The total weekly cost (in dollars) incurred by Lincoln Records in pressing x compact discs is given by the following function.
C(x) = 2000 + 2x - 0.0001x^2 (0≤x≤6000)
(a) What is the actual cost incurred in producing the 1071st and the 1891st disc? (Round your answers to the nearest cent.)
1071st disc=$
1891st disc=$
(b) What is the marginal cost when x = 1070 and 1890? (Round your answers to the nearest cent.)
1070=$
1890=$
a) Plug in x = 1071, and x = 1891 into the equation C(x) to find the actual cost at these values.
b) The marginal cost is the cost of producing one more extra good, so the marginal cost when x = 1070 is
C(x=1071) - C(x=1070)
for 1890 it is
C(x=1891) - C(x=1890)
Im not sure if this is right.
To find the actual cost incurred in producing the 1071st and 1891st discs, we need to substitute the respective values for x into the function C(x) and round to the nearest cent.
(a) For the 1071st disc, x = 1071:
C(1071) = 2000 + 2(1071) - 0.0001(1071)^2
C(1071) = 2000 + 2142 - 0.0001(1146241)
C(1071) = 2000 + 2142 - 114.6241
C(1071) = 4142 - 114.6241
C(1071) ≈ $4027.38
The actual cost incurred in producing the 1071st disc is approximately $4027.38.
For the 1891st disc, x = 1891:
C(1891) = 2000 + 2(1891) - 0.0001(1891)^2
C(1891) = 2000 + 3782 - 0.0001(3575881)
C(1891) = 2000 + 3782 - 357.5881
C(1891) = 5782 - 357.5881
C(1891) ≈ $5424.41
The actual cost incurred in producing the 1891st disc is approximately $5424.41.
(b) The marginal cost represents the rate of change of the cost function with respect to x. To find the marginal cost at x = 1070 and x = 1890, we need to find the derivative of the cost function, which gives us the rate of change.
C(x) = 2000 + 2x - 0.0001x^2
Differentiate the function C(x) with respect to x:
C'(x) = 2 - 0.0002x
For x = 1070:
C'(1070) = 2 - 0.0002(1070)
C'(1070) = 2 - 0.214
C'(1070) ≈ $1.786
The marginal cost when x = 1070 is approximately $1.786.
For x = 1890:
C'(1890) = 2 - 0.0002(1890)
C'(1890) = 2 - 0.378
C'(1890) ≈ $1.622
The marginal cost when x = 1890 is approximately $1.622.
To determine the cost incurred for producing the 1071st and the 1891st disc, we need to substitute the given x-values into the cost function C(x).
(a)
To find the cost incurred for the 1071st disc:
C(1071) = 2000 + 2(1071) - 0.0001(1071)^2
To calculate this, you can use a calculator or follow these steps:
1. Calculate the square of 1071:
1071^2 = 1,146,241
2. Multiply the squared value by 0.0001:
0.0001 * 1,146,241 = 114.6241
3. Multiply 2 by 1071:
2 * 1071 = 2142
4. Add up all the values:
2000 + 2142 - 114.6241 = 4027.3759
Therefore, the cost for producing the 1071st disc is approximately $4027.38 when rounded to the nearest cent.
Now let's do the same for the 1891st disc:
C(1891) = 2000 + 2(1891) - 0.0001(1891)^2
1. Calculate the square of 1891:
1891^2 = 3,573,881
2. Multiply the squared value by 0.0001:
0.0001 * 3,573,881 = 357.3881
3. Multiply 2 by 1891:
2 * 1891 = 3782
4. Add up all the values:
2000 + 3782 - 357.3881 = 4424.6119
Therefore, the cost for producing the 1891st disc is approximately $4424.61 when rounded to the nearest cent.
(b)
To find the marginal cost, we need to calculate the derivative of the cost function C(x) with respect to x and then substitute the given x-values into the derivative function.
First, let's find the derivative of C(x) by differentiating each term separately:
C'(x) = 2 - 0.0001(2x) = 2 - 0.0002x
To calculate the marginal cost for x = 1070:
C'(1070) = 2 - 0.0002(1070)
1. Multiply 0.0002 by 1070:
0.0002 * 1070 = 0.214
2. Subtract the result from 2:
2 - 0.214 = 1.786
Therefore, the marginal cost when x = 1070 is approximately $1.79 when rounded to the nearest cent.
Now for x = 1890:
C'(1890) = 2 - 0.0002(1890)
1. Multiply 0.0002 by 1890:
0.0002 * 1890 = 0.378
2. Subtract the result from 2:
2 - 0.378 = 1.622
Therefore, the marginal cost when x = 1890 is approximately $1.62 when rounded to the nearest cent.