barrys bagel emporium sells a dozen bagels for $5.00. this price is no longer high enough to create a…

barrys bagel emporium sells a dozen bagels for $5.00. this price is no longer high enough to create a profit. the owner decides to raise the price. he does not want to alarm his customers with too large of an increase. he is considering four different plans.\n\nplan a: raise the price by $0.05 each week until the price reaches $8.00.\n\nplan b: raise the price by 10 percent each week until the price reaches $8.00.\n\nplan c: raise the price by the same amount each week for 6 weeks, so that in the sixth week the price is $8.00.\n\nplan d: raise the price by $0.25 each week until the price reaches $8.00.\n\nwhich plan will result in the price of the bagels reaching $8.00 fastest?\n\nplan a\n\nplan b\n\nplan c\n\nplan d

barrys bagel emporium sells a dozen bagels for $5.00. this price is no longer high enough to create a profit. the owner decides to raise the price. he does not want to alarm his customers with too large of an increase. he is considering four different plans.\n\nplan a: raise the price by $0.05 each week until the price reaches $8.00.\n\nplan b: raise the price by 10 percent each week until the price reaches $8.00.\n\nplan c: raise the price by the same amount each week for 6 weeks, so that in the sixth week the price is $8.00.\n\nplan d: raise the price by $0.25 each week until the price reaches $8.00.\n\nwhich plan will result in the price of the bagels reaching $8.00 fastest?\n\nplan a\n\nplan b\n\nplan c\n\nplan d

Answer

Answer:

B. plan B

Explanation:

Step1: Analyze Plan A

Initial price $P_0 = 5$, increase $\Delta P_A=0.05$. Let $n_A$ be the number of weeks. We solve $5 + 0.05n_A=8$. So $0.05n_A=8 - 5=3$, and $n_A=\frac{3}{0.05}=60$ weeks.

Step2: Analyze Plan B

Initial price $P_0 = 5$, growth factor $r = 1 + 0.1=1.1$. We solve $5\times(1.1)^n_B=8$. Then $(1.1)^n_B=\frac{8}{5} = 1.6$. Taking the logarithm of both sides, $n_B=\frac{\ln(1.6)}{\ln(1.1)}\approx \frac{0.47}{0.095}\approx 4.95$ weeks.

Step3: Analyze Plan C

Initial price $P_0 = 5$, in 6 weeks it reaches 8. Let the weekly - increase be $\Delta P_C$. Using the formula $P = P_0+\Delta P_C\times n$, $8 = 5+\Delta P_C\times6$. So $\Delta P_C=\frac{8 - 5}{6}=0.5$, and it takes 6 weeks.

Step4: Analyze Plan D

Initial price $P_0 = 5$, increase $\Delta P_D = 0.25$. Let $n_D$ be the number of weeks. We solve $5+0.25n_D=8$. So $0.25n_D=8 - 5 = 3$, and $n_D=\frac{3}{0.25}=12$ weeks.

Since $4.95<6<12<60$, Plan B reaches $8$ fastest.