Eng. Math ProjectMade by:

Hassan Ahmed Hassan

Khalifa Abdulwahab

Class: 12-14

Supervised by: Ms.Banan

Goals Of the Project How to find the work done by a variable force using Hooke’s

Law

How to find the volume of the solid of revolution by using the washer method

Recognize and solve differential equation that can be solved by separation of variables with an application to wildlife population

Approximating a definite integral using Simpson’s rule with an application to approximate the work done on an object.

Task 1 Hooke’s Law Application – WORK –

The concept of work is important to scientists and engineers for determining the energy needed to perform various jobs. For instance, it is useful to know the amount of work done when a spring is compressed.

Robert Hook developed a law in physics called Hooke’s law which says that the force required to compress or stretch a spring within its elastic limits is proportional to the distance that the spring is compressed from its original length; That is

F= K x

Where K is the spring constant and x is the distance that the spring is compressed or stretched.

Task 1

The work done by the force F as the object moved from x = a to x = b is:

1. A force of 750 pounds compresses a spring 3 inches from its natural length of 15 inches. Find the work done in compressing the spring from x = 3 to x = 6 inches.

Task 1 Solution

Task 1

2. To determine the size of the motor required to operate a press, a company must know the amount of work done when the press moves an object linearly 5 feet. The variable force to move the object is where F is given in pounds and x gives the position of the unit in feet.  

Use Simpson’s rule with n = 12 to approximate the work W done through one cycle.

Task 1 Solution

Task 2

Mercator’s world map

Mercator’s map greatly simplified navigation: even over long distances, straight lines on the map corresponded to fixed compass bearings. This was accomplished by distorting distances. On Mercator’s map, the distance of an object from the equator to a parallel at latitude θ is given by

Where k is a constant of proportionality

Task 2

1. Verify that

Task 2

2. Los Angeles has latitude of 34°03'N (the 03' represents 3 minutes of latitude, each minute of latitude is 1/60 of a degree. if Los Angeles is to be 7 inches from the equator on a Mercator map, how far from the equator should we place Abu Dhabi which has a latitude of 24°28'N.

Task 2 Solution

Task 3

Wildlife Population

The rate of change of the number of coyotes N (t) in a population is directly proportional to 650 – N (t) that is

where t is the time in years. When t=0 the population is 300 and when t=2 the population has increased to 500

Task 3

1. Use the separation of variables procedure to find the general solution of the differential equation

Task 3 2. Find the population when t=3

The End

In the end I would like to thank you all for Listening and paying attention at our presentation