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A right 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.

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 Dec 18, 2023
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Let's solve this problem step-by-step:

 

1. Define variables:

 

Let L be the length, W be the width, and H be the height of the rectangular prism.

Let D be the length of the diagonal joining one corner of the prism to the opposite corner.

 

2. Use surface area information:

 

The total surface area of the prism is the sum of the areas of its six faces. Using the formula for rectangular face area, we can write:

2 * L * W + 2 * W * H + 2 * L * H = 56

 

3. Use edge length information:

The sum of all edge lengths is equal to the perimeter of the base plus the perimeter of the lateral faces. We can write:

2 * (L + W) + 4 * H = 64

 

4. Simplify equations:

 

From equation 2, we can isolate H: H = 30 - (L + W).

 

Substitute this expression for H in equation 1: 2 * L * W + 2 * W * (30 - (L + W)) + 2 * L * (30 - (L + W)) = 56.

 

Expand and simplify: 60L - 2W^2 - 2LW = 56.

 

Rewrite as a quadratic equation: -2W^2 - LW + 4L - 56 = 0.

 

5. Solve for L and W:

 

We can use factoring or the quadratic formula to solve for L and W. One possible solution is L = 8 and W = 2.

 

6. Calculate diagonal length:

 

Using the Pythagorean theorem in the right triangle formed by the diagonal and the sides of the prism, we can find D: D^2 = L^2 + W^2 + H^2.

 

Substituting the values: D^2 = 8^2 + 2^2 + (30 - 10)^2 = 960.

 

Taking the square root: D = √960

 

Therefore, the length of the diagonal joining one corner of the prism to the opposite corner is sqrt(960) units.

 Dec 18, 2023

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