Pizza Place Promotion

Task

In order to gain popularity among students, a new pizza place near school plans to offer a special promotion. The cost of a large pizza (in dollars) at the pizza place as a function of time (measured in days since February 10th) may be described as

(Assume $t$ only takes whole number values.)

1. If you want to give their pizza a try, on what date(s) should you buy a large pizza in order to get the best price?

2. How much will a large pizza cost on Feb. 18th?

3. On what date, if any, will a large pizza cost 13 dollars?

4. Write an expression that describes the sentence "The cost of a large pizza is at least $A$ dollars $B$ days into the promotion," using function notation and mathematical symbols only.

5. Calculate $C(9) - C(8)$ and interpret its meaning in the context of the problem.

6. On average, the cost of a large pizza goes up about 85 cents per day during the first two weeks of the promotion period. Which of the following equations best describes this statement?

   • $\frac{C(13)+C(0)}{2}=0.85$
   • $\frac{C(13)-C(0)}{13}=0.85$
   • $\frac{C(13)}{13}=0.85$
   • $\frac{C(\text{Feb.23})-C(\text{Feb.10})}{13}=0.85$

Solution

1. Based on the function above, the lowest price that the promotion offers for a large pizza is 9 dollars. This is the cost of the pizza when $t=0$, $t=1$, $t=2$, and $t=3$. We know that $t$ denotes the number of days since February 10^th. Thus, $t=0$ corresponds to February 10^th, $t=1$ corresponds to February 11^th, $t=2$ corresponds to February 12^th, and $t=3$ corresponds to February 13^th. Therefore, the best days to give the new pizza place a try in order to get the best price are February 10, February 11, February 12 and February 13.

2. February 18^th is eight days after February 10^th, corresponding to $t=8$. Based on the function above, the cost of a large pizza in dollars is given by $C(t)=9+t$ when $3 \lt t \leq 8$. Then, when $t=8$, $C(t)=9+8=17$ dollars. Thus, the cost of a large pizza on February 18th is 17 dollars.

3. We know that $13 \neq 9$ which implies that a large pizza cannot cost 13 dollars when $0 \leq t \leq 3$ because for these values of $t$, $C(t) = 9$ based on the function above. Similarly, we know that $13 \neq 20$, which implies that a large pizza cannot cost 13 dollars when $8 \lt t \lt 28$ because $C(t)=20$ for these values. Thus, we know that the only time a large pizza could cost 13 dollars is when $t$ is in the interval $3 \lt t \leq 8$, for which $C(t)=9+t$. In order to find out which date a large pizza will cost 13 dollars we must plug 13 into this equation as our cost and solve for $t$:

   $$ \begin{align} 13 &= 9 + t \\ t &= 13 - 9 \\ t &= 4 \end{align} $$

   This means that a large pizza costs 13 dollars when $t=4$. We know that $t$ denotes the days since February 10^th so $t=4$ corresponds to February 14^th. Thus, a large pizza will cost 13 dollars on February 14^th.

4. The statement that a pizza is at least $A$ dollars $B$ days into the promotion means that the cost of a large pizza $B$ days into the promotion, denoted $C(B)$, is greater than or equal to $A$ dollars. Thus, an expression that describes this sentence using function notation and mathematical symbols is simply

   $$ C(B) \geq A $$

5. $C(9)=20$ and $C(8)=9+8=17$ based on the equation above. Thus, $ C(9)-C(8)=20-17=3$. $C(9)=20$ corresponds to the cost of a large pizza 9 days after February 10^th, or February 19^th. $C(8)=17$ corresponds to the cost of a large pizza 8 days after February 10^th, or February 18^th. Thus, the meaning of $C(9)-C(8)=3$ in the context of the problem is that on February 19^th, a large pizza will be 3 dollars more expensive than a large pizza on February 18^th.

6. The first two weeks of the promotion take place from February 10^th, when $t=0$, to February 23^rd, when $t=13$. The average rate of change is given by:

   $$ \frac{C(13)-C(0)}{13-0} = \frac{20-9}{13} \approx 0.85 $$

   The expression above shows that the cost of a large pizza goes up about 0.85 dollars, or 85 cents, per day during the first two weeks of the promotion period. Therefore, the expression that best describes this statement is:

$$ \frac{C(13)-C(0)}{13} $$