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
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
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
To determine the minimum number of units that must be produced and sold daily to ensure that there is no loss to the company, we need to find the minimum point on the cost function.
Given that the cost function is 𝒄 = 𝟎. 𝟎𝟐𝟓𝒙𝟐 + 𝟏𝟑𝒙 + 𝟏𝟎𝟎 and each unit is sold for Ksh 20, we can express the revenue function as 𝑟 = 𝟐𝟎𝑥.
To find the minimum point, we need to find the value of x that makes the cost and revenue functions equal:
𝒄 = 𝑟
𝟎. 𝟎𝟐𝟓𝒙𝟐 + 𝟏𝟑𝒙 + 𝟏𝟎𝟎 = 𝟐𝟎𝑥
Now, let's solve for x.
𝟎. 𝟎𝟐𝟓𝒙𝟐 + 𝟏𝟑𝒙 + 𝟏𝟎𝟎 = 𝟐𝟎𝑥
𝟎. 𝟎𝟐𝟓𝒙𝟐 - 𝟐𝟎𝑥 + 𝟏𝟑𝒙 + 𝟏𝟎𝟎 = 𝟎
𝟎. 𝟎𝟐𝟓𝒙𝟐 + 𝟏𝟑𝒙 - 𝟐𝟎𝑥 + 𝟏𝟎𝟎 = 𝟎
𝟎. 𝟎𝟐𝟓𝒙𝟐 + (𝟏𝟑𝒙 - 𝟐𝟎𝑥) + 𝟏𝟎𝟎 = 𝟎
𝟑𝒙 - 𝟑𝟎𝑥 + 𝟏𝟎𝟎 = 𝟎
𝟑𝒙 - 𝟑𝟎𝑥 = -𝟏𝟎𝟎
𝒙(𝟑 - 𝟑𝟎) = -𝟏𝟎𝟎
𝒙(𝟐𝟗) = -𝟏𝟎𝟎
𝒙 = -𝟏𝟎𝟎 / 𝟐𝟗
Therefore, x = -10/29.
Since you cannot produce a negative number of units, we round up to the nearest whole number. Therefore, the minimum number of units that must be produced and sold daily to ensure that there is no loss to the company is 1 unit.