Daphne begins pouring water into a swimming pool at a constant rate. The linear equation y=1.5x+12

models the height of the water in pool, in inches, after Daphne has been pouring water for
minutes.

Based on the linear model, which statement is true?
Daphne has been pouring water for 12 minutes.
Daphne is pouring water into the pool at a rate of 12 inches per minute.
The height of the water in the pool is increasing at a rate of 1.5 inches per minute.
The height of the water in the pool was 1.5 inches when Daphne started pouring water.

1 answer

To analyze the given linear equation \( y = 1.5x + 12 \), we can identify the components of the equation:

  • \( y \) represents the height of the water in the pool in inches.
  • \( x \) represents the time in minutes after Daphne has begun pouring water.
  • The coefficient of \( x \), which is 1.5, represents the rate at which the height of the water increases per minute.
  • The constant term (12) represents the initial height of the water in the pool when \( x = 0 \).

Now, let's evaluate the statements:

  1. Daphne has been pouring water for 12 minutes.
    This statement is incorrect because there's no indication from the equation that Daphne has been pouring water for exactly 12 minutes.

  2. Daphne is pouring water into the pool at a rate of 12 inches per minute.
    This statement is incorrect because the rate is 1.5 inches per minute, not 12 inches per minute.

  3. The height of the water in the pool is increasing at a rate of 1.5 inches per minute.
    This statement is correct, as the coefficient of \( x \) (1.5) indicates the rate of increase in height per minute.

  4. The height of the water in the pool was 1.5 inches when Daphne started pouring water.
    This statement is incorrect. When \( x = 0 \), the height of the water \( y \) is \( 1.5(0) + 12 = 12 \) inches, not 1.5 inches.

Therefore, the true statement based on the linear model is:
The height of the water in the pool is increasing at a rate of 1.5 inches per minute.