Questions: The average price of a ticket to a baseball game can be approximated by p(x)=0.03 x^2+0.59 x+5.49, where x is the number of years after 1991 and p(x) is in dollars. a) Find p(3). b) Find p(13). c) Find p(13)-p(3). d) Find (p(13)-p(3))/(13-3), and interpret this result. a) p(3)= (Simplify your answer.)

The average price of a ticket to a baseball game can be approximated by p(x)=0.03 x^2+0.59 x+5.49, where x is the number of years after 1991 and p(x) is in dollars.
a) Find p(3).
b) Find p(13).
c) Find p(13)-p(3).
d) Find (p(13)-p(3))/(13-3), and interpret this result.
a) p(3)= (Simplify your answer.)

Transcript text: The average price of a ticket to a baseball game can be approximated by $p(x)=0.03 x^{2}+0.59 x+5.49$, where $x$ is the number of years after 1991 and $p(x)$ is in dollars. a) Find $p(3)$. b) Find $p(13)$. c) Find $p(13)-p(3)$. d) Find $\frac{p(13)-p(3)}{13-3}$, and interpret this result. a) $p(3)=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

Step 1: Evaluate the quadratic function at specific points

To find $p(x_1)$ and $p(x_2)$, we substitute $x_1 = 3$ and $x_2 = 13$ into the quadratic function $p(x) = 0.03x^2 + 0.59x + 5.49$. Thus, $p(x_1) = 0.03(3)^2 + 0.59(3) + 5.49 = 7.53$ and $p(x_2) = 0.03(13)^2 + 0.59(13) + 5.49 = 18.23$.

Step 2: Find the difference in function values

The difference $p(x_2) - p(x_1)$ is calculated as $18.23 - 7.53 = 10.7$.

Step 3: Calculate the average rate of change

The average rate of change over the interval from $x_1$ to $x_2$ is $\dfrac{p(x_2) - p(x_1)}{x_2 - x_1} = \dfrac{10.7}{10} = 1.07$.

Interpretation

From 1994 to 2004, the average price for a ticket increased on average by 1.07 per year.

Final Answer:

The function value at $x_1$ is 7.53, at $x_2$ is 18.23, the difference in function values is 10.7, and the average rate of change over the interval is 1.07 per unit.

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