# Add Mahs Sba

Add maths SBA

Mrs. Ramnarine

5s

Title: To find the maximum volume of a box using the method of differentiation. Problem statement: Mr. Lee, owner of a private cake company, sells a square 5 inch cake in a box made from 50 x 50 cm sheets of material. He would like to put a bigger square 8 inch cake in a box made from the same 50 x 50 com sheets of material. He decided to use the method of differentiation to help him with his task. Method:

1. Three squares measuring 50 x 50 cm were cut from bristol board sheets using rulers, set squares, pencils and scissors. It is from this square that the smaller squares of sides (x) will be cut from the edge. 2. The differential of the volume of the box was found, and the value of (x) that would give the maximum volume was found by substituting the (x) values into the second differential. 3. Then smaller squares of size (x), which was found to be 8.33 x 8.33 cm, were cut from the edges of the 50 x 50 cm square. The cut shape was then folded and taped to provide the box with the maximum volume. 4. A square of sides 2 cm was cut from the edge of the 50 x 50 cm sheet. The flat shape was also folded and taped to produce a box. 5. A square of sides 20 cm was also cut from the edge of the 50 x 50 cm sheet. The flat shape was also folded to produce a box. 6. Appropriate calculations were made to prove that the square 2 cm and the square 20 cm did not produce a box with the maximum volume.

The concept used to solve the problem

To calculate the volume of the cube, length x breadth x height was utilized. This gave a cubic equation. This equation was then differentiated which gave a quadratic formula. dydx was then equated = 0. The quadratic was then solved using the quadratic formula ( x=-b±b2-4ac2a) to obtain two values of (x). These values were then substituted into the second differential (d2ydx2). If the value substituted produced a negative value, then that will be the length of...

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