A rectangular piece of cardboard is 2 units longer than it is wide?
From each of its corner a square piece 2 units on a side is cut out.The flaps are then turned up to form an open box that has a volume of 70 cubic units.Find the length and width of the original piece of cardboard.
Did you make a sketch?
width of cardboard --- x
length of cardboard -- x+2
width of box = x-4
length of box = x-2
height of box = x
volume = x(x-2)(x-4) = 70
x^3 - 6x^2 + 8x - 70 = 0
That was easy so far, but now we have a nasty cubic.
I tried factors of 70 , no luck
so I used Wolfram to find
x = appr 6.4442
original widht = 6.442
original length = 8.442
http://www.wolframalpha.com/input/?i=x%5E3+-+6x%5E2+%2B+8x+-+70+%3D+0
check:
original widht = 6.442
original length = 8.442
box base is 2.4442 by 4.4442
volume = 2.4442 x 4.4442 x 6.4442 = 70.00002 , not bad
To solve this problem, we need to follow these steps:
Step 1: Define variables
Let's assign variables to the dimensions of the original piece of cardboard. Let's call the width of the cardboard "w" units. Since the cardboard is 2 units longer than it is wide, the length of the cardboard will be w + 2 units.
Step 2: Calculate the dimensions of the box
When squares measuring 2 units are cut out from each corner of the cardboard, the length and width of the resulting box will be decreased by 4 units. Therefore, the length of the box (after folding the flaps) will be (w + 2) - 4 = w - 2 units, and the width of the box will be w - 4 units.
Step 3: Calculate the volume of the box
The volume of a rectangular box is calculated by multiplying its length by its width by its height. In this case, the height of the box will be 2 units. Therefore, the volume of the box is (w - 2) * (w - 4) * 2 = 70 cubic units.
Step 4: Solve the equation
Simplify the equation: 2(w - 2)(w - 4) = 70
Expand: 2(w^2 - 6w + 8) = 70
Simplify further: 2w^2 - 12w + 16 = 70
Rearrange the equation: 2w^2 - 12w + 16 - 70 = 0
Simplify: 2w^2 - 12w - 54 = 0
Divide by 2: w^2 - 6w - 27 = 0
Step 5: Factor or solve the quadratic equation
Factoring the quadratic equation w^2 - 6w - 27 = 0 is not easily done, so let's solve it using the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Applying the formula to our equation (w^2 - 6w - 27 = 0), we get:
w = (6 ± √((-6)^2 - 4 * 1 * (-27))) / (2 * 1)
w = (6 ± √(36 + 108)) / 2
w = (6 ± √144) / 2
w = (6 ± 12) / 2
Simplifying further, we have two possible solutions:
w = (6 + 12) / 2 = 18 / 2 = 9
w = (6 - 12) / 2 = -6 / 2 = -3
Step 6: Evaluate the solution
Since we cannot have a negative width for the cardboard, the only valid solution is w = 9.
Step 7: Calculate the length
Using the width value we found, we can calculate the length of the cardboard:
Length = width + 2 = 9 + 2 = 11
Therefore, the original piece of cardboard is 9 units wide and 11 units long.
Let's suppose the width of the original rectangular piece of cardboard is x units.
According to the given information, the length of the cardboard is 2 units longer than its width. Therefore, the length of the cardboard can be represented as (x+2) units.
When the squares are cut out from each corner, the width of the resulting box will be (x - 4) units (2 units from each side) and the length will be ((x+2) - 4) units.
The height of the box is assumed to be 2 units, as the square pieces that were cut out are also 2 units on each side.
The formula for the volume of a rectangular box is V = length × width × height. Therefore, we have the equation: V = (x-4) × ((x+2)-4) × 2.
Given that the volume is 70 cubic units, we can substitute the value into the equation: 70 = (x-4) × (x-2) × 2.
Simplifying this equation, we get: 70 = 2(x^2 - 2x - 4x + 8).
Expanding further, the equation becomes: 70 = 2(x^2 - 6x + 8).
Dividing both sides of the equation by 2, we get: 35 = x^2 - 6x + 8.
Rearranging the equation, we get: x^2 - 6x - 27 = 0.
To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring this equation gives us: (x - 9)(x + 3) = 0.
So, x can be either 9 or -3.
However, since the width cannot be negative, we discard -3 as the solution.
Therefore, the original width of the cardboard is 9 units.
Substituting this value into the equation for the length, we have: length = (x + 2) = 9 + 2 = 11.
Hence, the length of the original piece of cardboard is 11 units, and the width is 9 units.