In planning for a new item, a manufacturer assumes that the number of items produced x and the cost in dollars C of producing these items are related by a linear equation.
Projections are that 100 items will cost $10,000 to produce and that 300 items will cost $22,000 to produce. Find the equation that relates C and x.
Show work.
I will be happy to critique your work on this.
10000=100 a+b
22000=300 a+b
To find the equation that relates the cost of producing the items and the number of items produced, we can set up a system of equations using the given information.
Let's assume that the equation relating the number of items produced (x) and the cost in dollars (C) is of the form C = ax + b, where a and b are constants that we need to determine.
Given that 100 items cost $10,000 and 300 items cost $22,000, we can write the following equations:
10,000 = 100a + b
22,000 = 300a + b
Let's solve this system of equations step by step.
First, let's multiply the first equation by 3 to eliminate the variable b:
3(10,000) = 3(100a + b)
30,000 = 300a + 3b
Now, let's subtract the second equation from this new equation to eliminate the variable b:
30,000 - 22,000 = (300a + 3b) - (300a + b)
8,000 = 2b
Divide both sides by 2 to solve for b:
8,000/2 = 2b/2
b = 4,000
Now, substitute the value of b into the first equation:
10,000 = 100a + 4,000
Subtract 4,000 from both sides:
10,000 - 4,000 = 100a
6,000 = 100a
Divide both sides by 100 to solve for a:
6,000/100 = 100a/100
60 = a
Now, we have found the values of a and b. The equation that relates the cost in dollars C and the number of items produced x is:
C = 60x + 4,000
So, the equation is C = 60x + 4,000.
You can verify this equation by substituting the values of x and checking if the cost C matches the given information.