The table below shows the time, in seconds, that it takes to fill 20-ounce bottles with water.
Time, x (in seconds) # of bottles filled, y
0 0
30 6
60 12
90 18
120 24
Determine whether y varies directly with x and explain how you came to that conclusion.
If this is direct variation, find the constant of variation, k, and write the equation.
To determine whether y varies directly with x, we need to check if the ratio of y to x remains constant.
Looking at the table, we can see that as x increases by 30 seconds, the corresponding value of y increases by 6.
Therefore, the ratio of y to x is constant, which means that y varies directly with x.
To find the constant of variation, k, we can take the ratio of any y and x values from the table. Let's use the first two values:
k = y/x = 6/30 = 1/5
Therefore, the constant of variation is 1/5.
The equation that represents this direct variation relationship between y and x is y = (1/5)x.
To determine whether y varies directly with x, we need to check if the ratio of y to x remains constant.
First, let's calculate the ratio of y to x for each pair of values:
- For the first pair, (0, 0), the ratio is 0/0, which is indeterminate.
- For the second pair, (30, 6), the ratio is 6/30 = 0.2.
- For the third pair, (60, 12), the ratio is 12/60 = 0.2.
- For the fourth pair, (90, 18), the ratio is 18/90 = 0.2.
- For the fifth pair, (120, 24), the ratio is 24/120 = 0.2.
Since the ratio of y to x remains constant at 0.2 for each pair of values, we can conclude that y varies directly with x.
To find the constant of variation (k), we can choose any pair of values and divide y by x. Let's use the second pair, (30, 6):
k = y / x = 6 / 30 = 0.2
The equation for this direct variation relationship is:
y = kx
Therefore, the equation is y = 0.2x.