Question

questions 23 through 25 are gridded response items that require you to write your answers in the boxes provided on your answer sheet. write only one number or symbol in each box and fill in the circle in each column that matches what you have printed. fill in only one circle in each column. 23 every ten years, the census counts how many people are living in every town in the united states. - the 2010 census showed that 1,000 people were living in appleville, and 4,000 people were living in bridgetown. - the population of appleville is predicted to double every ten years. - the population of bridgetown is predicted to increase by 1,000 every ten years. if the predictions come true, what will be the first census year that will show appleville with a larger population than bridgetown? 24 two stores have movies to rent. - the first store charges a $12.50 - per - month membership fee plus $1.50 per movie rented. - the second store has no membership fee but charges $3.50 per movie rented. what is the minimum number of movies a person would need to rent in a month for the first store to be a better deal?

Explanation:

Question 23

Step 1: Define the population functions

Let \( t \) be the number of 10 - year periods after 2010. The population of Appleville \( A(t) \) starts at 1000 and doubles every 10 years, so \( A(t)=1000\times2^{t} \). The population of Bridgetown \( B(t) \) starts at 4000 and increases by 1000 every 10 years, so \( B(t) = 4000+1000t \).

Step 2: Test \( t = 1 \) (2020)

For \( t = 1 \) (2020), \( A(1)=1000\times2^{1}=2000 \), \( B(1)=4000 + 1000\times1=5000 \). Since \( 2000<5000 \), Appleville's population is less.

Step 3: Test \( t = 2 \) (2030)

For \( t = 2 \) (2030), \( A(2)=1000\times2^{2}=4000 \), \( B(2)=4000+1000\times2 = 6000 \). Since \( 4000<6000 \), Appleville's population is less.

Step 4: Test \( t = 3 \) (2040)

For \( t = 3 \) (2040), \( A(3)=1000\times2^{3}=8000 \), \( B(3)=4000+1000\times3=7000 \). Since \( 8000 > 7000 \), Appleville's population is larger.

Step 1: Set up the inequality

Let \( x \) be the number of movies rented. The cost of the first store \( C_1=12.50 + 1.50x \), and the cost of the second store \( C_2 = 3.50x \). We want \( C_1

Step 2: Solve the inequality

Subtract \( 1.50x \) from both sides: \( 12.50<3.50x - 1.50x \), which simplifies to \( 12.50<2x \). Then divide both sides by 2: \( x>\frac{12.50}{2}=6.25 \).

Step 3: Find the minimum integer \( x \)

Since \( x \) must be an integer and \( x>6.25 \), the minimum value of \( x \) is 7.

Answer:

2040

Question 24