Q. 173.8( 10 Votes )

A person wants to

Answer :

The cost of planting trees is given by


C(x) = x3 - 45x2 + 600x where x is number of trees


We have to find the number of trees x such that the cost C(x) is minimum which means we have to find at what value of x we get minimum value for C(x)


Let us find where C(x) attains its minimum value


For this let us first identify where C(x) is strictly increasing/decreasing


We know that if C’(x) > 0 then C(x) is strictly increasing and if C’(x) < 0 then C(x) is strictly decreasing


Let us find C’(x)


C’(x) = 3x2 – 90x + 600


C’(x) = 3x2 – 30x – 60x + 600


C’(x) = 3x (x – 10) – 60(x – 10)


C’(x) = (3x – 60) (x – 10)


C’(x) = 3(x – 20) (x – 10)


C’(x) is 0 at x = 10 and x = 20


Now we will check whether C’(x) is greater than 0 or less than 0 in domain (-∞, 10), (10, 20) and (20, ∞)



In (-∞, 10) C’(x) > 0 hence C(x) is strictly increasing in (-∞, 10)


In (10, 20) C’(x) < 0 hence C(x) is strictly decreasing in (10, 20)


In (20, ∞) C’(x) > 0 hence C(x) is strictly increasing in (20, ∞)


Observe that as C(x) is strictly decreasing in (10, 20) the graph of C(x) will be going downwards as one goes from 10 to 20 hence the minimum value of C(x) will be at 20


Hence the person should order for 20 trees to pay the minimum


As local authority has imposed a restriction that it can plant 10 to 20 trees in one community park 20 is a feasible number hence 20 people should order 20 trees


Now cost of 20 trees that is C(x) at x = 20 C (20)


C (20) = 203 – 45(20)2 + 600(20)


C (20) = 8000 – 45 × 400 + 12000


C (20) = 20000 – 18000


C (20) = Rs 2000


Hence the person should order 20 trees which will cost the minimum Rs 2000.


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

If the sum of theMathematics - Board Papers

A metal box with Mathematics - Board Papers

Show that aRD Sharma - Volume 1

Find the local maMathematics - Board Papers

Prove that the seMathematics - Board Papers

Prove that the raMathematics - Board Papers

Prove that the leMathematics - Board Papers