longest diagonal of a right rectangular prism.?
The surface area of a right rectangular prism is 48 square feet, and the sum of its length, width, and height is 13 feet. What is the length of the longest diagonal connecting two corners of the box?
could you please explain how did you get it?
i need the solution.
many thanks..
lets say length is x and width y and height z
using area given we get 2xy + 2yz + 2xz = 48 or xy +z(x+y) = 24..... eqn 1
also x + y +z =13
so z = 13-(x+y)... substitute in eqn 1
xy + (13-x-y)(x+y) = 24
13x+13y-x^2-y^2-xy = 24
now differentiate, 13 + 13dy/dx -2x-2dy/dx-(xdy/dx +y) = 0
dy/dx(13-2-x) = 2x+y-13
dy/dx = (2x+y-13)/(11-x) = 0
so y = 13-2x, but why really is 13-x-z from given information, so we conclude x and z are same
now we can solve for the longest diagonal as sqrt(x^2+x^2+y^2)
substitute y as 13-2x and you get a quadratic eqn on simplifying in the form
6x^2-52x+48=0
x = 1.05038......
and y = 10.8992...
longest diagonal is sqrt(x^2+x^2+y^2)
= sqrt(121) = 11
Geometry - Diagonal of a Right Hexagonal Prism
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