At different times on a certain day, a manager at a theme park recorded the number of visitors. An exponential model gives the estimated number of visitors, V, at the theme park t hours after the theme park opened, where t ≤ 3. Based on the model, 80 visitors were estimated to be at the theme park when it opened, and every 30 minutes the estimated number of visitors increased by 150% of the estimated number of visitors 30 minutes earlier. Which of the following equations best represents this model?

At different times on a certain day, a manager at a theme park recorded the number of... | SAT Exponential functions Practice

Answer: A. $V = 80(2.50)^{2t}$

Explanation:

$1$ . Identify the components of the exponential model $V = A \cdot B^k$ :

- A = Initial value.

- B = Growth factor.

- k = Number of growth periods.

$2$ . Determine the initial value (A) and growth factor (B):

- The initial number of visitors is $80$ , so $A = 80$ .

- An increase of $150\%$ means the new amount is $100\% + 150\% = 250\%$ of the previous amount. The growth factor is $250\%$ as a decimal, so $B = 2.50$ .

$3$ . Determine the number of growth periods (k):

- The growth occurs every $30$ minutes, or $0.5$ hours.

- The variable t is in hours. The number of growth periods in t hours is $\dfrac{t}{0.5} = 2t$ . So, $k = 2t$ .

$4$ . Assemble the equation:

- $V = A \cdot B^k = 80(2.50)^{2t}$ .

$5$ . Conclusion:

- The equation that best represents the model is $V = 80(2.50)^{2t}$ , which corresponds to option A.

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SAT Exponential functions practice question

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