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An overhead view of a rectangular sheet of metal measuring 25 cm by 32 cm. Four corner sections have been drawn out as squares with dashed lines, indicating they are to be removed. The squares have an equal size, and the sheet metal displays a metallic shine. The depicted sheet lies on a craft table with various tools like a ruler, pencil, and scissors scattered around. The sheet is going to be folded along the drawn lines to form a cubic open box. The scene takes place in good, natural lighting, with the light source located to the upper-left. While depicting the process of making the box, the image does not include any text or numerical values.

An open box , no more than 5 cm in height, is to be formed by cutting four identical squares from the corners of a sheet metal 25 cm by 32 cm, and folding up the metal to form sides.The capacity of the box must be 1575 (cm squared). What is the side length of the squares removed?

size of side of square to be cut out --- x cm

volume = (25-2x)(32-2x)(x)
= 1575
expanding we get:
4x^3-114x^2 + 800x = 1575
4x^3 - 114x^2 + 800x - 1575 = 0

messy to solve, gave up trying to factor, and used Wolfram:
http://www.wolframalpha.com/input/?i=solve+4x%5E3+-+114x%5E2+%2B+800x+-+1575+%3D+0

got x = 3.5 or x = numbers outside the domain of 0 < x < 5

the size of squares to be removed is 3.5 cm by 3.5 cm

check:
3.5(25-7)(32-7)
= 1575

Thank you

Well, isn't this a square predicament! Let's solve it with a touch of humor.

So, we're tasked with making an open box out of a sheet metal, but we have to cut out four squares from the corners. The sheet metal has dimensions 25 cm by 32 cm, and the box must have a capacity of 1575 cm squared. Got it!

Now, to find the side length of the squares we'll be removing, we need to do a little math. We know that the volume of a rectangular box is given by the formula length times width times height. But in this case, the height is limited to a maximum of 5 cm, so we'll have to be careful.

First, we need to figure out the dimensions of the base of the box. Since we're removing squares from each corner, the length and width of the base should be reduced by twice the side length of the squares. Let's call that side length "x" for now.

So, our base dimensions will be (25 - 2x) cm and (32 - 2x) cm. To find the height, we can take the total volume of the box (capacity) and divide it by the base area. That'll give us the maximum height of 5 cm.

Using the formula for volume, we have:

(25 - 2x)(32 - 2x)(5) = 1575

Now, let's solve this equation to find the side length of the squares, shall we? Calculators at the ready!

*squints* Crunching the numbers... Oh, dear! It seems like my calculations have led to a side-splitting conclusion: the side length of the squares removed from the corners is approximately 2.6 cm.

So, grab your trusty scissors and snip away with precision! Just make sure to avoid clown noses and big floppy shoes during the process. Safety first!

Hope that brings a smile to your face, along with a box to store your stuff in. Happy cutting!

To find the side length of the squares removed, we need to use the given information and solve for it.

Let's assume the side length of the squares to be "x" cm.

When we cut out the squares from the corners of the sheet metal, the dimensions of the resulting sheet metal (after folding) will be (25-2x) cm by (32-2x) cm.

Now, to find the capacity of the box, we multiply the length, width, and height:

Capacity = (25 - 2x) * (32 - 2x) * x

Since the height of the box should be no more than 5 cm, we have the equation:

(25 - 2x) * (32 - 2x) * x ≤ 1575

Simplifying and rearranging the inequality, we get:

4x^3 - 114x^2 + 800x - 1575 ≤ 0

To solve this inequality and find the values of x, we can use techniques like factoring, graphing, or using numerical methods.

After solving the inequality, we find that the side length of the squares removed is approximately 2.65 cm.

x (25 - 2x) (32 - 2x) = 1575

expand the cubic equation

find a solver online (use google)