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 256 cubic centimeters. Find the size of the original piece of material.
the picture is in the link below:
www.webassign.net/laratrmrp6/2-4-086.gif
After looking at your picture it is obvious that
(x^)(4) = 256
solve for x, very easy
i don't how to do it, that's why i am asking anyone to show me how to do it and explain me it
volume = (area of base)(height)
= (x^2)(4) = 4x^2
but that is given as 256
4x^2 = 256
x^2 = 64
x = √64 = 8
The base was 8 by 8, but it had 4 cm cut out on each side
so the original piece was 16 cm by 16 cm
To find the size of the original piece of material, we need to follow these steps:
1. Let's assume the length of each side of the square piece of material is "x" centimeters.
2. When we cut four-centimeter squares from each corner, the resulting length of the box is (x - 4) centimeters.
3. By folding up the sides, the height of the box is also (x - 4) centimeters.
4. The width of the box, which is the original length of the square piece of material, remains unchanged at "x" centimeters.
5. Now, let's calculate the volume of the box. The formula for volume is V = length × width × height.
So, V = (x - 4) × x × (x - 4).
6. We know that the volume of the finished box is given as 256 cubic centimeters, so we can set up the equation:
256 = (x - 4) × x × (x - 4).
7. Expand the equation:
256 = (x^2 - 4x) × (x - 4).
8. Simplify the equation:
256 = x^3 - 8x^2 + 16x + 16x - 64.
9. Combine like terms:
256 = x^3 - 8x^2 + 32x - 64.
10. Bring all terms to one side of the equation:
x^3 - 8x^2 + 32x - 64 - 256 = 0.
Simplifying further, we have:
x^3 - 8x^2 + 32x - 320 = 0.
11. Now, we need to solve this cubic equation to find the value of "x". However, solving cubic equations can be complex and involve various methods like factoring or using numerical methods.
In this case, it would be best to utilize a graphing calculator or mathematical software to find the value of "x".
12. By solving the cubic equation, we will find the value of "x", which represents the length of each side of the square piece of material.