the total cost c of a firm is partly constant and partly varies as the square of the quantity x of goods produced if 20 units of goods cost#30,000 and 45 units cost #60,000. find the cost of producing 120 units
c = a + b x^2
30,000 = a + b (400)
60,000 = a + b (2025)
------------------------------- subtract
-30,000 = - 1625 b
b = 18.4615
30,000 = a + 18.4615 (400)
a = 30,000 - 7,384.62 = 22,615.38
so
c = 22,615.38 + 18.4615 (120)^2
= 288,461.
Help me solve the question
To solve this problem, we can break down the cost into its constant and variable components. Let's denote the constant cost as "a" and the variable cost as "bx^2", where "b" is the coefficient of the variable cost.
1. From the given information, we can set up two equations using the cost and quantity values provided:
- When 20 units of goods are produced, the cost is #30,000:
c = a + b(20^2) -> Equation 1
- When 45 units of goods are produced, the cost is #60,000:
c = a + b(45^2) -> Equation 2
2. To find the values of "a" and "b", we can solve these two equations simultaneously. Subtracting Equation 1 from Equation 2 eliminates the constant cost "a":
a + b(45^2) - a - b(20^2) = #60,000 - #30,000
b(45^2 - 20^2) = #30,000
3. Simplifying the equation gives us:
2025b - 400b = #30,000
1625b = #30,000
4. Solving for "b":
b = #30,000 / 1625
b ≈ #18.4615
5. Now, we can substitute the value of "b" back into Equation 1 (or Equation 2) to find the value of "a". Let's use Equation 1:
#30,000 = a + (#18.4615)(20^2)
#30,000 = a + 7392.6
a = #30,000 - 7392.6
a ≈ #22,607.4
6. Finally, we can find the cost of producing 120 units by substituting the values of "a" and "b" into the cost equation:
c = #22,607.4 + (#18.4615)(120^2)
c ≈ #22,607.4 + 266,153.8
c ≈ #288,761.2
Therefore, the cost of producing 120 units is approximately #288,761.2.
To find the cost of producing 120 units, we need to determine the constant cost and the variable cost component of the total cost.
First, let's consider the given information. We have two sets of data:
1. When 20 units of goods are produced, the cost is #30,000.
2. When 45 units of goods are produced, the cost is #60,000.
Let's assign variables to these values:
- For the quantity produced (x), we have:
- x₁ = 20 units
- x₂ = 45 units
- For the corresponding costs (c), we have:
- c₁ = #30,000
- c₂ = #60,000
Now we can use this information to set up two equations.
Equation 1: c = a + bx²
Here, 'a' represents the constant cost and 'b' represents the variable cost component.
Using the first set of data (x₁, c₁):
c₁ = a + bx₁²
Substituting the values:
#30,000 = a + b(20)²
#30,000 = a + 400b
Equation 1a: a = #30,000 - 400b
Using the second set of data (x₂, c₂):
c₂ = a + bx₂²
Substituting the values:
#60,000 = a + b(45)²
#60,000 = a + 2025b
Equation 1b: a = #60,000 - 2025b
Now we can equate the two expressions for 'a':
#30,000 - 400b = #60,000 - 2025b
Simplifying the equation:
-400b + 2025b = #60,000 - #30,000
1625b = #30,000
Solving for 'b':
b = #30,000 / 1625
b ≈ #18.46
Now that we have the value of 'b', we can substitute it back into either Equation 1a or Equation 1b to find 'a'.
Using Equation 1b:
a = #60,000 - 2025b
a = #60,000 - 2025(#18.46)
a ≈ #60,000 - #37,357.50
a ≈ #22,642.50
So, the equation for the total cost is: c = #22,642.50 + #18.46x²
To find the cost of producing 120 units (c₃):
c₃ = #22,642.50 + #18.46(120)²
c₃ ≈ #22,642.50 + #18.46(14,400)
c₃ ≈ #22,642.50 + #265,824
c₃ ≈ #288,466.50
Therefore, the cost of producing 120 units is approximately #288,466.50.