A polling firm determines that the percent of likely voters who support a certain mayoral candidate can be represented by a function \( P(t) \) where \( t \) is measured in weeks since the candidate entered the race. Interpret the given equations, including units. (a) \( P(4)=28 \) (b) \( P^{\prime}(4)=7 \) (a) Choose the correct answer below. A. The percentage of likely voters to support the candidate 28 days after entering the race is \( 4 \% \). B. The percentage of likely voters to not support the candidate 28 days after entering the race is \( 4 \% \). C. The percentage of likely voters to not support the candidate 4 weeks after entering the race is \( 28 \% \). D. The percentage of likely voters to support the candidate 4 weeks after entering the race is \( 28 \% \). (b) Choose the correct answer below. A. 7 weeks after entering the race, support for the candidate increased by approximately 4 percentage points in the next week. B. 4 weeks after entering the race, support for the candidate increased by approximately 7 percentage points in the \( { }^{\circ} \) next week. C. 7 weeks after entering the race, support for the candidate decreased by approximately 4 percentage points in the next week. D. 4 weeks after entering the race, support for the candidate decreased by approximately 7 percentage points in the next week.

To interpret the given equations, we need to analyze each part step by step. ### Part (a): \( P(4) = 28 \) This equation indicates that the function \( P(t) \) gives the percentage of likely voters who support the candidate at a specific time \( t \). Here, \( t = 4 \) represents 4 weeks since the candidate entered the race. - **Interpretation**: \( P(4) = 28 \) means that 28% of likely voters support the candidate 4 weeks after entering the race. Now, let's evaluate the answer choices: - A. The percentage of likely voters to support the candidate 28 days after entering the race is \( 4 \% \). (Incorrect, as it refers to 28 days instead of 4 weeks) - B. The percentage of likely voters to not support the candidate 28 days after entering the race is \( 4 \% \). (Incorrect, same reason) - C. The percentage of likely voters to not support the candidate 4 weeks after entering the race is \( 28 \% \). (Incorrect, it states the percentage of support, not non-support) - D. The percentage of likely voters to support the candidate 4 weeks after entering the race is \( 28 \% \). (Correct) Thus, the correct answer for part (a) is **D**. ### Part (b): \( P^{\prime}(4) = 7 \) This equation represents the derivative of the function \( P(t) \) at \( t = 4 \). The derivative \( P^{\prime}(t) \) indicates the rate of change of the percentage of support with respect to time. - **Interpretation**: \( P^{\prime}(4) = 7 \) means that at 4 weeks after entering the race, the support for the candidate is increasing at a rate of 7 percentage points per week. Now, let's evaluate the answer choices: - A. 7 weeks after entering the race, support for the candidate increased by approximately 4 percentage points in the next week. (Incorrect, it refers to 7 weeks instead of 4 weeks) - B. 4 weeks after entering the race, support for the candidate increased by approximately 7 percentage points in the next week. (Correct, as it accurately reflects the interpretation of the derivative) - C. 7 weeks after entering the race, support for the candidate decreased by approximately 4 percentage points in the next week. (Incorrect, it states a decrease and refers to 7 weeks) - D. 4 weeks after entering the race, support for the candidate decreased by approximately 7 percentage points in the next week. (Incorrect, it states a decrease) Thus, the correct answer for part (b) is **B**. ### Summary of Answers: - (a) **D**: The percentage of likely voters to support the candidate 4 weeks after entering the race is \( 28 \% \). - (b) **B**: 4 weeks after entering the race, support for the candidate increased by approximately 7 percentage points in the next week.