three numbers are given whose sum is 180 and the first of these number is twice the second number.if the product of the number is greatest, find the numbers.
Answers
Out of three given numbers, the first number is twice the second and thrice the third. If the average of these three numbers is 154, then what is the difference between the first and the third numbers?
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Let the numbers be a,b and c
ATQ
a=2b……….(I)
a=3c……….(ii)
a+b+c=462 (avg is 154)
a+(a/2)+(a/3)=462 ( from I and ii )
11 a = 462 * 6
a= 252
So, b = 126 and c= 84
Therefore, a-c= 252–84 =168.
Answer
The three numbers are 40, 80 and 60
Step-by-step explanation:
Let the numbers be x, y and z.
Thus, x + y + z = 180 (given)----->1
Also, 2x = y (given)----->2
Thus the product becomes : -
= xyz
= x(2x)(180 - 3x) ( using equation 1 and 2)
= 360x^2 - 6x^3 = P (let)
Thus for P being maximum : -
dP/dx = 0
=> 720x - 18x^2 = 0
=> x(720 - 18x) = 0
Thus x = 0 or x = 720/18 = 40
But for maxima d^2P/dx^2 < 0
d^2P/d^2 = 720 - 36x
Thus when x = 0 , d^2P/d^2 = 720
And when x = 40, d^2P/d^2 = -720
Thus d^2P/d^2 being less than zero is valid only when x = 40.
Thus y = 80 and z = 60
( using equation 1 and 2)
Thus three numbers are : 40, 80 and 60
Hope that it is clear to you all.
Best of luck !!!