An open box contains 80cm^3 and is made from a square piece of tinplate with 3cm squares cut from each of its four corners. Find the dimensions of the original piece of tinplate.
A=80
A=(length)(width)(height)
since length and width are the same, and when the box is folded the height will be 3cm,
80=(w^2)(3)
Since the width has 2 3cm sections cut out of it, I said:
w=x+6
To find x, I tried subbing w=x+6 into my previous equation:
80=((x+6)^2)*3
Which expands as:
(3x^2)+(36x)+28=0
What do I do next? I tried the quadratic formula to get x, but ended up with two negative numbers, which is not a valid solution. Please help me... Was my process wrong?
I posted this a few hours ago but I wasn't answered.
Your process was correct up to the point where you tried substituting the width, w, with x+6. However, there was a small mistake in your expansion of the equation.
Let's go through the problem again step by step:
1. We have an open box made from a square piece of tinplate, with 3cm squares cut from each of its four corners. Let's call the side length of the original square tinplate piece 'x' cm.
2. When the box is formed, the width and length will be equal, so each side of the box will be (x - 6) cm.
3. The height of the box will be 3 cm.
4. The volume of the box is given as 80 cm^3. We can set up the equation:
Volume = (length) * (width) * (height)
80 = (x - 6) * (x - 6) * 3
5. Expanding the equation:
80 = (x - 6)^2 * 3
80 = (x^2 - 12x + 36) * 3
80 = 3x^2 - 36x + 108
6. Rearranging the equation and simplifying:
3x^2 - 36x + 108 - 80 = 0
3x^2 - 36x + 28 = 0
Now, we need to solve this quadratic equation to find the value of x.
Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 3, b = -36, and c = 28.
Calculating the discriminant, b^2 - 4ac:
Discriminant = (-36)^2 - 4 * 3 * 28 = 1296 - 336 = 960
Since the discriminant is positive, we will have two real solutions for x.
Using the quadratic formula:
x = (-(-36) ± √960) / (2 * 3)
x = (36 ± √960) / 6
Simplifying further:
x = (36 ± 4√60) / 6
x = 6 ± 2√60
Therefore, we have two possible values for x: 6 + 2√60 and 6 - 2√60.
To validate which solution is appropriate, we need to check if (x - 6) is smaller than or equal to 3 cm since the side length of the cut-out squares is 3 cm.
For 6 + 2√60:
(x - 6) = (6 + 2√60 - 6) = 2√60 ≈ 24.65 cm
For 6 - 2√60:
(x - 6) = (6 - 2√60 - 6) = -2√60 ≈ -24.65 cm
Since a negative value doesn't make sense in this context, the appropriate solution is x = 6 + 2√60.
So, the dimensions of the original tinplate piece are approximately 6 + 2√60 cm by 6 + 2√60 cm.