course MTH 271 12/15, 1500 assignment #026026. `query 26
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14:37:19 **** Query 3.5.12 find the price per unit p for maximum profit P if C = 35x+500, p=50-.1`sqrt(x) **** What price per unit produces the maximum profit?
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RESPONSE --> P=R-C R=xp C=35x+500 P=x(50 -.1sqrtx) - 35x - 500 P=50x - .1 x^(3/2) -35x -500 P=15x -.1x^(3/2) - 500 P'=15 - .15x^(1/2) 0=15 -.15x^(1/2) -15=-.15x^(1/2) 100=x^(1/2) 10,000=x p=50- .1 sqrt(10,000) p=50- .1(100) p=40 confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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14:37:26 Revenus is price * number sold: R = xp. Since p = 50 - .1 sqrt(x) we have R = x(50 - .1 `sqrt (x)) = 50x - .1x^(3/2) Price is revenue - cost: P = R - C = 50x - .1 x^(3/2) - 35x - 500. Simplifying: P = 15x - .1x^(3/2) - 500 Derivative of profit P is P ' = 15 -.15 x^(1/2). Derivative is zero when 15 - .15 x^(1/2) = 0; solving we get x = 10,000. 2d derivative is .075 x^-(1/2), which is negative, implying that x = 10000 gives a max. When x = 10,000 we get price p = 50 - .1 sqrt(x) = 50 - .1 * sqrt(10,000) = 40. Price is $40. **
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RESPONSE --> ------------------------------------------------ Self-critique Rating:ent: 3
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15:03:05 According to my note here amount deposited A is proportional to the square of interest rate r so A = k r^2 for some proportionality constant k. The interest paid at rate r on amount A is A * r. The bank can reinvest at 12% so it gets return A * .12. The bank therefore nets .12 * A - r * A = (.12 - r) * A. Since A = k r^2 the bank nets profit P = (.12 - r) * (k r^2) = k * (.12 r^2 - r^3). We maximize this expression with respect to r: dP/dr = k * (.24 r - 3 r^2). dP/dr = 0 when .24 r - 3 r^2 = 0, when 3 r ( .08 - r) = 0, i.e., when r = 0 or r = .08. The second derivative is -6 r + .24, which is negative for r > .06. This shows that the critical point at r = .08 is a maximum. The max profit is thus P = (.12 * .08 - .08^3) * k = (.096 - .0016) k = .080 k. In order to find the optimal interest rate it is not necessary to find the proportionality constant k. However if the proportionality constant was known we could find the max profit. **
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RESPONSE --> For some reason this problem completely baffled me. I was trying to use the equations we used before involving interest, but they were inadequate b/c they relied on knowing the type of interest compound/simple, time of investment, etc... After going over the explanation, the process becomes a little more clear, but remains still a little mysterious as business/economical math is far from my strong suit. ------------------------------------------------ Self-critique Rating:ent: 2
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