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.
Model it with this equation:
cost= K1* n + K2
where K2, K1 are constants, and n is the number of items.
Now plug the two equations in (for n=100 and n=300) and solve for K1 and K2.
To find the equation that relates the cost (C) and the number of items produced (x), we can use the given model equation:
cost = K1 * n + K2
Let's begin by plugging in the first set of values for n=100 and C=10,000:
10,000 = K1 * 100 + K2
Similarly, we plug in the second set of values for n=300 and C=22,000:
22,000 = K1 * 300 + K2
Now, we have a system of two linear equations with two unknowns, K1 and K2. We can solve this system of equations to find the values of K1 and K2.
To solve the system, we can use any method, such as substitution or elimination. Here, we'll use the elimination method.
First, let's multiply the first equation by -3 to eliminate K1:
-3 * (10,000 = K1 * 100 + K2)
-30,000 = -300K1 - 3K2
Now, we subtract this modified equation from the second equation:
22,000 - (-30,000) = K1 * 300 + K2 - (-300K1 - 3K2)
52,000 = K1 * 300 + 300K1 - 4K2
Combining like terms:
52,000 = 300K1 + 300K1 - 4K2
52,000 = 600K1 - 4K2
Now we can rewrite this equation as:
600K1 - 4K2 = 52,000
Great! We now have a new equation (equation 3) derived from the given information.
The equation relating C and x is:
cost = 600K1 * n - 4K2
So, the equation relating the cost and the number of items produced is cost = 600K1 * n - 4K2, where K1 and K2 are constants that can be found by solving the system of equations.