A 12cm by 8cm rectangular piece of metal is to be made into an open-top box by cutting a sqaure from corner and folding up the resulting flaps (sides). If the volume of the lidless box is 36 cm what are the integer dimensions of the box?
Square to be cut out : x cm by x cm
then the height of the box is x, and the base is
12-2x by 8-2x, where 0 < x < 4
volume = x(12-2x)(8-2x)
= 36
x(6-x)(4-x) = 9
x(24 - 10x + x^2) = 9
x^3 - 10x^2 + 24x - 9 = 0
I tried x = ±1, ±3 and x = 3 was a solution.
by synthetic division, I got
(x-3)(x^2 - 7x + 3) = 0
the quadratic has no real solutions, so
x = 3 and the box will be
6 by 2 by 3
hi,
how did you go from 36=x(12-2x)(8-2x) to 9=x(6-x)(4-x)?
To solve this problem, we need to follow these steps:
Step 1: Find the dimensions of the square to be cut from the corners.
Let's assume the side length of the square to be cut from each corner is "x" cm.
Step 2: Calculate the dimensions of the box base.
After cutting the square and folding up the resulting flaps, the base of the box will have dimensions:
Length = 12 cm - 2x
Width = 8 cm - 2x
Height = x
Step 3: Calculate the volume of the box.
The volume of the box is given as 36 cm³:
Volume = Length × Width × Height
36 = (12 - 2x) × (8 - 2x) × x
Step 4: Solve the equation.
Let's solve the equation to find the value of "x".
36 = (12 - 2x) × (8 - 2x) × x
36 = (96 - 36x - 16x + 4x²) × x
36 = (4x² - 52x + 96) × x
36 = 4x³ - 52x² + 96x
Step 5: Simplify the equation.
Bring the equation to standard form:
4x³ - 52x² + 96x - 36 = 0
Step 6: Factor the equation.
After factoring, we get:
(x - 3)(x - 4)(4x - 3) = 0
Step 7: Find the values of "x" that satisfy the equation.
From the factored equation, we have three possible solutions:
x = 3, x = 4, and x = 3/4 (0.75).
Step 8: Calculate the dimensions of the box.
Using the values of "x" we found, we can calculate the dimensions of the box as follows:
For x = 3:
Length = 12 - 2(3) = 6 cm
Width = 8 - 2(3) = 2 cm
Height = 3 cm
For x = 4:
Length = 12 - 2(4) = 4 cm
Width = 8 - 2(4) = 0 cm (not possible)
Height = 4 cm
For x = 3/4:
Length = 12 - 2(3/4) = 10.5 cm (not an integer)
Therefore, the integer dimensions of the box are:
Length = 6 cm
Width = 2 cm
Height = 3 cm.
To determine the integer dimensions of the box, we need to follow these steps:
1. Visualize the rectangular piece of metal:
The rectangular piece of metal is 12 cm long and 8 cm wide.
2. Cut squares from the corners:
Cut identical squares from each corner of the metal sheet. Let's assume that the size of each side of the square is x cm.
3. Fold up the resulting flaps:
Fold up the flaps to create the sides of the open-top box.
4. Determine the dimensions of the box:
The length of the box will be reduced by 2x cm due to folding up the flaps, and the width of the box will be reduced by 2x cm as well. The height of the box will be equal to x cm.
5. Calculate the volume of the box:
The volume of the box can be calculated by multiplying the length, width, and height: Volume = length × width × height.
In this case, the volume is given as 36 cm³.
6. Write the equation to solve for x:
36 = (12 - 2x) × (8 - 2x) × x
7. Solve for x:
Expand the equation, collect like terms, and rearrange to solve for x.
8. Find the integer dimensions of the box:
Once you find the value of x, substitute it back into (12 - 2x) and (8 - 2x) to determine the integer dimensions of the box.