MAXIMISING VOLUMES – EXTENSION 2

A Rectangular piece of material 4m by 3m is to be folded into an open ended Isosceles Triangular Prism ( Like a Tent ), See diagram below.

4 metres

x =1 metre

The dotted lines indicate folds, remember the rectangle can be folded in 2 ways, along the length or width. Calculate the volume created with the folds shown.

3 metres

x = 1 metre

Calculate the volume for other values of x. Plot the information in a graph. Try to predict the maximum volume from the graph. Generalise this process into an algebraic function. Graph the function using a graphics calculator or ( Graph paper ) use the Trace function to confirm your solutions or find the maximum volume. Reflect on the problem solving process, explain what you have discovered and how it might be used in other ways.

3 metres

DESIGNING A WATER TANK

2 m

2 metres

A Square metal sheet ( 2 metres by 2 metres ) is to made into an open topped water tank by cutting squares from each corner of the sheet ( See diagrams ) and bending the 4 remaining rectangular pieces up, to form the sides of the tank. These edges are then welded together.

We want to maximise the Volume of the final Water Tank.

How will the final volume of the tank depend upon the size of the squares cut out from the corners. Describe your answers by :

a) Making different sized scale models.b) Making calculationsc) Sketching Graphsd) Explaining the shape of your graph in words.e) Trying to find an Algebraic fomula

What is the biggest Volume possible, justify your answer.

DESIGNING A WATER TANK – SOME HINTS

Imagine cutting very small squares from each corner of the metal sheet. In your mind fold the sheet up to make an open top box. Will the Volume be large or small ? Why ?

Now imagine cutting out larger and larger squares… What will happen to the Volume ?

What are the largest squares you can cut ? What will be Volume be then ?

Sketch a rough graph of Volume against side of square cut ( see Diagram below ) to describe yout thoughts and explain fully in words what you might expect to happen.

Volume

Side of square cut.

In order to find a formula, imagine cutting a square of size x metres by x meters from each corner of the sheet. Find an algebraic expressions for the sides and hence the Volume of the resulting Water Tank.

Now try plotting an accurate graph using calculations or a Formula.

Use your graph to find the best cut to get the greastest Volume of the water tank.

Think about how you can improve the accuracy of your results.

WATER TANK – MAXIMISING VOLUMES – EXTENSION 1

You have now found the maximum volume for a water tank of side length 2m.

You now need to generalise your findings for a water tank of any size say a m

a m

best cut ? m

a m

To do this you need to find a rule for the length of cut from a sheet of metal of any size say a m, you need to look for a pattern between the length of one side of the sheet of metal a m and the size of the cut needed to get the maximum volume.

The answer may be obvious to you!!! In which case check your theory, explain your reasoning and find an algebraic expression for the biggest volume possible for a water tank when the sheet of metal is a m each side.

If the answer is not immediately obvious then you will have to work out the maximum volumes for sheets of metal of 3m by 3m, 4m by 4m etc. and then tabulate your results. In one column put size of sheet of metal in the other column put the best size cut and look for a rule.

Finally write an algebraic expression in terms of a for the maximum volume possible for any sheet of metal. If you can’t do this show your theory by working out the maximum volume possible for a tank made from a sheet of metal 60m by 60m.

Where possible write algebraic expressions using fractions, not decimals

MYP STUDENT HELP SHEET – DESIGNING A WATER TANK ( MAXIMISING VOLUMES )

Project Title : Maximising Volumes : Description : The project is in 3 parts. Part 1, students calculate volumes of a cuboid by varying squares to be cut out of each corner, graph results to approximate the maximum volume. Part 2, students generalise the process into an algebraic formula and try to find a more accure answer for the maximum volume. Part 3 is a similar problem but this time students have to maximise the volume of a triangular prism.

Criterion A – Knowledge & Understanding - Level Indicators.Score Number / Calculations Algebra Graph Work Measurement1 – 2 Does simple calculations

without aid of a calculator.Measures Accurately.

3 - 4 Does complex calculations without a calculator.

Uses algebraic symbols to represent quantities.Tries to write simple formula for a length. using symbols.

Graphs information generated from numerical data.

Calculates missing lengths from known lengths.Draws simple scale diagrams.

5 - 6 Tries to write complex formula for Volume using symbols.

Can draw a practical graph based on calculations of volume.

Calculates volume of cuboids.Uses Pythagoras to work out lengths.

7 – 8 Uses a calculator effectively to resolve complex calculations correctly.

Finds an algebraic expression for lengths, areas and volumes of the water tank or triangular prism in terms of a given variable.

Graphs simple algebraic Functions using tables of values.Finds maximum volume using numerical data.

Calculates volumes of Prisms.Uses Trigonometry to work out required lengths.

9 - 10 Expands double brackets to simply algebraic expressions.Combines pythagoras and algebra to find an expression for the volume of a Triangular Prism.

Finds maximum volumes by graphing complex algebraic functions.

Checks the dimensions of a formula using dimensional analysis.

Score Criterion B : Reasoning & Application : Level Indicators Specific1 – 2 Demonstrates understanding of the problem and generates some results practically.

With help from the teacher, draws accurate diagrams of boxes for given values of squares to be cut out.Writes about and explains the processes involved in calculations of lengths, areas and volumes.

3 – 4 Recognises patterns and suggests rules.Uses trial and error tecniques.

5 - 6 Graphs data generated from diagrams.Describes a rule in words or attempts to find a rule using symbols.Uses advanced trial and error techiques to increase accuracy.

7 – 8 Student uses a graph to find the maximum value for the volume of the water tank and gives reasons why the result is sensible.Student begins algebraic approach to problem and finds an algebraic expression for the Water Tank.Justifies why the algebraic expression for the Water Tank is correct by using a graphical analysis.Numerical analysis of Triangular Prism Problem.

9 - 10 Finds maximum volumes for Triangular Prism by graphing algebraic functions and interprets correctly specific features of the graph in terms of the original problem. Justifies why the algebraic expressions are correct by using graphical and dimensional analysis approaches.Uses appropriate technology, graphics calculators or graph plotters to find or check algebraic solutions.

MYP STUDENT HELP SHEET – DESIGNING A WATER TANK ( MAXIMISING VOLUMES )

Score Criterion C : Communication : Level Indicators Specific1 – 2 Student uses approprite symbols for lengths and volumes.

Some information is presented clearly.Writes about and explains the processes involved in calculations of lengths, areas and volumes.

3 - 4 Student uses and recognises a range of mathematical symbols including length, area, volume & Algebraic symbols such as powers and roots.Explains and interprets results.Tabulates results in a systematic way.Produces accurate graphs and interprets results.

5 - 6 The student uses and recognises a wide range of mathematical symbols related to measures, algebraic symbols such as powers and roots. Expressions with trigonometry where applicable.Students are clearly explained and interpreted.Information is displayed in the most appropriate way using the correct technology where applicable.

Score Criterion D : Reflection & Evaluation : Level Indicators Specific1 – 2 The student tries to give reasons why the method they used is appropriate

The student discusses the reliability of the results.3 - 4 The student gives good reasons for the method used in solving the problem.

The student evaluates the reliability of the findings.5 - 6 The student presents a reasoned argument why the methods used are appropriate to solving the problem.

Correctly interprets the results and evaluates reliability of the findings.Considers other approaches to solving the problem.

7 - 8 Student presents concise arguments why their findings are valid or not. A thorough evaluation of the reliability of the results.A complete interpretation of results. Compares the numerical and algebraic approaches.Discusses how maximisation techniques might be applied to other types of real problems.