A rectangular piece of cardboard is 2 in. longer than it is wide. A square piece 3 in. on a side

A rectangular piece of cardboard is 2 in. longer than it is...

Key Concepts

Volume Calculation of 3D Objects

Calculating volume is central when constructing three-dimensional shapes, such as an open-top box made from a folded cardboard piece. This concept involves using the product of new length, width, and height, which are derived from the original dimensions after transformation.

Algebraic Equation Formation and Solving

This concept involves translating the geometric relationships and volume requirements into an algebraic equation. Establishing a relationship between the modified dimensions and the known volume leads to a polynomial equation, which is then solved using algebraic techniques to find the original dimensions.

Dimensional Relationships

This concept involves understanding how measurements are interrelated, such as one dimension being defined in terms of another. When one dimension is described relative to a second, the relationship allows you to express the unknown variables with fewer parameters, which simplifies solving the problem.

Geometric Transformation by Cutting and Folding

This concept covers the process of modifying a two-dimensional shape into a three-dimensional object. When squares are cut from the corners of a rectangular piece and the sides are folded up, the dimensions of the usable base change, and recognizing these new dimensions is key to formulating the problem.

Transcript

00:01 Hi everyone so in this given question we have given a rectangle so let's say this is a rectangle okay and we are said that let i'm assuming it's this breath to be x and it is given the okay the length is two meter more than its breath okay so its breath will be x minus 2 and its length will be x we can assume anything actually it's in our hand so after that what is done we have cut it for corners okay four rectangle four square corner of three inches inside okay so the result is that it will look like this we are left with this portion and the remaining portion is lifted upwards so it will form like kind of open box so you can assume it like this let's say this is the our base of the books and attached to it it will look like this these are the four sides which are lifted upwards within the box okay so now we are given that the volume of this box is given and we are asked to find these length and initial breaths okay so what is the left like what is this side which is left it will be nothing but x minus 2 minus this plus this which is both as 3 and 3 so we will subtract x minus 2 with this so we will be left with x minus 8 so this will be equal to x minus 8 now what will be this now we have to find this as well okay so this will be nothing but x minus 3 so this will be equal to x minus 3 minus 3 so we are actually removing these two lengths from x okay so we are left with x minus 6 okay so now what we have to do is we have to now find its volume and height as you know it is 3 okay because this is the portion which we have removed okay so we have our length which is equal to x minus 8 our breath which is equal to x minus 6 and our height which is equal to 3 and we know that volume for this cuboid is equal to length time bread time height will be equal to its volume so volume of this is actually given as 765 765 which is equal to x minus 8 x minus 6 times 3 so we can cut 3 from here it will result in 3 to 6 5 times 5 okay so we are left with this we are now left with a quadratic so let's solve this quadratic so this quadratic will be x square minus times of 14x plus 48 minus 255 is equal to 0.

03:33 So it will result in x square minus 14 x plus sorry it will be minus okay minus times of when we remove 48 we are left with 207.

03:48 So now we will use the general formula of quality equation to solve it which state that if quality equation is this then we can find x by using this formula minus b plus minus under b square minus 4 c divided by 2a okay so now for this we can write the same let's compare the terms between these two equations so here a will be 1 b will be minus 14 and c will be minus 207 so our x from here will be minus time of minus 14 which will go to 14 plus minus under root minus 14 square is actually okay 14 square will be equal to will be equal to 196 okay so 196 minus times 4 a is 1 it will be minus 207 okay and in the denominator we are left in 2 and 1 so let's simplify it further so x will be equal to 7 plus minus under root 4 times of 9 minus so it should be plus 207 okay so where x will be 7 plus minus when we add this we will get 256 so you can you recall it will be under it will be square of 18 okay so let's no sorry it will not be a square of 18 actually so actually will be a square root of 16 okay so our x will result in 7 plus minus 16.

05:41 Since negative i are not a perceptive, i will go with positive.

05:44 So it will result in 23...

Please give Ace some feedback