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# Geometry

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A rectangular prism has a length of \$3\$, a width of \$5\$, and a height of \$11\$. What is the distance between two opposite corners of the prism?

May 1, 2024

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To find the distance between two opposite corners of the rectangular prism, we can visualize a diagonal line going through the entire prism. This diagonal will form a right triangle with sides that correspond to the lengths, width, and height of the prism.

We can use the Pythagorean Theorem to find the length of this diagonal.

Pythagorean Theorem:

a^2 + b^2 = c^2

where:

a and b are the lengths of the two legs of a right triangle.

c is the length of the hypotenuse (the diagonal in this case).

Applying the theorem:

In our case:

a = length of the prism = 3 units

b = width of the prism = 5 units

c = diagonal length (what we want to find)

Solve for c (diagonal length):

c^2 = a^2 + b^2

c^2 = 3^2 + 5^2

c^2 = 9 + 25

c^2 = 34

Taking the square root of both sides (remembering there might be positive and negative square roots), we get:

c = ±√34

Since c represents a distance (and cannot be negative), we take the positive square root:

c = √34 units

Therefore, the distance between two opposite corners of the rectangular prism is √34 units. (This is the exact answer, but you can approximate it to a decimal value if needed.)

May 1, 2024