A township has a population of at least 83,000 people and grows at a rate of 1.5% each year. The city’s water supply can currently serve at most 88,000 people, and the water company is increasing the water supply at a constant rate of 2,000 people each year.

Question

Let x=
number of years and y=
population in the 10,000s. Which of the following sets of inequalities represents this situation?

(1 point)
Responses

y≥8.3(1.015)x

y≤8.8+0.02x

y is greater than or equal to 8 point 3 times 1 point 0 1 5 to the x th power y is less than or equal to 8 point 8 plus 0 point 0 2 x

y≥8.3(1.015)x

y≥8.8+0.02x

y is greater than or equal to 8 point 3 times 1 point 0 1 5 to the x th power y is greater than or equal to 8 point 8 plus 0 point 0 2 x

y≤8.3(1.015)x

y≤8.8+0.02x

y is less than or equal to 8 point 3 times 1 point 0 1 5 to the x th power y is less than or equal to 8 point 8 plus 0 point 0 2 x

y≤83(1.015)x

y≤88+0.02x

y is less than or equal to 83 times 1 point 0 1 5 to the x th power y is less than or equal to 88 plus 0 point 0 2 x

Answer 1

To formulate the inequalities based on the given information, we start with the population growth and the water supply adjustments.

1. Population Growth:

   • The initial population is at least 83,000, which in terms of \(y\) (in 10,000s) is represented as \(y \geq 8.3\).
   • Given a growth rate of 1.5% per year, the population can be modeled by: \[ y \geq 8.3(1.015)^x \]

2. Water Supply:

   • The maximum water supply is capable of serving at most 88,000 people, which translates to \(y \leq 8.8\) in 10,000s.
   • The water supply is increasing by 2,000 people each year, or 0.2 in 10,000s. The water supply over time can be modeled as: \[ y \leq 8.8 + 0.02x \]

Combining these two components, we can form the following set of inequalities:

• For the population, \[ y \geq 8.3(1.015)^x \]

• For the water supply, \[ y \leq 8.8 + 0.02x \]

Thus, the correct answer is:

• \( y \geq 8.3(1.015)^x \)
• \( y \leq 8.8 + 0.02x \)