A rectangular prism has a total surface area of $56.$ Also, the sum of all the edges of the prism is $64.$ Find the length of the diagonal joining one corner of the prism to the opposite corner. *no pic*

Guest Apr 13, 2023

#1**0 **

Let l, w, and h be the length, width, and height of the rectangular prism, respectively. We know that the total surface area is 56, so we can use the formula for the surface area of a rectangular prism to get the following equation:

2lw+2wh+2lh=56

We also know that the sum of all the edges is 64, so we can use the formula for the sum of the edges of a rectangular prism to get the following equation:

2l+2w+2h=64

We can solve the first equation for l to get:

l=2w56−2wh−2lh

We can then substitute this into the second equation to get:

2w56−2wh−2lh+2w+2h=64

We can then simplify this equation to get:

2wh+2lh−56=0

We can then factor this equation to get:

(w−4)(2h−14)=0

We can then solve for w and h to get:

w=4 h=214=7

We can then use these values to find the length of the diagonal by using the Pythagorean theorem:

d2=l2+w2+h2=42+72+72=108

d=sqrt(108)=6*sqrt(3)

Therefore, the length of the diagonal joining one corner of the prism to the opposite corner is 6*sqrt(3).

Guest Apr 13, 2023