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?
Your process is correct. However, there seems to be an error in your expansion of the equation (3x^2)+(36x)+28=0.
To solve the quadratic equation, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, the equation is in the form (3x^2) + (36x) + 28 = 0, where a = 3, b = 36, and c = 28.
Plugging the values into the quadratic formula:
x = (-36 ± √((36)^2 - 4(3)(28))) / (2(3))
Simplifying further:
x = (-36 ± √(1296 - 336)) / 6
x = (-36 ± √(960)) / 6
x = (-36 ± √(16 * 60)) / 6
x = (-36 ± 4√(60)) / 6
x = (-6 ± √(60))
Now you have two solutions for x: x = (-6 + √(60)) and x = (-6 - √(60)).
You mentioned that you obtained two negative numbers using the quadratic formula, but that is not the case. Both of these solutions are valid, and you can use them to calculate the dimensions of the original piece of tinplate.
Let's substitute one of the solutions into the expression w = x + 6:
w = (-6 + √(60)) + 6
w = √(60)
Similarly, the width would be:
w = (-6 - √(60)) + 6
w = -√(60)
However, the dimensions cannot be negative, so we disregard the negative solution.
Therefore, the dimensions of the original piece of tinplate are:
Length = √(60)
Width = √(60)
Height = 3 cm
Your process is correct up until this point. However, there seems to be an error in your expansion of the equation. Let me guide you through the correct steps to find the dimensions of the original piece of tinplate.
1. Start with the equation you derived: 80 = (w^2) * 3.
2. Substitute w = x + 6 (since width = length + 6).
3. Square both sides of the equation:
(80) = (3) * [(x + 6)^2]
80 = 3 * (x + 6)^2
4. Divide both sides of the equation by 3:
80/3 = (x + 6)^2
5. Take the square root of both sides of the equation:
√((80/3)) = x + 6
6. Subtract 6 from both sides of the equation:
√((80/3)) - 6 = x
Now you can calculate the value of x using a calculator or by simplifying the square root. Plug in this value of x back into w = x + 6 to find the width of the original piece of tinplate.
Once you find the value of x, you can find the dimensions of the original piece of tinplate by adding 6 to x to get the width and length.