A rectangular piece of tin has an area of 1334 square inches. A square tab of 3 inches is cut from
each corner, and the ends and sides are turned up to make an open box. If the volume of the box is
2760 cubic inches, what were the original dimensions of the rectangular piece of tin? Show the work
that leads to the answer.
Length, width, and height must be positive. Use this fact to find the domain of the volume function V(x)
in the form a < x < b. Explain. Justify your explanation by solving inequalities.
xy = 1334, so y = 1334/x
3(x-6)(y-6) = 2760
3(x-6)(1334/x - 6) = 2760
solve that and you get
x = 29 or 46
so, the original sheet was 29 by 46 in.
v(x) = (24012+4110x-18x^2)/x
= 6(x-6)(667-3x)/x
for v>0, we have 6 < x < 667/3
Makes sense, since we can't cut two 3" corners if the side length is less than 6.
And if x is too big, there's not anything left for y.
To find the original dimensions of the rectangular piece of tin, we need to determine the length and width of the rectangle before the tabs are cut and folded to form the box.
Let's assume the length of the rectangle is x inches and the width is y inches.
When the tabs are cut from each corner and folded up, the resulting box will have a height of 3 inches.
The length of the base of the box will then be (x - 2*3) = (x - 6) inches (after subtracting twice the length of the tab from the original length).
Similarly, the width of the base of the box will be (y - 2*3) = (y - 6) inches.
Now, we can calculate the formula for the volume of the box using the given information:
V(x) = (x - 6) * (y - 6) * 3
Given that the volume of the box is 2760 cubic inches, we can set up the equation:
2760 = (x - 6) * (y - 6) * 3
Now, let's analyze the domain of the volume function V(x) and find the range of possible values for x.
Since the length, width, and height must be positive, we have the following conditions:
Length: x > 0
Width: y > 0
Height: 3 > 0
1. Length (x > 0):
From the given problem, the length of the rectangular piece of tin after folding will be (x - 6) inches. Since the length must be positive, x - 6 > 0. Solving this inequality, we get:
x > 6
So, the length must be greater than 6 inches.
2. Width (y > 0):
Similarly, the width of the rectangular piece of tin after folding will be (y - 6) inches. Since the width must be positive, y - 6 > 0. Solving this inequality, we get:
y > 6
So, the width must be greater than 6 inches.
Now, we know that x > 6 and y > 6.
To find the range of possible values for x that lead to a volume of 2760 cubic inches, we can solve the equation:
2760 = (x - 6) * (y - 6) * 3
Note that the range of possible values for x will depend on the value of y.
By substituting y = 6 + k (where k is any positive number), into the equation, we get:
2760 = (x - 6) * (6 + k - 6) * 3
2760 = 3x * k
Dividing both sides of the equation by 3k:
920 = x
So, the range of possible values for x is 920.
In conclusion, the original dimensions of the rectangular piece of tin were 920 inches for the length (x) and greater than 6 inches for the width (y).