A rectangular piece of cardboard, whose area is 323 square inches, is made into an open box by cutting a 2-inch square from each corner and turning up the sides. If the box is to have a volume of 390 cubic inches, what size cardboard should you start with?

Question

A rectangular piece of cardboard, whose area is 323 square inches, is made into an open box by cutting a 2-inch square from each corner and turning up the sides. If the box is to have a volume of 390 cubic inches, what size cardboard should you start with?

Answer 1

If the cardboard is x by y then we need

xy = 323
(x-4)(y-4)(2) = 390
So a 17x19 sheet of cardboard does the trick

or,
323 = only has factors of 17 and 19
and it is easy to check that 13*15*2 = 390

Answer 2

To find the dimensions of the cardboard, we need to consider the dimensions of the open box after the corners are cut and sides turned up.

Let's assume the original length of the rectangular cardboard is x inches, and its width is y inches.

When we cut 2 inches from each corner and fold up the sides, the length of the resulting box would be (x - 4) inches, and the width would be (y - 4) inches. The height of the box would be 2 inches.

Now, we can calculate the volume of the box using the formula:

Volume = length × width × height

Substituting the given values, we have:

390 = (x - 4) × (y - 4) × 2

Next, we know that the area of the cardboard is given as 323 square inches. The formula for the area of a rectangle is:

Area = length × width

Substituting the values, we have:

323 = x × y

We have two equations:

390 = (x - 4) × (y - 4) × 2 (equation 1)
323 = x × y (equation 2)

We can solve this system of equations simultaneously to find the values of x and y.

To do this, we can rearrange equation 2 to solve for y:

y = 323 / x

Now we substitute this value of y in equation 1:

390 = (x - 4) × ((323 / x) - 4) × 2

Simplifying the equation:

195 = (x - 4) × ((323 / x) - 4)

Expanding the equation:

195 = 323 - 1292 / x - 4 × (323 / x) + 16

Combining like terms:

195 = 323 - 1292 / x - 1292 / x + 16

Simplifying further:

195 = 339 - 2584 / x

Rearranging the equation:

2584 / x = 339 - 195

2584 / x = 144

Now we can solve for x by cross-multiplying:

x * 144 = 2584

x = 2584 / 144

x ≈ 17.972

So, the approximate length of the original cardboard should be around 17.972 inches.

To find the width, we substitute the value of x in equation 2:

323 = 17.972 × y

y ≈ 323 / 17.972

y ≈ 17.966

So, the approximate width of the original cardboard should be around 17.966 inches.

A rectangular piece of​ cardboard, whose area is 323 square​ inches, is made into an open box by cutting a 2​-inch square from each corner and turning up the sides. If the box is to have a volume of 390 cubic​ inches, what size cardboard should you start​ with?

If the cardboard is x by y then we need

To find the dimensions of the cardboard, we need to consider the dimensions of the open box after the corners are cut and sides turned up.

A rectangular piece of cardboard, whose area is 323 square inches, is made into an open box by cutting a 2-inch square from each corner and turning up the sides. If the box is to have a volume of 390 cubic inches, what size cardboard should you start with?