A supermarket employee wants to construct an open-top box...

Name: A supermarket employee wants to construct an open-top box from a 14 by 30 in piece of cardboard. To do this, the employee plans to cut out squares of equal size from the four corners so the four sides can be bent upwards. What size should the squares be in order to create a box with the largest possible volume?
Uploaded: 2022-11-04T05:12:08-08:00
Duration: 4 min 42 s
Channel: Khushbu Rani
Description: A supermarket employee wants to construct an open-top box from a 14 by 30 in piece of cardboard. To do this, the employee plans to cut out squares of equal size from the four corners so the four sides can be bent upwards. What size should the squares be in order to create a box with the largest possible volume?

A supermarket employee wants to construct an open-top box from a 14 by 30 in piece of cardboard. To do this, the employee plans to cut out squares of equal size from the four corners so the four sides can be bent upwards. What size should the squares be in order to create a box with the largest possible volume?

Transcript

00:01 Hi, there is a question we say that a supermarket employee wants to construct an open top box from a 14 by 30 inches piece of cardboard this is 13 inch and we have 14 inch to do this employee plans to cut out squares of equal size from four corners and bent upwards what size should be should the squares be in order to create the box with largest possible volume so let us say x sides of dimension x are being cut from all sides so this length will be 30 minus 2x now because x will cut from this side x will cut from this side and the shape must be like this and this length resultant length will be 14 minus 2x all are in inches the volume will be length 30 minus 2x into width 14 minus 2x into now this will be bent so height will be x only let us take 2 and 2 common from here so 4 15 minus x into 7 minus x into x so velocity as a function of x sorry volume as a function of x would be 4 15 minus x and if you multiply it it will be 7x minus x squared now v prime x must be equal to 0 for critical point because this needs to be maximized 4 into 15 minus x product rule keeping first differentiating second 7 minus 2x plus differentiating first keeping second this must be equal to 0 dividing both sides by 4 4 will get cancelled out let us open it up 105 minus 30 x minus 30 x minus minus 7x plus 2x squared minus 7x plus x square will be equal to 0 3x squared minus 44 x plus 105 will be equal to 0 so using quality formula x will be minus b minus 44 plus minus b squared minus 4 plus minus b squared minus 4 ac divide by 2 into 3 and use the formula x equal to minus b plus minus b squared minus 4ac and the root by 2a for quality equation a x squared plus bx equal to 0 so this is 44 plus minus 44 square 44 square minus 4 into 105 1516 square root 38 .9358 divide by 6.

03:39 If you take a negative sign it will be 44 minus 38 .9358 divide by 6.

03:49 Okay, so 0 .844...

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