Find the Face Diagonal of a Rectangular Prism with Base Diagonal X and Length 3X

A rectangular prism has a square base with a diagonal length of X.

The prism has a length of 3X.

How long is the diagonal of the rectangular face of the prism.

Video Solution

Solution Steps

00:00 Calculate the diagonal length of the rectangular faces in the box

00:03 Box is square according to the given data, therefore sides are equal

00:07 In a square all angles are right angles

00:12 We'll use the Pythagorean theorem in triangle A1D1C1 to find A1C1

00:15 We'll substitute appropriate values according to the given data and solve to find X

00:22 This is the length of diagonal face X

00:28 We'll express Y using X

00:38 Each face in the box is a rectangle, therefore all angles are right angles

00:47 We'll use the Pythagorean theorem in triangle AA1D1 to find AD1

00:58 We'll substitute appropriate values according to the given data and solve to find AD1

01:18 Group terms

01:22 Take the square root

01:26 This is the length of the diagonal of the rectangular faces in the box

01:29 And this is the solution to the problem

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Determine the side length of the square base
Step 2: Calculate the diagonal of the rectangular face using the Pythagorean theorem

Now, let's work through each step:
Step 1: The side length $s$ of the square base is calculated from its diagonal $X$ as $s = \frac{X}{\sqrt{2}}$ .
Step 2: The diagonal $d$ of the rectangular face is found using $d^2 = s^2 + (3X)^2$ , which simplifies to $d = X\sqrt{9.5}$ .

The solution to the problem is $x\sqrt{9.5}$ .

Find the Face Diagonal of a Rectangular Prism with Base Diagonal X and Length 3X

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