A gift box is made from a rectangular piece of cardboard that is three times as long as it is wide. 5 cm squares are cut from each corner and the ends are then folded up to make the box. If the box's volume is 4340 cm^3, find the length and width of the cardboard.
5 (w - 10) (3w - 10) = 4340
3 w^2 - 40 w - 768 = 0
To find the length and width of the cardboard, we need to break down the problem into steps:
Step 1: Define the variables
Let's say the width of the cardboard is "w" cm.
Then the length of the cardboard would be three times the width, so the length = 3w cm.
Step 2: Calculate the volume of the box
The volume of a rectangular box is given by the formula: V = l × w × h, where V is the volume, l is the length, w is the width, and h is the height.
In this case, the height of the box is 5 cm since we cut squares from each corner, and the resulting shape is folded up.
So we have: V = (3w) × w × 5
Given that the volume is 4340 cm^3, we can set up the equation:
4340 = 3w × w × 5
Step 3: Solve the equation
Simplifying the equation, we get:
4340 = 15w^2
Divide both sides of the equation by 15:
w^2 = 289.33
Take the square root of both sides to find the value of w:
w ≈ 17 cm
Step 4: Find the length
Since the length is three times the width, we can calculate it:
Length = 3w
Length = 3 × 17
Length ≈ 51 cm
So, the width of the cardboard is approximately 17 cm and the length is approximately 51 cm.