Cuboid Dimensions and Diagonal Calculation

We are given the length and breadth of a cuboid and the length of its diagonal. Our goal is to find the volume of the cuboid.

• Length ($l$) = 50 cm
• Breadth ($b$) = 75 cm
• Diagonal ($d$) = $17\sqrt{29}$ cm

Finding the Cuboid Height

The relationship between the diagonal ($d$), length ($l$), breadth ($b$), and height ($h$) of a cuboid is given by the formula:

$$d^2 = l^2 + b^2 + h^2$$

First, let's calculate the square of the given diagonal:

$$d^2 = (17\sqrt{29})^2 = 17^2 \times (\sqrt{29})^2 = 289 \times 29$$

Calculating the product:

$$289 \times 29 = 8381$$

So, $d^2 = 8381$ cm$^2$.

Next, let's calculate the squares of the length and breadth:

$$l^2 = 50^2 = 2500 \text{ cm}^2$$

$$b^2 = 75^2 = 5625 \text{ cm}^2$$

Now, substitute these values into the diagonal formula to find $h^2$:

$$8381 = 2500 + 5625 + h^2$$

Combine the known squares:

$$8381 = 8125 + h^2$$

Isolate $h^2$:

$$h^2 = 8381 - 8125$$

$$h^2 = 256 \text{ cm}^2$$

To find the height, take the square root:

$$h = \sqrt{256}$$

$$h = 16 \text{ cm}$$

Calculating the Cuboid Volume

The volume ($V$) of a cuboid is calculated using the formula:

$$V = l \times b \times h$$

Substitute the known values of length, breadth, and the calculated height:

$$V = 50 \text{ cm} \times 75 \text{ cm} \times 16 \text{ cm}$$

Perform the multiplication:

$$V = (50 \times 75) \times 16$$

$$V = 3750 \times 16$$

$$V = 60000 \text{ cm}^3$$

Therefore, the volume of the cuboid is 60,000 cm$^3$.