# 5.2 Solving problems

 Page 1 / 2

## Using differential calculus to solve problems

We have seen that differential calculus can be used to determine the stationary points of functions, in order to sketch their graphs. However, determining stationary points also lends itself to the solution of problems that require some variable to be optimised .

For example, if fuel used by a car is defined by:

$f\left(v\right)=\frac{3}{80}{v}^{2}-6v+245$

where $v$ is the travelling speed, what is the most economical speed (that means the speed that uses the least fuel)?

If we draw the graph of this function we find that the graph has a minimum. The speed at the minimum would then give the most economical speed.

We have seen that the coordinates of the turning point can be calculated by differentiating the function and finding the $x$ -coordinate (speed in the case of the example) for which the derivative is 0.

Differentiating [link] , we get: ${f}^{\text{'}}\left(v\right)=\frac{3}{40}v-6$ If we set ${f}^{\text{'}}\left(v\right)=0$ we can calculate the speed that corresponds to the turning point.

$\begin{array}{ccc}\hfill {f}^{\text{'}}\left(v\right)& =& \frac{3}{40}v-6\hfill \\ \hfill 0& =& \frac{3}{40}v-6\hfill \\ \hfill v& =& \frac{6×40}{3}\hfill \\ & =& 80\hfill \end{array}$

This means that the most economical speed is 80 km $·$ hr ${}^{-1}$ .

The sum of two positive numbers is 10. One of the numbers is multiplied by the square of the other. If each number is greater than 0, find the numbers that make this product a maximum.

1. Let the two numbers be $a$ and $b$ . Then we have:

$a+b=10$

We are required to minimise the product of $a$ and $b$ . Call the product $P$ . Then:

$P=a·b$

We can solve for $b$ from [link] to get:

$b=10-a$

Substitute this into [link] to write $P$ in terms of $a$ only.

$P=a\left(10-a\right)=10a-{a}^{2}$

2. The derivative of [link] is: ${P}^{\text{'}}\left(a\right)=10-2a$

3. Set ${P}^{\text{'}}\left(a\right)=0$ to find the value of $a$ which makes $P$ a maximum.

$\begin{array}{ccc}\hfill {P}^{\text{'}}\left(a\right)& =& 10-2a\hfill \\ \hfill 0& =& 10-2a\hfill \\ \hfill 2a& =& 10\hfill \\ \hfill a& =& \frac{10}{2}\hfill \\ \hfill a& =& 5\hfill \end{array}$

Substitute into [link] to solve for the width.

$\begin{array}{ccc}\hfill b& =& 10-a\hfill \\ & =& 10-5\hfill \\ & =& 5\hfill \end{array}$
4. The product is maximised if $a$ and $b$ are both equal to 5.

Michael wants to start a vegetable garden, which he decides to fence off in the shape of a rectangle from the rest of the garden. Michael only has 160 m of fencing, so he decides to use a wall as one border of the vegetable garden. Calculate the width and length of the garden that corresponds to largest possible area that Michael can fence off.

1. The important pieces of information given are related to the area and modified perimeter of the garden. We know that the area of the garden is: $A=w·l$ We are also told that the fence covers only 3 sides and the three sides should add up to 160 m. This can be written as: $160=w+l+l$

However, we can use [link] to write $w$ in terms of $l$ : $w=160-2l$ Substitute [link] into [link] to get: $A=\left(160-2l\right)l=160l-2{l}^{2}$

2. Since we are interested in maximising the area, we differentiate [link] to get: ${A}^{\text{'}}\left(l\right)=160-4l$

3. To find the stationary point, we set ${A}^{\text{'}}\left(l\right)=0$ and solve for the value of $l$ that maximises the area.

$\begin{array}{ccc}\hfill {A}^{\text{'}}\left(l\right)& =& 160-4l\hfill \\ \hfill 0& =& 160-4l\hfill \\ \hfill \therefore 4l& =& 160\hfill \\ \hfill l& =& \frac{160}{4}\hfill \\ \hfill l& =& 40\mathrm{m}\hfill \end{array}$

Substitute into [link] to solve for the width.

$\begin{array}{ccc}\hfill w& =& 160-2l\hfill \\ & =& 160-2\left(40\right)\hfill \\ & =& 160-80\hfill \\ & =& 80\mathrm{m}\hfill \end{array}$
4. A width of 80 m and a length of 40 m will yield the maximal area fenced off.

## Solving optimisation problems using differential calculus

1. The sum of two positive numbers is 20. One of the numbers is multiplied by the square of the other. Find the numbers that make this product a maximum.
2. A wooden block is made as shown in the diagram. The ends are right-angled triangles having sides $3x$ , $4x$ and $5x$ . The length of the block is $y$ . The total surface area of the block is $3600{\mathrm{cm}}^{2}$ .
1. Show that $y=\frac{300-{x}^{2}}{x}$ .
2. Find the value of $x$ for which the block will have a maximum volume. (Volume = area of base $×$ height.)
3. The diagram shows the plan for a verandah which is to be built on the corner of a cottage. A railing $ABCDE$ is to be constructed around the four edges of the verandah. If $AB=DE=x$ and $BC=CD=y$ , and the length of the railing must be 30 metres, find the values of $x$ and $y$ for which the verandah will have a maximum area.

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!