Mathematical Investigation

Differential Calculus – Cake Tin

Noah Nishihara HG: E05

The purpose of this investigation will be to investigate the method to maximise the volume of varying cake tinsa cake tin manufacturer will make. “Tiny” to “gigantic” sizes are prescribed for the range of cake tins. The shapes of the unfolded net of the tins will be square or rectangular based. For both types, the aim is to maximise the volume of each cake tin. The size of the square piece of tinplate (represented by the x variable) will be analysed through the use of various mathematical calculations involving calculus and the quadratic formula. Then a conjecture will be made from the results of the optimum open top cake tin which maximises the volume. This conjecture will show the relationship between the length and width of the tin plate and the side lengths of the cut-out square pieces. In addition, the rectangular based tin dimensions will be compared with the square-based tin dimensions. Finally, the conjectures for both types of cake tins will be potentially proven.

Optimisation has uses in economics, engineering, and also in the finding of the right price to yield maximum profit. Efficient and accurate tin plates are required for various cake dimensions, and by using differential calculus, the optimisation of the tinplate dimensionsis expected to be successful. Part A of this investigation will centre on the square piece of tinplate.A square (with x as both the length and width) from each corner of a square piece of tinplate (with side lengths l cm) will be cut off.The sides will then be folded to produce an open top cake tin. From the volume formula , a cubic function is derived, of which the derivative is found and solved to find the stationary point(s) by applying . The quadratic formula will then be used to solve for x. The roots will be checked with the domain previously worked out; those outside the domain are eliminated. The resulting side length of x cm determines the size of the squares to be cut to maximise volume.

To optimisethe cake tin, the length of each side of the tinplate will be supposed as 5cm. It will then be shown that the volume of the cake tin can be expressed as cm³. , or the derivative function will then be determined. Then the exact value of x which will maximise the volume of the cake tin will be calculated using the quadratic formula. Furthermore, the exact value of x for other values of l cm (the side lengths of the original square tinplate) will then be calculated and shown in table format. From this, a conjecture will be presented regarding any square piece of tinplate of side length l cm, which when a square is cut from each corner of length x cm, a maximum volume for the resulting open top cake tin will be obtained.

InPart B, the open top cake tin will be made by cutting a square (xcm by xcm) from each corner of a rectangular based tin plate. The sides of the rectangle are in a ratio r:s. The value of x that gives the maximum volume for the cake tin will be found. This process will then be repeated for rectangular tinplates with sides in two other ratios. To aid in the process, exact solutions for x will also be found.

A conjecture regarding the relationship between x (the cut to be made for the square) and the length of each side of the rectangle such that the cake tin has a maximum volume will be developed.

The first limitation of this investigation occurs as an error relating to the number of decimal places taken for a particular x value. This is because some functions result in quadratic formula calculations which involve square roots. Thus, the eventual conjecture based on the decimal version of the answer will be affected. If the fraction version of the answer was used this limitation would be overcome.

The algebraic calculations performed by hand had the possibly problematic consequence of minor errors and mistakes. However, as the answers were eventually checked using technology, namely using graphics calculators, this limitation did not impact the results of this investigation greatly. However, this limited the precision of the answers and resulted in conjectures for which the underlying reason was difficult to determine. To address this issue, a software package with greater capabilities and exactness of number values could have been used. In addition, the conjecture involving x, r, and s had different relationships for the changed ratios. Indeed this may be the case, however the inability to refer to more examples meant greater reliance on initial general facts. This investigation did not involve functions without derivatives. Functions are not always continuous, for instance, the function must follow this condition: . In other words, the graph must be smooth without sharp corners. These differential functions are able to describe in precise ways how related quantities change.

Conclusion

From this investigation, the most efficient dimensions for the tinplate to maximise its volume was found for both square and rectangular based cake tins. The process taken to achieve this outcome began with the analysisof a square piece of tinplate. From the volume formula, a cubic function was derived, of which the derivative was found and solved to find the stationary point(s) by applying Fermat’s Theorem. The quadratic formula was then used to solve for x. The roots were then tested with the domain previously worked out. The resulting side length of x cm determined the size of the squares to be cut to maximise volume. The various results from different side lengths then allowed a conjecture to be stated. Subsequently, the of which will maximise volume of the cake tin with side length .

This investigation proves that calculations involving differential calculuscan successfully determine the dimensions necessary for cake tins (rectangular prisms) to be optimised. It has determined an accurate and effective conjecture for all cake tins. The equation:

was found to have give the x value for the maximum possible volume. It could be stated that this proof reflects the quadratic formula in a minor way. This equation represents the optimal volume.

Due to the necessity of cake tins which maximise volume, the calculation (to find and prove the relationship between side lengths of the net) involving differential calculusand the quadratic formula was attempted. Through this investigation, the extent of the effectiveness of differential calculus in finding the maximum volume has been found. This includes an understanding of the relationship between . The quadratic formula was necessary as part of the method to ultimately prove the two conjectures. This can be seen from the final which depicts the relationship between the side lengths r cm and s cm with x cm.

The investigation results indicated that the method followed taken to achieve the optimum dimensions is the most efficient. To increase the accuracy of functions with certain limitations, technology can be used. The use of the quadratic formula instead of technology allowed the conjectures and final proofs to be made. The square based tinplate was the simplest, and allowed familiarisation with the method to follow and the patterns to expect. The rectangular prism tin required the setting of specific ratios to work out the optimum dimensions and the resulting patterns. Finally, an algebraic method was pursued because of the limitations involved with propositions and conjecturing based on patterns observed.

Further research into volume optimisation through side length calculations could involve an investigation of the pattern found with rectangular prisms which are progressively reduced in terms of the number of faces it has. The volume in that case increases by .