marginal cost a division of ditton industries manufactures the futura model microwave oven. the daily cost…
marginal cost a division of ditton industries manufactures the futura model microwave oven. the daily cost (in dollars) of producing these microwave ovens given by the following function, where x stands for the number of units produced. c(x)=0.0002x³ - 0.06x² + 120x + 5000 (a) what is the actual cost incurred in manufacturing the 111st oven? the 191st oven? the 321st oven? (round your answers to the nearest cent.)
Answer
Explanation:
Step1: Calculate cost for (n) units
The cost function is (C(x)=0.0002x^{3}-0.06x^{2}+120x + 5000).
Step2: Calculate cost for 111 units
Substitute (x = 111) into (C(x)): [ \begin{align*} C(111)&=0.0002\times(111)^{3}-0.06\times(111)^{2}+120\times111 + 5000\ &=0.0002\times1367631-0.06\times12321+13320 + 5000\ &=273.5262-739.26+13320+5000\ &=17854.2662\approx17854.27 \end{align*} ]
Step3: Calculate cost for 110 units
Substitute (x = 110) into (C(x)): [ \begin{align*} C(110)&=0.0002\times(110)^{3}-0.06\times(110)^{2}+120\times110 + 5000\ &=0.0002\times1331000-0.06\times12100+13200+5000\ &=266.2 - 726+13200+5000\ &=17740.2 \end{align*} ]
Step4: Calculate cost of 111th unit
The cost of the 111th unit is (C(111)-C(110)=17854.27 - 17740.2=114.07)
Step5: Calculate cost for 191 units
Substitute (x = 191) into (C(x)): [ \begin{align*} C(191)&=0.0002\times(191)^{3}-0.06\times(191)^{2}+120\times191+5000\ &=0.0002\times6967871-0.06\times36481+22920+5000\ &=1393.5742-2188.86+22920+5000\ &=27124.7142\approx27124.71 \end{align*} ]
Step6: Calculate cost for 190 units
Substitute (x = 190) into (C(x)): [ \begin{align*} C(190)&=0.0002\times(190)^{3}-0.06\times(190)^{2}+120\times190+5000\ &=0.0002\times6859000-0.06\times36100+22800+5000\ &=1371.8-2166+22800+5000\ &=27005.8 \end{align*} ]
Step7: Calculate cost of 191st unit
The cost of the 191st unit is (C(191)-C(190)=27124.71 - 27005.8 = 118.91)
Step8: Calculate cost for 321 units
Substitute (x = 321) into (C(x)): [ \begin{align*} C(321)&=0.0002\times(321)^{3}-0.06\times(321)^{2}+120\times321+5000\ &=0.0002\times32977661-0.06\times103041+38520+5000\ &=6595.5322-6182.46+38520+5000\ &=43933.0722\approx43933.07 \end{align*} ]
Step9: Calculate cost for 320 units
Substitute (x = 320) into (C(x)): [ \begin{align*} C(320)&=0.0002\times(320)^{3}-0.06\times(320)^{2}+120\times320+5000\ &=0.0002\times32768000-0.06\times102400+38400+5000\ &=6553.6-6144+38400+5000\ &=43809.6 \end{align*} ]
Step10: Calculate cost of 321st unit
The cost of the 321st unit is (C(321)-C(320)=43933.07 - 43809.6=123.47)
Answer:
111st unit: $$114.07$ 191st unit: $$118.91$ 321st unit: $$123.47$