Question

The length of the longest rod is 13 cm in a box of 12 cm long and 4 cm wide. The height of the box is ______.

Options

• 2 cm

• 3 cm

• 4 cm

• 5 cm

Answer 1

The length of the longest rod is 13 cm in a box of 12 cm long and 4 cm wide. The height of the box is 3 cm.

Explanation:

We are given a box with dimensions:

• Length = 12 cm
• Width = 4 cm 
• The longest rod that can fit in the box has a length of 13 cm.

We are asked to find the height of the box.

Step 1: Use the Pythagorean theorem 

The longest rod that can fit in the box forms the diagonal of the box. The diagonal of a rectangular box can be calculated using the Pythagorean theorem in three dimensions:

`d = sqrt(l^2 + w^2 + h^2)`

Where:

• l is the length of the box = 12 cm 
• w is the width of the box = 4 cm
• h is the height of the box (which we need to find) 
• d is the diagonal, which is the length of the longest rod = 13 cm

Step 2: Set up the equation

Substitute the known values into the equation:

`13 = sqrt(12^2 + 4^2 + h^2)`

Step 3: Simplify the equation

`13 = sqrt(144 + 16 + h^2)`

`13 = sqrt(160 + h^2)`

Square both sides to eliminate the square root:

169 = 160 + h²

Step 4: Solve for h²

h² = 169 – 160 = 9

h = `sqrt(9)` = 3 cm