1- Saud Sulaiman

2- Khalifa Salem

3- Saeed Mohammed

Grade 12-01

This Project for: Mr. Amer

The Use of Differentiation In Real-life Applications

**Task 1**

Differentiation is very important in many fields, and has made great contributions in these fields, which are still noticed nowadays. In economics, differentiation can be used to calculate the rate of change of the ordering and transportation cost for the components used to manufacture a product, in which the cost is dependent on the order size. For example, the transportation and order cost of the components used to manufacture a product can be calculated by using the function below.

Where C is the order and transportation cost, and x is the order size. The derivative of the function above can be used to calculate the rate of change of the order and transportation cost as the order size increases. By doing so, the company or the cooperation can predict the order size in which the cost will start decreasing or increasing.

Also, differentiation is used in biology to evaluate the rate at which a bacteria population is growing. For instance, the growth in number of a population of 500 bacteria that was introduced into a culture can be calculated using the equation below.

Where t is the time measure in hours. The derivative of the equation above can be used to calculate the rate at which the population grows, which can allow biologists to predict whether the population is growing fast and counter measures should be taken to slow down the growth. In the same way, the rate of growth of some viruses can be calculated by using different equations. Knowing the rate of growth could help biologists to predict the time it takes for the virus to spread in an area or a city.

Examples of Real-life Applications of Differentiation

**Task 2**

Calculus Speed Trap

Three students used a

distance measure app to

measure the distance

between the gate of the

school and the road, which

was found to be 78m. One of the students extended his arm and followed different passing cars until it hits a 45-degree angle. The other two students timed how long it took for each passing car and averaged the times. The times that the cars took to travel that distance are presented in the table below

Average time = 4.97s

The average rate of change of his arm was calculated by dividing the 45-degree angle by the average time. The average rate of change was evaluated and was found to be 0.158 rad/s. Knowing that the student stopped at 45-degree angle, that will form an isosceles triangle. Therefore, the distance that the cars traveled along the road till the student reached 45-degree angle is the same as the distance between the gate and the road. The average speed of the cars was calculated by dividing the distance they traveled along the road by the average time they took to travel the same distance. The average speed of the cars is 15.7 m/s.

**Task 3**

Coffee Maker

In the process of making coffee, coffee is draining from a conical filter into a cylindrical coffeepot at a rate of 10 in ^3/min as show in the figure below.

The equation used to find the volume of a cylinder is

Since the radius is constant which is 3in, the volume of the cylinder in this situation could be written as V=9πh. The derivative of the volume was found which is The rate at which the level of the coffee in the pot was rising was evaluated and is 0.35 in/min. The ratio between the depth of the coffee in the filter and it’s radius at that depth is The following equation is used to find the volume of a cone is Since the radius of the coffee at a certain height is half that height, the equation can be rewritten as The derivative of the volume is The level of the coffee in the cone is falling at a rate of 0.51 in/min when the coffee in the cones is 5 in. deep.

**Task 4**

Consuming Gas Mileage

The gas mileage of an automobile is affected by many factors like road surface, tire type, velocity, fuel octane rating and the speed and direction of the wind. The effect of the velocity on the gas mileage of a certain car is expressed in the function below

The maximum or minimum gas mileage is obtained when it’s rate of change is 0. The rate of change of the gas mileage is 0 when the velocity is 103.62km/h or 38.6km/h. The gas mileage when the velocity is 103.62km/h is 11.51mil/gal, while it’s 28.528mil/gal when the velocity is 38.6km/h. Therefore, to obtain the best gas mileage when driving this car a velocity of 103.62km/h should be maintained.

**Task 5**

Response of a Plant to Daylight changes

In London, the number of hours of daylight can be found using:

L(t)=12- 4.5cos(t)

Where t is the time measured in units, 2π corresponds to one year. A certain plant unfolds it’s leaves in the spring in response to changes in day length. The rate of change of the day length is expressed by:

The graphs of L(t) and are presented in the figures below.

From the figure on the lift slide, it can be seen from the graph of L(t) that the number of hours of daylight is increasing in the interval (0,π), which corresponds to the interval of January and June, since 2π is considered as a one year. Similarly, the number of hours of daylight is decreasing in the interval (π,2π), which correspond the interval, June and December. The longest day will come in June, while the shortest will occur in December. It’s obvious from the graph that the longest day has 16.5 hours of daylight, while the shortest has 7.5 hours. It can be seen from the graph of that at March and September it would be easiest to the plant mentioned above to detect changes in day length, since they are maximum and minimum points.

**The End**

Thank You

Pictures and videos While we were working.