A box is to be formed from a rectangular piece of sheet metal by cutting squares measuring 5 inches on a side and then folding the sides. The piece of sheet metal is twice as long as it is wide. If the volume of the box is to be 1760 in^3, what are the dimensions of the original piece of sheet metal?
height = 5
(L-10)(w-10)5 = 1760
(L-10)(w-10) = 352
L = 2 w
(2w-10)(w-10) = 352
(w-5)(w-10) = 176
w^2 - 15 w + 50 = 176
w^2 - 15 w - 126 = 0
(w-21)(w+6) = 0
so
w = 21
L = 42
Let's call the width of the original piece of sheet metal "w".
Given that the length of the sheet metal is twice its width, the length would be 2w.
To form the box, squares measuring 5 inches on a side are cut from each corner. Hence the length of the resulting box would be (2w - 10) inches (since two 5-inch squares are removed from each side).
Similarly, the width of the resulting box would be (w - 10) inches.
The height of the box created by folding the sides would be 5 inches.
The volume of the box is given as 1760 in^3.
We can now calculate the volume of the box by multiplying its dimensions:
Volume = Length × Width × Height
1760 = (2w - 10) × (w - 10) × 5
Now, let's solve this equation to find the value of "w".
1760 = (2w - 10) × (w - 10) × 5
352 = (2w - 10) × (w - 10)
Expanding the equation:
352 = 2w^2 - 30w - 10w + 100
352 = 2w^2 - 40w + 100
2w^2 - 40w + 100 - 352 = 0
2w^2 - 40w - 252 = 0
Simplifying the equation by dividing through by 2:
w^2 - 20w - 126 = 0
Now we can solve this quadratic equation to find the value of "w".
Using the quadratic formula:
w = (-(-20) ± sqrt((-20)^2 - 4*1*(-126))) / (2*1)
Simplifying:
w = (20 ± sqrt(400 + 504)) / 2
w = (20 ± sqrt(904)) / 2
w = (20 ± 30.06) / 2
w ≈ (20 + 30.06) / 2 ≈ 25.03 (ignoring the negative solution)
So, the width of the original piece of sheet metal is approximately 25.03 inches.
The length of the original piece of sheet metal is twice the width, so:
Length = 2w ≈ 2 * 25.03 ≈ 50.06 inches
Therefore, the dimensions of the original piece of sheet metal are approximately 25.03 inches by 50.06 inches.
To solve this problem, we need to follow these steps:
Step 1: Understand the problem.
We have a rectangular piece of sheet metal, and we want to cut squares measuring 5 inches on each side from its corners. By folding the sides up, we can form a box. The volume of this box is given as 1760 in^3. We are asked to find the dimensions of the original piece of sheet metal.
Step 2: Assign variables.
Let's assign a variable to one side of the original piece of sheet metal. We'll call it "x" inches. Since the piece of sheet metal is twice as long as it is wide, the other side will be "2x" inches long.
Step 3: Determine the dimensions of the box.
When we cut squares measuring 5 inches from each corner, the resulting box will have a width of (2x - 2 * 5) inches and a length of (x - 2 * 5) inches.
Step 4: Calculate the volume of the box.
The volume of a box is calculated by multiplying its length, width, and height. In this case, the height is 5 inches. So, the volume of the box is (2x - 2 * 5) * (x - 2 * 5) * 5.
Step 5: Set up an equation.
We know that the volume of the box is given as 1760 in^3. So, we can set up the equation:
(2x - 2 * 5) * (x - 2 * 5) * 5 = 1760
Step 6: Solve the equation.
Multiply the terms inside the parentheses and then solve for x:
(2x - 10) * (x - 10) * 5 = 1760
(2x - 10) * (x - 10) = 1760 / 5
(2x - 10) * (x - 10) = 352
Expand and simplify the equation:
2x^2 - 20x - 10x + 100 = 352
2x^2 - 30x + 100 = 352
2x^2 - 30x - 252 = 0
Step 7: Apply the quadratic formula.
To solve the quadratic equation 2x^2 - 30x - 252 = 0, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = -30, and c = -252.
Using the quadratic formula, we can calculate the value of x.
Step 8: Calculate the value of x.
x = (-(-30) ± √((-30)^2 - 4 * 2 * (-252))) / (2 * 2)
x = (30 ± √(900 + 2016)) / 4
x = (30 ± √2916) / 4
x = (30 ± 54) / 4
Using both options, we have two possible values for x:
x1 = (30 + 54) / 4 = 84 / 4 = 21 inches
x2 = (30 - 54) / 4 = -24 / 4 = -6 inches
Since the length cannot be negative, we discard the value of x2.
Step 9: Find the other dimension.
The other dimension, which is twice the value of x, can be calculated as:
2x1 = 2 * 21 = 42 inches
Step 10: Verify the solution.
We can verify our solution by calculating the volume of the box using the dimensions we found:
Volume = (2x - 2 * 5) * (x - 2 * 5) * 5
= (42 - 2 * 5) * (21 - 2 * 5) * 5
= 32 * 11 * 5
= 1760 in^3
The volume matches the given value, so our solution is correct.
Therefore, the dimensions of the original piece of sheet metal are 21 inches by 42 inches.