Title: To find the maximum volume of a field utilizing the strategy of differentiation. Problem assertion: Mr. Lee, proprietor of a non-public cake company, sells a square 5 inch cake in a field produced from 50 x 50 cm sheets of fabric. He wish to put a bigger sq. 8 inch cake in a box created from the identical 50 x 50 com sheets of fabric. He decided to use the strategy of differentiation to assist him along with his task.

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Method:

1. Three squares measuring 50 x 50 cm were reduce from bristol board sheets utilizing rulers, set squares, pencils and scissors.

It is from this square that the smaller squares of sides (x) shall be reduce from the sting. 2. The differential of the amount of the box was found, and the worth of (x) that might give the utmost volume was discovered by substituting the (x) values into the second differential. 3. Then smaller squares of size (x), which was discovered to be 8.33 x 8.33 cm, have been cut from the sides of the 50 x 50 cm square.

The minimize form was then folded and taped to provide the field with the utmost quantity.

four. A sq. of sides 2 cm was reduce from the edge of the 50 x 50 cm sheet. The flat shape was additionally folded and taped to provide a box. 5. A sq. of sides 20 cm was additionally cut from the sting of the 50 x 50 cm sheet. The flat form was additionally folded to provide a field. 6. Appropriate calculations have been made to prove that the square 2 cm and the square 20 cm didn’t produce a field with the maximum volume.

The idea used to unravel the problem

To calculate the quantity of the cube, length x breadth x height was utilized. This gave a cubic equation. This equation was then differentiated which gave a quadratic method. dydx was then equated = zero. The quadratic was then solved using the quadratic formula ( x=-b±b2-4ac2a) to obtain two values of (x). These values have been then substituted into the second differential (d2ydx2). If the value substituted produced a unfavorable worth, then that will be the length of one side of the sq. to be reduce out from the perimeters of the 50 x 50 cm sq. to provide the maximum quantity of the field. If the value substituted produced a optimistic worth, it’s due to this fact not the size for use to supply the maximum volume of the box.

The mathematical equations that have been utilized.

1. dydx = the by-product of the volume of the box

2. d2ydx2 =the second by-product of the quantity of the box three. x=-b±b2-4ac2a = the quadratic formula

four. X=length of the aspect of the sq. to be cut/cm

5. l= length of cube/cm

6. w=width of the cube/cm

7. h=height of the cube/cm

eight. V= l × w × h= quantity of cube

Calculations V= l x w x h L= 50 – 2x

W= 50 – 2x

H= x

V= (50-2x) x (50-2x) x (x)

= 4×3 – 200×2 + 2500x

dydx = 12×2-200x +2500

dydx =0

12×2-200x +2500=0

Using the quadratic formula

x=-b±b2-4ac2a

x=400±4002-4250012 24

X=25 or x=8.33

Substitute the values of x into the second differential

d2ydx2 =24x-400

When x=25

d2vdx2=24(25)-400

=200

The value is positive when x=25, so therefore the quantity isn’t maximum.

When x=8.33

d2vdx2=24(8.33)-400

=-200

The worth is constructive when x=8.33, so therefore the value is most.

Justification of answer

Substitute the values of x into the volume

V= 4×3 – 200×2 + 2500x

Substitute x=20

=4(20)3-200(20)2+2500(20)

=2000

The box made when 20 cm squares had been reduce from the edges

The graph above reveals that were x=20 , the y coordinate is not the utmost on the curve. Therefore the volume isn’t the maximum.

Substitute x=2

V=4(2)3-200(2)2+2500(2)

=4232

The field made when 2 cm squares were minimize from the edges.

=4232

The graph above exhibits that had been x=20, the y coordinate is not the maximum on the curve. Therefore the amount isn’t the utmost.

Substitute x=8.33

V=4(813)3-200(813)2+2500(813)

=9259.333

The field made when eight.33 cm squares had been minimize for the edges.

the above graph shows that had been x=8.33, the y coordinate is the maximum on the curve. Therefore the amount is the maximum at x=8.33

The above graph shows the place of all the coordinates. It is clearly see here that when x=8.33, the curve is at a maximum.

Discussion

By taking this mathematical strategy, Mr. Lee was capable of efficiently remedy his drawback. He could now match an eight inch cake in a box made from the same amount of fabric used to make the 5 inch packing containers. This has many benefits. By using the same materials, profits are maximized. He can now promote an even bigger cake and make more earnings than he was making on the 5 inch cake. If he sells his 5 inch cake for $40 and he sells his 8 inch cake for $60, he can enhance his income by 50%. With this increased income that he’s making, he can: increase his enterprise, rent more workers, enhance his amenities, open new branches of his franchise and lots of other issues. He can even increase the dimensions of not solely his cakes, but anything that he makes use of to place within the boxes. Also if Mr. Lee exports his items, he can ship more of his items than he used to utilizing the smaller box.

This will aid him in receiving extra foreign income. Also if he’s promoting an even bigger cake, individuals will get a better buy for his or her money. This will enhance his recognition and more prospects will be drawn to shopping at Mr. Lee’s. This will benefit his enterprise greatly. Also Mr. Lee can put items that normally couldn’t have fit within the small box, in his greater field. This will lead to a greater variety in his products. He can not solely make extra cash from muffins, however from different objects. This will ultimately lead to an evolution of his firm.

He could increase his business from selling pastries to selling food and other objects. This use of a a lot bigger field can also help in the preservation of the environment. Because a a lot bigger cake is being put in an even bigger box created from the same amount of material. This will lead to much less materials being used and due to this fact much less air pollution. This methodology may not only be used by Mr. Lee and desserts, however for many other reasons. It can be used by manufacturing firms, packaging corporations, industries, native businesses, publish workplaces, and lots of others. This will ensure that their earnings are maximized and they’re working at most effectivity.