangle.

1. 👍 0
2. 👎 0
3. 👁 71
1. I suspect it will be 4x8, but let's see what the calculations show.

If triangle ABC has vertices at (0,0),(16,0) and (h,8) then the sloping sides are lines with the equations

y = 8/h x
y = 8(x-16)/(h-16)

Now, suppose the inscribed rectangle has height k. The the line y=k intersects the sides where x is

hk/8 and k(h-16)/8 + 16

The length of the rectangle is thus 16-2k

The area of the rectangle is

a = k(16-2k) = 16k-2k^2
The vertex of this parabola is where k=4.

Thus the inscribed rectangle of maximal area is half the height of the triangle, making its base half the base of the triangle. The rectangle is thus 4x8.

1. 👍 0
2. 👎 0
posted by Steve

## Similar Questions

1. ### college algebra

if the altitude and the base of a triangle are each increased by 4 inches the area of the triangle will be increased by 42 square inches. if the altitude is increased by 3 inches and the base is decreased by 2 inches the area will

asked by quintana on July 12, 2013
2. ### Differential calculus

A triangle has a base of 16 inches and an altitude of 8 inches. Find the dimentions of the largest rectangle that can be inscribed in the triangle if the base of the rectangle coincides with the base of the triangle

asked by Anonymous on October 2, 2018
3. ### math

A rectangle and a triangle have equal areas. The length of the rectangle is 12 inches, and its width is 8 inches. If the base of the triangle is 32 inches, what is the length, in inches, of the altitude drawn to the base?

4. ### Geometry

1.What is the perimeter of a rectangle with width 10 inches in length 13 inches? A.23inches B.36inches C.46inches*** D.130inches 2.What is the area of a triangle with base 19 inches and height 11 inchinches? A.209square inches

asked by GummyBears18 on September 6, 2016
5. ### math

find the lateral surface area of a right triangular prism if its altitude is five inches and its base is an isoceles right triangle with hypotenuse of length 10 inches

asked by mary on March 27, 2012
6. ### math

a rectangle and a triangle have equal areas. the lenght of the rectangle is 12 inches ,and its width is 8 inches. if the base of the triangle is 32inches,what is the lenght,in inches of the altitude drawn to the base? a)12 b)16

asked by JESSICA on December 2, 2007

Use familiar figures to to find the area of the figure shown. Show all work. Since I can't provide a picture, here is the description. It looks like a rocket ship. There is a triangle with a width of 4 inches. A rectangle with a

asked by Music Lover 14 on June 6, 2016
8. ### Math

The area of trapezoid is h( b1+b2)/5, where h is the altitude, and b1 and b2 are the lenghts of the parallel bases. If trapezoid has an altitude of 5 inches, an area of 55 square inches , and one base 12 inches long, what is the

asked by Nena on April 22, 2011
9. ### geometry

if the altitude of a triangle is 5 inches and the base is 10 inches long what is the area?

asked by mee on March 8, 2012
10. ### math

The area of trapezoid is h( b1+b2)/5,where h is the altitude, and b1 and b2 are the lenghts of the parallel bases. If trapezoid has an altitude of 5 inches, an area of 55 square inches , and one base 12 inches long, what is the