An open box is to be made from a square piece of material by cutting four-centimeter squares from each corner and turning up the sides (see figure). The volume of the finished box is to be 576 cubic centimeters. Find the size of the original piece of material.
size of original piece of material:
x cm by x cm
after cut-out
base is (x-8) by (x-8)
height = x
V = x(x-8)^2 = 576
x^3 - 16x^2 + 64x - 576 = 0
Nasty equation to solve.
Have you learned any methods to solve a cubic?
Newton's Method is probably the best way, but you have to know Calculus.
or, you could use Wolfram if you just want the answer.
http://www.wolframalpha.com/
To find the size of the original piece of material, we need to consider the dimensions of the box after the corners are cut and the sides are folded.
Let's assume that the size of each side of the square piece of material is "x" centimeters.
When we cut the four corners, each measuring 4 centimeters, the resulting dimensions of the box will be length = x - 2(4) = x - 8 and width = x - 2(4) = x - 8.
The height of the box will be the 4 centimeters cut from each corner. So, the height = 4 centimeters.
The volume of a rectangular box is given by the formula: volume = length x width x height.
Given that the volume of the finished box is 576 cubic centimeters, we can set up the equation:
576 = (x - 8) * (x - 8) * 4
Simplifying the equation:
576 = 4(x - 8)^2
Dividing both sides of the equation by 4:
144 = (x - 8)^2
Taking the square root of both sides:
±12 = x - 8
Since 12 is a positive value and represents the dimensions of the original square piece of material, we can ignore the negative solution.
x - 8 = 12
Adding 8 to both sides:
x = 20
Therefore, the size of the original piece of material is 20 centimeters by 20 centimeters.
To find the size of the original piece of material, we need to understand how volume is related to the dimensions of the box.
Let's assume that the side length of the square piece of material is x centimeters.
When we cut out four squares with side length 4 centimeters from each corner, the resulting shape will have dimensions of (x-8) centimeters by (x-8) centimeters. The height of the box will be 4 centimeters.
So, the volume of the finished box can be calculated as follows:
Volume = Length x Width x Height
= (x-8) cm x (x-8) cm x 4 cm
= 4(x-8)^2 cm^3
We are given that the volume of the finished box is 576 cubic centimeters, so we can set up an equation and solve for x:
4(x-8)^2 = 576
Now, let's solve this equation:
Divide both sides by 4:
(x-8)^2 = 144
Take the square root of both sides:
x-8 = ±12
Solve for x:
x-8 = 12 or x-8 = -12
Case 1: x-8 = 12
Add 8 to both sides:
x = 20
Case 2: x-8 = -12
Add 8 to both sides:
x = -4
Since the side length of a square cannot be negative, we can disregard the second case.
Therefore, the size of the original piece of material is 20 centimeters.