a bathtub in your home has a drain that leaks. When you fill you bathtub with 30 gallons of water you notice that after every 5 minutes you only have 20 gallons remaining, so the drain is leaking 2 gallons every minute which represents slope. what is the linear equation of the problem and what is the y-intercept and what does it mean in the context of the problem and what is the x-intercept in the context of the problem
tub = -2 t + 30
the y-intercept is 30 ... a full tub
the x-intercept is the time for the tub to empty
To find the linear equation of the problem, we can represent the leak rate as the slope (m) and the time as the x-axis variable (x). Since the drain is leaking 2 gallons every minute, we can set the slope (m) of the linear equation to -2.
The equation can be written as:
y = m*x + b
Where:
y = remaining gallons of water
m = slope (-2, which represents the leak rate of 2 gallons per minute)
x = time in minutes
b = y-intercept
To find the y-intercept (b), we can use the given information that when the bathtub is filled with 30 gallons, after 5 minutes, only 20 gallons remain. We plug in the values of x and y into the equation and solve for b:
20 = -2*5 + b
20 = -10 + b
b = 30
Therefore, the linear equation representing the problem is:
y = -2x + 30
The y-intercept (b = 30) in the context of the problem represents the initial amount of water in the tub when the time (x) is 0. It implies that when the tub is initially filled, there are 30 gallons of water remaining before the drain starts leaking.
The x-intercept in the context of the problem represents the time it takes for the tub to completely empty. To find the x-intercept, we set y to 0 and solve for x:
0 = -2x + 30
2x = 30
x = 15
Therefore, the x-intercept is 15, which means after 15 minutes, all the water in the bathtub will have leaked out.