Question

you use a garden hose to fill a wading pool. if the water level rises 14 centimeters every 3 minutes and you record the data point of (6,y), what is the value of y? use slope to justify your answer

the rate of change at which the water level rises is \\(\square\\) centimeters per minute. so, solving the equation \\(\square\\) for y gives a y - value equal to \\(\square\\)
(type integers or fractions. simplify your answers.)

\\(14y = 6\\)
\\(\frac{14}{3}=\frac{y}{6}\\)
\\(14y = 3\\)
\\(\frac{14}{6}=\frac{y}{3}\\)

Explanation:

Step1: Find the rate of change (slope)

The water level rises 14 centimeters every 3 minutes. The rate of change (slope) \( m \) is the change in height over the change in time, so \( m=\frac{14}{3} \) centimeters per minute.

Step2: Set up the proportion for slope

We have two points: (3, 14) and (6, y). The slope between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using the slope we found, \( \frac{14}{3}=\frac{y - 14}{6 - 3} \), but a simpler way is to use the proportion of slope (rate) being constant. So \( \frac{14}{3}=\frac{y}{6} \) (since slope is rise over time, so for time 6 minutes, the rise is y, and the ratio of rise to time should be equal to the rate \( \frac{14}{3} \)).

Step3: Solve for y

From \( \frac{14}{3}=\frac{y}{6} \), we can cross - multiply: \( 3y=14\times6 \). Then \( 3y = 84 \), and \( y=\frac{84}{3}=28 \).

Answer:

The rate of change at which the water level rises is \( \frac{14}{3} \) centimeters per minute. So, solving the equation \( \frac{14}{3}=\frac{y}{6} \) for y gives a y - value equal to 28.