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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.

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 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

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

1 answer

To represent the situation involving population growth and water supply in a township, we need to establish the correct inequalities based on the given information.

  1. Population growth:

    • The population is currently at least 83,000 people, which can be expressed as \( y \geq 8.3 \) (since \( 83,000 \) is \( 8.3 \) in terms of \( y \) being in units of 10,000).
    • The population grows at a rate of 1.5% each year. The formula for future population after \( x \) years is given by: \[ y \geq 8.3(1.015)^x \]

  2. Water supply:

    • The current water supply can serve at most 88,000 people, which can be expressed as \( y \leq 8.8 \) (since \( 88,000 \) is \( 8.8 \) in terms of \( y \)).
    • The water supply is increasing at a rate of 2,000 people per year. Given that the supply increases by 2,000 people per year, the inequality representing this situation becomes: \[ y \leq 8.8 + 0.02x \]

Now, combining these two inequalities, we have:

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

Based on the choices provided, the correct set of inequalities that represents this situation is:

y ≥ 8.3(1.015)^x and y ≤ 8.8 + 0.02x.

Thus, the response with these inequalities is: y ≥ 8.3(1.015)^x and y ≤ 8.8 + 0.02x.