Walter is making an open-top box. The length is 6 inches greater than the width, and the height is 2 inches. The area of the rectangular sheet of material that will be folded to make the box is 280 square inches.
What is the length of the box in inches?
(I really don't know where to start)
Well, it seems like Walter's box is quite an open book, or should I say, an open-top box! Let's start unraveling this mystery, shall we?
Let's assume that the width of the box is "w" inches. According to the given information, the length is 6 inches greater than the width. So, we can express the length as "w + 6" inches.
Now, we know that the area of the rectangular sheet of material is 280 square inches. The formula for the area of a rectangle is length times width. In this case, it would be (w + 6) * w = 280.
Now, let's put our mathematical skills to work and solve this equation:
w^2 + 6w = 280
Oh, that equation looks quite intimidating, but don't worry, we can handle it! By rearranging the terms, we get:
w^2 + 6w - 280 = 0
Now, we have a quadratic equation. We can either use some fancy quadratic formula or break out the clown calculator and do some guesswork. Let's go for the latter!
After some hilarious trial and error, I found out that the width, w, is 14 inches. And since the length is 6 inches greater, the length of the box is 20 inches. Ta-da!
So, the length of Walter's open-top box is 20 inches. Enjoy building that box, Walter, and remember, it's all about thinking outside the cardboard box!
To find the length of the box, we need to solve the problem step-by-step.
Step 1: Let's assign variables to the unknown quantities.
Let's say the width of the box is represented by "w" (in inches).
Step 2: Express the length in terms of the width.
According to the problem, the length is 6 inches greater than the width. Therefore, the length can be expressed as "w + 6" (in inches).
Step 3: Calculate the area.
The area of the rectangular sheet of material that will be folded to make the box is given as 280 square inches.
We can calculate the area of the sheet with the formula: Area = length × width.
In this case, we have: (w + 6) × w = 280.
Step 4: Solve the equation.
Let's solve the equation (w + 6) × w = 280 for the width of the box.
w^2 + 6w = 280
w^2 + 6w - 280 = 0
This is now a quadratic equation that can be factored or solved using the quadratic formula.
Step 5: Factor or use the quadratic formula to solve the equation.
Using the quadratic formula, we have:
w = (-6 ± sqrt(6^2 - 4 × 1 × -280)) / (2 × 1)
w = (-6 ± sqrt(36 + 1120)) / 2
w = (-6 ± sqrt(1156)) / 2
Step 6: Simplify the roots.
w = (-6 ± 34) / 2
w = (-6 + 34) / 2 or w = (-6 - 34) / 2
w = 28 / 2 or w = -40 / 2
w = 14 or w = -20
The width can't be negative, so the width of the box is 14 inches.
Step 7: Calculate the length.
Using the expression for the length: length = w + 6
length = 14 + 6
length = 20
Therefore, the length of the box is 20 inches.
To find the length of the box, we can start by setting up a simple equation using the given information.
Let's assume the width of the box is x inches. According to the problem, the length is 6 inches greater than the width. So, the length would be x + 6 inches.
Now, to find the area of the rectangular sheet of material, we need to multiply the length by the width. The formula for the area of a rectangle is length × width.
In this case, the area is given as 280 square inches. So we can write the equation as follows:
(x + 6) × x = 280
To solve this equation for x, we can expand and rearrange it:
x^2 + 6x = 280
Now, we have a quadratic equation. We can bring everything to one side to make the equation equal to zero:
x^2 + 6x - 280 = 0
To solve this quadratic equation, we can factor it, use the quadratic formula, or complete the square. In this case, let's factor the equation:
(x + 20)(x - 14) = 0
This equation can be true when either (x + 20) = 0 or (x - 14) = 0.
So, solving for x, we have two possibilities:
x + 20 = 0 => x = -20 (which is not possible since width cannot be negative)
OR
x - 14 = 0 => x = 14
Since the width of a box cannot be negative, we take x = 14 as the width.
Now, we can find the length by substituting the width value back into the equation for the length:
Length = x + 6 = 14 + 6 = 20 inches
Therefore, the length of the box is 20 inches.
just start by listing the facts:
width=w
length=w+6
height=2
area is the 4 sides + bottom
= 2(width*height)+2(length*height)+(length*width)
= 2(2w)+2(2(w+6))+w(w+6)
= 4w + 4w+24 + w^2+6w
= w^2 + 14w + 24
So, now we know that
w^2+14w+24 = 280
w^2+14w-256 = 0
Solve that for w, and the length is w+6