Mathematics and Mathematics and EngineeringEngineering

Mathematics and EngineeringMathematics and Engineering Engineering uses many fields of Engineering uses many fields of

mathematicsmathematics• CalculusCalculus• Geometry and TrigonometryGeometry and Trigonometry• OptimizationOptimization• AlgebraAlgebra• LogicLogic

Mathematics are used to determineMathematics are used to determine• The materials neededThe materials needed• How much time and how many people are How much time and how many people are

requiredrequired• How strong, fast, big, heavy, reliable, long-How strong, fast, big, heavy, reliable, long-

lasting, expensive the solution will belasting, expensive the solution will be

Ex.1: Calculus and Bridges Ex.1: Calculus and Bridges DurabilityDurability

A quarter-mile bridge rests on four A quarter-mile bridge rests on four concrete pillarsconcrete pillars

In the morning rush, the bridge is a traffic In the morning rush, the bridge is a traffic bottleneckbottleneck• (cars are stuck on the bridge)(cars are stuck on the bridge)

Every car weighs two metric tons and is Every car weighs two metric tons and is about 15 feet longabout 15 feet long

How much vehicle weight does each pillar How much vehicle weight does each pillar need to support?need to support?• Assume equal distribution of weight among Assume equal distribution of weight among

pillarspillars

Ex.2: Building a TunnelEx.2: Building a Tunnel RequirementsRequirements

• We need to build a tunnel that will fit two We need to build a tunnel that will fit two traffic lanes.traffic lanes.

• The tunnel will have the shape of a semi The tunnel will have the shape of a semi cylinder (for durability)cylinder (for durability)

• Each lane is 12 feet wide, and the tunnel must Each lane is 12 feet wide, and the tunnel must fit trucks of up to 4 metersfit trucks of up to 4 meters

• The length of the tunnel is 2 milesThe length of the tunnel is 2 miles• The average density of the soil is 1.6Mg/mThe average density of the soil is 1.6Mg/m33

• The tunnel boring machine can progress at a The tunnel boring machine can progress at a pace of 2m/hourpace of 2m/hour

Design questionsDesign questions• What should the diameter of the tunnel be?What should the diameter of the tunnel be?• How long will it take to dig the tunnel?How long will it take to dig the tunnel?• How much soil has to be dug and removed (in How much soil has to be dug and removed (in

weight)?weight)?

Ex.3 Building an aqueduct Ex.3 Building an aqueduct We have a source of fresh water We have a source of fresh water

(reservoir)(reservoir)• Cold SpringsCold Springs

We have two cities:We have two cities:• Sun City, with a demand of 1 mSun City, with a demand of 1 m33/s/s• Sun Valley, with a demand of 2 mSun Valley, with a demand of 2 m33/s /s

We need to install water pipes We need to install water pipes from Cold Springs that will supply from Cold Springs that will supply Sun City and Sun ValleySun City and Sun Valley

The cost of pipelines isThe cost of pipelines is• Cost (1 mCost (1 m33/s) = $10,000 / mile /s) = $10,000 / mile • Cost (2 mCost (2 m33/s) = $20,000 / mile /s) = $20,000 / mile • Cost (3 mCost (3 m33/s) = $30,000 / mile/s) = $30,000 / mile

Problem:Problem:• Find the cheapest pipeline networkFind the cheapest pipeline network

SunCity

SunValley

Cold Springs

60 miles 60 miles

60 miles

1 m3/s 2 m3/s

Math Origami ExerciseMath Origami Exercise

Fold a square piece Fold a square piece of paper intoof paper into• A square of half the A square of half the

surfacesurface• An isosceles triangleAn isosceles triangle• An equilateral An equilateral

triangletriangle