a) f(x) = 4.1x +92.16
b) f(x) = 17.9x + 19.36
c) f(x) = 8.8x + 55.4
ANSWERS: a) 4.8 b) 0.93 c) 2.2, i just have no clue how they got these answers
EXTRA INFO:
The demand function for a new product is p(x)= - 4x +42.5, where x is the quantity sold in thousands and p is the price in dollars. the company that manufactures the product is planning to buy a new machine for the plant. there are 3 diff. types of machines. the cost function for each machine is shown (ABOVE). Investigate the break even quantities for each machine.
typo: f(x) = 4.1x + 92.16, not 4.1x^2
If the demand is met, the total revenue is price per unit * number of units
for machine (a)
revenue = p(x)*x = 42.5x - 4x^2
cost = f(x)*x = 4.1x^2 + 92.16
breakeven is where cost = revenue
42.5x - 4x^2 = 4.1x + 92.16
4x^2 -38.4x + 92.16 = 0
x^2 - 9.6x + 23.04 = 0
x = 4.8
So, for unit (a) the breakeven occurs at 4800 units
Go forth and do likewise for (b) and (c)
Well, let's break it down!
To find the break-even quantity for each machine, we need to determine at what quantity the cost function equals the demand function. In other words, we want to find the value of x that satisfies the equation f(x) = p(x).
For machine a:
f(x) = 4.1x + 92.16
p(x) = -4x + 42.5
Setting them equal to each other, we get:
4.1x + 92.16 = -4x + 42.5
Now, let's solve for x. I'll leave that math to you.
For machine b and c, you can follow the same steps:
For machine b:
f(x) = 17.9x + 19.36
p(x) = -4x + 42.5
For machine c:
f(x) = 8.8x + 55.4
p(x) = -4x + 42.5
Solving these equations will give you the break-even quantities for each machine.
Remember, finding the break-even point is important if you don't want to be left juggling your finances! Good luck!
To calculate the break-even quantity for each machine, we need to find the value of x where the cost function equals the demand function.
Let's start with machine a):
a) f(x) = 4.1x + 92.16
The cost function for machine a is given by f(x).
The demand function is p(x) = -4x + 42.5, where p(x) represents the price.
To find the break-even quantity, we set the cost function equal to the demand function:
4.1x + 92.16 = -4x + 42.5
To solve for x, we can start by isolating the x variable:
4.1x + 4x = 42.5 - 92.16
8.1x = -49.66
Divide both sides of the equation by 8.1:
x = -49.66 / 8.1
By calculating this, we find that x ≈ -6.13
However, since the quantity should be a positive value, we can conclude that it is not possible to have a break-even point for machine a (x cannot be negative in this context).
Therefore, there is no break-even quantity for machine a (f(x) = 4.8 is not correct).
Now, let's move to machine b):
b) f(x) = 17.9x + 19.36
The cost function for machine b is given by f(x).
Using the demand function p(x) = -4x + 42.5, we can set up the equation:
17.9x + 19.36 = -4x + 42.5
Now, let's isolate the x variable:
17.9x + 4x = 42.5 - 19.36
21.9x = 23.14
Divide both sides of the equation by 21.9:
x = 23.14 / 21.9
By calculating this, we find that x ≈ 1.06
Therefore, the break-even quantity for machine b is approximately 1.06 (f(x) = 0.93 is correct).
Finally, let's calculate the break-even quantity for machine c:
c) f(x) = 8.8x + 55.4
The cost function for machine c is given by f(x).
Using the demand function p(x) = -4x + 42.5, we can set up the equation:
8.8x + 55.4 = -4x + 42.5
Isolate the x variable:
8.8x + 4x = 42.5 - 55.4
12.8x = -12.9
Divide both sides of the equation by 12.8:
x = -12.9 / 12.8
By calculating this, we find that x ≈ -1.01
Since the quantity sold cannot be negative, we can conclude that it is not possible to have a break-even point for machine c (x cannot be negative in this context).
Therefore, there is no break-even quantity for machine c (f(x) = 2.2 is not correct).
To summarize, the break-even quantities for the machines are:
a) There is no break-even quantity for machine a.
b) The break-even quantity for machine b is approximately 1.06.
c) There is no break-even quantity for machine c.
To find the break-even quantity for each machine, we need to determine at what quantity the total revenue (TR) equals the total cost (TC). The break-even quantity is the point where there is no profit or loss, which means the TR and TC are equal.
The total revenue (TR) of a product can be calculated by multiplying the quantity sold (x) by the price (p(x)). In this case, the price function is given as p(x) = -4x + 42.5. So, the total revenue function (TR(x)) is: TR(x) = (x) * (-4x + 42.5).
To find the cost function for each machine, we can use the given cost equations a), b), and c).
a) The cost function for machine a is f(x) = 4.1x + 92.16.
b) The cost function for machine b is f(x) = 17.9x + 19.36.
c) The cost function for machine c is f(x) = 8.8x + 55.4.
Now, let's set up the equation for each break-even scenario:
a) TR(x) = f(x) -> (x) * (-4x + 42.5) = 4.1x + 92.16
b) TR(x) = f(x) -> (x) * (-4x + 42.5) = 17.9x + 19.36
c) TR(x) = f(x) -> (x) * (-4x + 42.5) = 8.8x + 55.4
To solve each equation:
a) Simplify the equation and solve for x.
b) Simplify the equation and solve for x.
c) Simplify the equation and solve for x.
Once you substitute the found x values back into the total revenue function (TR(x)), you will get the break-even quantities for each machine.