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.
Find and simplify the difference quotient of y = V(x). That is, find (௫ା)ି(௫)
. (This will get a little
messy.) Then, substitute h with 0 and simplify. The expression you obtained is called the derivative of
y = V(x). (You will study derivatives in calculus.) Now, find the zeros of the derivative accurate to three
decimal places. What do you notice?
To find the original dimensions of the rectangular piece of tin, we need to break down the problem into steps:
Step 1: Define variables
Let's define the original length of the tin as L inches and the original width of the tin as W inches.
Step 2: Calculate the dimensions after cutting out the tabs
When you cut out a 3-inch square tab from each corner, the length and width of the base of the box decrease by 6 inches each (3 inches from each end). So the length and width of the base of the box would be (L - 6) inches and (W - 6) inches, respectively.
Step 3: Calculate the height of the box
The height of the box is the height of the tabs that were folded up. We are given that the volume of the box is 2760 cubic inches. The volume of a rectangular box can be calculated by multiplying its length, width, and height. Therefore, we can write the equation: (L - 6)(W - 6)h = 2760, where h is the height of the box.
Step 4: Calculate the area of the tin
The area of a rectangle can be calculated by multiplying its length and width. We are given that the area of the tin is 1334 square inches. Therefore, we can write the equation: L * W = 1334.
Step 5: Simplify the equations
Using the equation from Step 4, we can rewrite it as L = 1334/W.
Step 6: Substitute the simplified equation into the height equation
Substituting L = 1334/W into the height equation from Step 3, we get (1334/W - 6)(W - 6)h = 2760.
Step 7: Solve for the dimensions
Now, we need to solve the equation from Step 6 to find the value of W. We can use algebraic methods or numerical methods to find the value of W. Once we have W, we can substitute it back into the equation L = 1334/W to find the value of L.