Answer:
(a) 11.75%
(b) Profit decreases by $5.88 per calculator.
Step-by-step explanation:
(a) The percentage of failures with time is given by the following expression:
[tex]f(x) = 0.125*e^{-0.125x}[/tex]
Integrating this function from x = 0 to x = 1 year, gives us the percentage of failures in the first year:
[tex]\int\limits^1_0 {f(x)} \, dx = F(x)=-e^{-0.125x}|_0^1\\F(1) = -e^{-0.125*1}-(-e^{-0.125*0}) = 1-e^{-0.125}\\F(1) =0.1175[/tex]
11.75% of the calculators will fail within the warranty period.
(b) If the cost of a calculator is $50, and the profit per sale is $25, the average revenue per calculator is $75. Considering no income in failed calculators, the new cost per calculator is:
[tex]C =\$50*(1+0.1175)\\C=\$55.88[/tex]
The effect of warranty replacement on profit is given by the difference in cost per calculator:
[tex]\Delta P= \$50-\$55.88=-\$5.88[/tex]
Profit decreases by $5.88 per calculator.
The percentage that fails within the warranty period is 11.75%, and the effect of warranty replacement on profit is a decrease in profit
The probability distribution function is given as:
[tex]f(x) = 0.125e^{-0.125x}[/tex], x > 0
The warranty is one year.
So, the percentage of the calculators that will fail within the warranty period must fall within 0 to 1 year.
This is represented as:
[tex]F(x) = \int\limits^1_0 {f(x)} \, dx[/tex]
This gives
[tex]F(x) = \int\limits^1_0 {0.125e^{-0.125x}} \, dx[/tex]
Factor out 0.125
[tex]F(x) = 0.125 \int\limits^1_0 {e^{-0.125x}} \, dx[/tex]
Integrate
[tex]F(x) = \frac{0.125}{0.125}* [-{e^{-0.125x}]|\limits^1_0[/tex]
This gives
[tex]F(x) = [-{e^{-0.125x}]|\limits^1_0[/tex]
Expand
[tex]F(x) = -e^{-0.125(1)} +e^{-0.125(0)[/tex]
[tex]F(x) = -e^{-0.125(1)} +1[/tex]
Solve the expression
[tex]F(x) = 0.1175[/tex]
Express as percentage
[tex]F(x) = 11.75\%[/tex]
The manufacturing cost of a calculator is $50, and the profit per sale is $25.
So, we have the new cost (C) to be:
[tex]C = 50 + 50 * 11.75\%[/tex]
[tex]C = 55.875[/tex]
Calculate the change in cost
[tex]\Delta C = 50 - 55.875[/tex]
[tex]\Delta C = - 5.875[/tex]
-5.875 is less than 0.
Hence, the effect of warranty replacement on profit is a decrease in profit
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