Question

temperature ( t ) (in degrees celsius).

\begin{array}{c|ccc} t & 18 & 20 & 27 \\ hline v(t) & 10 & 18 & 46 end{array}

when does the number of visitors in the park increase faster?

choose 1 answer:

a between outside temperatures of 18 degrees celsius and 20 degrees celsius

b between outside temperatures of 20 degrees celsius and 27 degrees celsius

c the number of visitors increases at the same rate over both intervals

Explanation:

Step1: Calculate rate between 18 and 20

To find the rate of change (slope) between \( t = 18 \) and \( t = 20 \), use the formula for slope: \( \frac{\Delta V}{\Delta t}=\frac{V(20)-V(18)}{20 - 18} \). Substituting the values \( V(20)=18 \), \( V(18)=10 \), we get \( \frac{18 - 10}{20 - 18}=\frac{8}{2}=4 \).

Step2: Calculate rate between 20 and 27

For the interval \( t = 20 \) to \( t = 27 \), use the slope formula: \( \frac{\Delta V}{\Delta t}=\frac{V(27)-V(20)}{27 - 20} \). Substituting \( V(27)=46 \), \( V(20)=18 \), we get \( \frac{46 - 18}{27 - 20}=\frac{28}{7}=4 \)? Wait, no, wait: \( 46 - 18 = 28 \), \( 27 - 20 = 7 \), \( 28\div7 = 4 \)? Wait, no, that can't be. Wait, no, wait, I made a mistake. Wait, \( 46 - 18 = 28 \), \( 27 - 20 = 7 \), \( 28\div7 = 4 \)? Wait, but that's the same as the first interval? Wait, no, wait, let's recalculate. Wait, \( V(18)=10 \), \( V(20)=18 \): change in \( V \) is \( 18 - 10 = 8 \), change in \( t \) is \( 20 - 18 = 2 \), so \( 8/2 = 4 \). Then \( V(20)=18 \), \( V(27)=46 \): change in \( V \) is \( 46 - 18 = 28 \), change in \( t \) is \( 27 - 20 = 7 \), \( 28/7 = 4 \). Wait, so both rates are 4? But that would mean option C. But wait, maybe I miscalculated. Wait, no, \( 18 - 10 = 8 \), \( 20 - 18 = 2 \), 8/2=4. \( 46 - 18 = 28 \), \( 27 - 20 = 7 \), 28/7=4. So both intervals have a rate of 4. Wait, but that contradicts? Wait, no, let's check again. \( V(18)=10 \), \( V(20)=18 \): difference in V is 8, difference in t is 2, 8/2=4. \( V(20)=18 \), \( V(27)=46 \): difference in V is 28, difference in t is 7, 28/7=4. So both rates are equal. So the number of visitors increases at the same rate over both intervals.

Answer:

C. The number of visitors increases at the same rate over both intervals