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Geometry/Direct and Inverse Variation

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QUESTION: Hello sir

Sir below are eg of  direct an inverse variation.

direct variation--> An electric pole ,14 meters high, casts a shadow of 10 meters . Find the height of a tree that casts a shadow of  15 meters unders similar condition.

sol let the height of a tree be x meter

Height of tree--   14  x
lenght of shadow-- 10  15

 Note the more the height of object , the more would be its lenght of its shadow

there for it is a case of direct variation

=  that is X1 / Y1= X2 / y2

= 14/ 10=x/15

=14/10 x 15  =x

14 x 3/ 2 = x

21= x

Inverse variation

If 15 workers can buld a wall in 48 hrs , how many workers will be required to do the same work in 30 hrs

Let the no of workers employed to build the wall in 30 hours be y

no. of hours----- 48  30
no. of workers--- 15  y

This is a case of inverse variation. that is x1 y1=x2 y2
48 x 15 30 x y

there fore  48 x 15
         -------  = y
         30

or y =24

Sir on basis of example please tell me why we do different in both variation and why

ANSWER: Hi Aishwarya,

Direct variation involves a common ratio.  This is usually pretty clear to see.  As you say in your example, the taller an object, the longer its shadow.

Indirect variation is a bit trickier in that it involves a common product (multiplication).  In the example you gave, we have worker-hours, which is a measure of total effort.  15 hours times 48 workers means (15)(48)=720 worker-hours.  If you want to do the same work, in 30 hours, we will get (30)y=720 worker-hours, which means that y=24 workers.
You could string it all together like this:

(15)(48)=720=(30)(y)

But the 720 in the middle does not matter.  We know the left side must equal to the right side, so we can reduce it to:

(15)(48)=(30)(y)

And that is the format of the equation you provided.

Hope this helps,
Azeem

---------- FOLLOW-UP ----------

QUESTION: Hello sir

Sir iam extremely sorry t5o irritate you behind direct and inverse variation  but sir my question is not cleared. My confusion is what is in the inverse that we do common product but why not in direct variation. If we do direct variation method in inverse why it is wrong . Why do we apply different formula for direct and diferent for inverse.

Answer
Hi Aishwarya,

In general, why do you use multiplication sometimes and division others?  Why do you add sometimes, and subtract sometimes?

They are different situations and they call for different operations or different ways of solving them.  It is up to you to determine from the formulation of the problem which is the correct way to solve it.

Regards,
Azeem

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Azeem Hussain

Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

Publications
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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