Can you find X value? | (Rectangular Box) | |

[00:00:00]Welcome to PreMath. In this video, we have got this closed rectangular box, as you can see in this given diagram, such that the area of this front rectangle has been given to us as 9 * x cm squared. The area of this top rectangle has been given to us as 6 * x cm squared. And finally, the area of this right rectangle has been given to us as 4 * x cm squared as well. And furthermore, the volume of this box has been given to us as 1,728 cm cubed. And now our task is to find the value of x. Please don't forget to give a thumbs up and subscribe. And please keep in mind that this figure may not be 100% true to the scale. Let's go ahead and get started, and here's our very first step. Let's focus on this closed box.

[00:01:05]And our very first task is to label its side lengths. I'm going to label this side as lower case a.

[00:01:17]This side length as lower case b.

[00:01:22]And finally, this side length as lower case c.

[00:01:27]And here's our next step. Let's focus on this front rectangle. And we can see the area of this front rectangle has been given to us as 9 * x cm squared. And now let's recall the area of a rectangle formula. Area is always equal to length * the width of the rectangle. And here we can see the dimensions of this front rectangles are a and b.

[00:01:55]And the area of this uh front rectangle is 9 * x. So, therefore, by using this formula, we can write uh this uh as uh a * b = 2 9 * x. I'm going to call this as our equation number one.

[00:02:15]And now, let's focus on this uh top uh rectangle. And we can see its uh dimensions are uh a and uh c.

[00:02:24]And the area of this uh top rectangle has been given to us as 6 * x. So, therefore, we could write this one as uh a * c = 2 6 * x. And I'm going to label this one as our equation number two.

[00:02:45]And now, finally, let's focus on this uh right uh rectangle. And we can see its dimensions are uh b and uh c. And the area of this uh right rectangle has been given to us as 4 * x. So, therefore, we could write uh b * c = 2 4 * x. And I'm going to label this one as our equation number three. And now, we can see we are ended up with this uh system of uh three equations. The first equation is a * b = 9 * x.

[00:03:25]a * c is uh 6 * x. And finally, b * c is 4 * x. And now, we are going to multiply out uh all these three equations. On the left-hand side, we are going to multiply out uh these left-hand sides. So, therefore, we could write uh a * b * a * c * b * c is going to be equal to another right-hand side. I'm going to multiply out uh these one as well. So, therefore, we could write uh 9 * X uh times 6 times uh X and finally 4 times X.

[00:04:13]And now on the left-hand side, we could see A times A is going to give us uh A squared, B times B is going to give us B squared, and likewise, C times C is going to give us uh C squared. So, therefore, we could write this one as uh A squared times B squared times C squared equals to Let's go ahead and multiply out uh the right-hand side as well. So, that is going to give us uh 216 X power 3.

[00:04:49]And now we are going to tweak the left-hand side. This could be written as uh A times B times C whole squared is going to be equal to 216 X power 3. And I'm going to label this one as our equation number 4. And now let's make an observation. We can see the volume of this closed box has been given to us as uh 1,728.

[00:05:20]And now let's recall the volume formula.

[00:05:23]Volume is always equal to length times the width times the height.

[00:05:28]And here we can see the dimensions for this closed box uh are lower case A, lower case B, and lower case C. So, let's go ahead and fill in the blanks in this formula. So, therefore, we could write uh A times B times C is going to be equal to this given volume which is 1,728.

[00:05:56]And now let's focus once again on this equation 4.

[00:06:00]And here I have copied down our equation 4. And we know our A times B times C value is 1,728.

[00:06:09]So I'm going to substitute that A times B times C value 1,728 over here. So therefore we could write 1,728 whole square equals to 216 X power 3. And we know the square of to be equal to 290 8 59 84 is going to be equal to 216 X power 3. And now I'm going to divide both sides by 216. This 216 and 216 is gone. So therefore we could see that our X power 3 value is going to be equal to 1 3 8 2 4. And now I'm going to take the cube root on both sides of this equation. And here we can see this cube and cube root undo each other. So therefore our X value is going to be equal to the cube root of this whole thing is going to be 24.

[00:07:37]So thus after all the calculations and manipulations, the X value turns out to be 24. So, thus we could see that our X value turns out to be 24.

[00:07:52]And that's our final answer.

[00:07:55]Thanks for watching and please don't forget to subscribe to my channel for more exciting videos. Bye.