Questions: A piece of sheet metal measuring 50 cm × 40 cm has squares cut out of each corner so that it can be bent and formed into an open box (with no lid) with a base area of 1344 cm^2. Find the dimensions of the box.

A piece of sheet metal measuring 50 cm × 40 cm has squares cut out of each corner so that it can be bent and formed into an open box (with no lid) with a base area of 1344 cm^2. Find the dimensions of the box.

Transcript text: 7 A piece of sheet metal measuring $50 \mathrm{~cm} \times 40 \mathrm{~cm}$ has squares cut out of each corner so that it can be bent and formed into an open box (with no lid) with a base area of $1344 \mathrm{~cm}^{2}$. Find the dimensions of the box.

Solution

Solution Steps

To find the dimensions of the box, we first need to determine the side length of the squares cut out from each corner. Then, we can calculate the dimensions of the box by subtracting twice the side length from the original dimensions of the sheet metal.

Let the side length of the squares cut out be $x$. The base area of the box is equal to the area of the sheet metal after the squares are cut out, which can be expressed as $(50-2x)(40-2x) = 1344$. We can solve this equation to find the value of $x$, and then calculate the dimensions of the box.

Step 1: Find the side length of the squares cut out

Given the solutions to the equation $ (40 - 2x)(50 - 2x) - 1344 = 0 $ are $ x = 4 $ and $ x = 41 $, we choose $ x = 4 $ since it represents a valid side length for the squares cut out.

Step 2: Calculate the dimensions of the box

Substitute $ x = 4 $ into the expressions for length and width: