no loss means you want a positive profit
profit = revenue - cost
so you want to solve
20x - (0.025x^2 + 13x + 100) > 0
The roots of this equation are
x = 140±20√39
you know from the shape of the parabola that it is positive between the roots. So,
15.1 < x < 264.9
a) A manufacturing firm finds that the daily costs of producing x units of a product is given
by:
𝒄 = 𝟎. 𝟎𝟐𝟓𝒙𝟐 + 𝟏𝟑𝒙 + 𝟏𝟎𝟎
a. If each of the units is sold for Ksh 20, determine the minimum numbers that
must be produced and sold daily to ensure that there is no loss to the company
2 answers
Given cost function
C= 0.025x^2 +13x+100
Each unit is sold for ksh 20.
Hence revenue function , R= 20x
Where, X is the no. Of products
A) to ensure there is no loss, the cost function should be equal to revenue function.
That is 0.025x^2+ 13x + 100 = 20x
0.025x^2 + 13x -20x +100 =0
0.025x^2 - 7x +100= 0
Dividing the equation by 0.025
x^2 -(7/0.025) x + 100 /0.025 = 0
x^2 - 280x + 4000 = 0
By solving using the formula
x = (-b +- √(b^2 - 4ac))/2a
Where a= , b= -280 , c= 4000
we get x= 265 , 15
To ensure no loss 265 is selected.
Hence minimum 265 unit should be produced to ensure there is no loss occur.
B) selling price is incresed by 45%
Hence new selling price = 20+ 20*45/100
= 20+ 9 = 29
Hence revenue function = 29x
To ensure no loss = . 025x^2 + 13x + 100 = 29x
Tht is 0.025x^2 +13x +100-29x =0
0.025x^2 -16x + 1000=0
By dividing throughout by 0.025
x^2 - 640x + 4000= 0 by solving
With a=1 , b= -640 and c= 4000
We get x = 634 , 6.
To ensure no loss 634 is selected.
Hence to ensure no loss production level should be 634 products
C= 0.025x^2 +13x+100
Each unit is sold for ksh 20.
Hence revenue function , R= 20x
Where, X is the no. Of products
A) to ensure there is no loss, the cost function should be equal to revenue function.
That is 0.025x^2+ 13x + 100 = 20x
0.025x^2 + 13x -20x +100 =0
0.025x^2 - 7x +100= 0
Dividing the equation by 0.025
x^2 -(7/0.025) x + 100 /0.025 = 0
x^2 - 280x + 4000 = 0
By solving using the formula
x = (-b +- √(b^2 - 4ac))/2a
Where a= , b= -280 , c= 4000
we get x= 265 , 15
To ensure no loss 265 is selected.
Hence minimum 265 unit should be produced to ensure there is no loss occur.
B) selling price is incresed by 45%
Hence new selling price = 20+ 20*45/100
= 20+ 9 = 29
Hence revenue function = 29x
To ensure no loss = . 025x^2 + 13x + 100 = 29x
Tht is 0.025x^2 +13x +100-29x =0
0.025x^2 -16x + 1000=0
By dividing throughout by 0.025
x^2 - 640x + 4000= 0 by solving
With a=1 , b= -640 and c= 4000
We get x = 634 , 6.
To ensure no loss 634 is selected.
Hence to ensure no loss production level should be 634 products