To solve the given problem, we need to calculate the rate of change in the number of tickets sold between 10:00 a.m. and 1:00 p.m. using the given function [tex]\( T(h) \)[/tex].
First, let's understand the function [tex]\( T(h) \)[/tex]:
[tex]\[ T(h) = \frac{386 + 110h}{h} \][/tex]
### Step 1: Calculate [tex]\( T(h) \)[/tex] at 10:00 a.m. and 1:00 p.m.
#### 10:00 a.m.
7:00 a.m. is when the park opens, so 10:00 a.m. is 3 hours after the park opens:
[tex]\[ h_1 = 10 - 7 = 3 \][/tex]
Substitute [tex]\( h_1 = 3 \)[/tex] into the function:
[tex]\[ T(3) = \frac{386 + 110 \cdot 3}{3} \][/tex]
[tex]\[ T(3) = \frac{386 + 330}{3} \][/tex]
[tex]\[ T(3) = \frac{716}{3} \][/tex]
[tex]\[ T(3) \approx 238.67 \][/tex]
#### 1:00 p.m.
1:00 p.m. is 6 hours after the park opens:
[tex]\[ h_2 = 13 - 7 = 6 \][/tex]
Substitute [tex]\( h_2 = 6 \)[/tex] into the function:
[tex]\[ T(6) = \frac{386 + 110 \cdot 6}{6} \][/tex]
[tex]\[ T(6) = \frac{386 + 660}{6} \][/tex]
[tex]\[ T(6) = \frac{1046}{6} \][/tex]
[tex]\[ T(6) \approx 174.33 \][/tex]
### Step 2: Calculate the rate of change
The rate of change in the number of tickets sold between 10:00 a.m. and 1:00 p.m. is given by the difference in tickets sold at these times divided by the change in time. In other words:
[tex]\[ \text{Rate of Change} = \frac{T(h_2) - T(h_1)}{h_2 - h_1} \][/tex]
Substitute the calculated values:
[tex]\[ \text{Rate of Change} = \frac{174.33 - 238.67}{6 - 3} \][/tex]
[tex]\[ \text{Rate of Change} = \frac{-64.34}{3} \][/tex]
[tex]\[ \text{Rate of Change} \approx -21.44 \][/tex]
### Step 3: Select the correct answer
From the given choices, the closest answer to [tex]\(-21.44\)[/tex] tickets per hour is:
C. -21 tickets per hour
Therefore, the correct answer is:
[tex]\[ \boxed{-21} \][/tex] tickets per hour.
