1. Water is draining from a tank at a constant rate. After 3 minutes, the tank contains 1557 gallons.
After 10 minutes, the tank contains 1494 gallons.
(a) Find a linear equation that gives the amount of water in the tank, G(x), after x minutes have
elapsed. Write the equation in slope-intercept form. Remember to explain what you are doing
at each step and why you are doing it.
(b) Based on your work in part (a), what is the slope of the linear model? Interpret what it means
in the context of the situation.
(c) Use the linear model you created in part (a) to predict the amount of water in the tank after
half an hour. Remember to explain what you are doing and why.
(d) Find the x-intercept of the linear model and interpret what it means in the context of the
situation. Remember to explain what you are doing and why.
(e) Find the y-intercept of the linear model and interpret what it means in the context of the
situation. Remember to explain what you are doing and why
amount = change/min * time + initial amount
G = r x + Gi
G(3) = 1557
G(10) =1494
so r = (1494-1557) / (10-3) = - 9 = slope in gallons / minute
put a point in to get Gi
1557 = -9 * 3 + Gi
Gi = 1557 + 27 = 1584
so
G = -9 x + 1584
Your data consists of two ordered pairs (3,1557) and (10,1494)
where x is the number of minutes and y is the number of gallons
slope = (1557-1494)/(3-10) = - 63/7 = - 9
so y = -9x + b, with (3,1557) satisfying that equation
1557 = -9(3) + b
b = 1584
y = -9x + 1584
From the work I have done for you, I am sure you can find the
conclusions and answers for each of the parts of your problem.
your turn now :)
To solve this problem, we need to find a linear equation that represents the amount of water in the tank after a certain number of minutes.
Step 1: Define variables
Let G(x) represent the amount of water in the tank after x minutes have elapsed.
Step 2: Determine the slope
The slope represents the rate at which the water is draining from the tank. In this case, the water is draining at a constant rate. To find the slope, we can use the formula:
slope = (change in y)/(change in x)
The change in y is the difference in the amount of water between two data points, and the change in x is the difference in time between those two data points.
Given:
After 3 minutes, the tank contains 1557 gallons.
After 10 minutes, the tank contains 1494 gallons.
Change in y = 1494 - 1557 = -63 gallons
Change in x = 10 - 3 = 7 minutes
slope = (-63 gallons)/(7 minutes) = -9 gallons/minute
Step 3: Write the equation
We can use the slope-intercept form of a linear equation, which is:
y = mx + b
In our case:
G(x) = -9x + b
Step 4: Find the y-intercept
The y-intercept, represented by b, is the initial amount of water in the tank when x = 0. We can find it by substituting the given values into the equation. We have two data points to choose from, let's use the one with 3 minutes:
1557 = -9(3) + b
1557 = -27 + b
b = 1557 + 27
b = 1584
So, the equation becomes:
G(x) = -9x + 1584
(a) The linear equation in slope-intercept form that gives the amount of water in the tank, G(x), after x minutes have elapsed is:
G(x) = -9x + 1584
(b) The slope of the linear model is -9. In this context, it means that the tank is losing water at a rate of 9 gallons per minute.
(c) To predict the amount of water in the tank after half an hour (30 minutes), we substitute x = 30 into the equation:
G(x) = -9(30) + 1584
G(x) = -270 + 1584
G(x) = 1314 gallons
So, the predicted amount of water in the tank after half an hour is 1314 gallons.
(d) To find the x-intercept of the linear model, we set G(x) equal to zero:
0 = -9x + 1584
9x = 1584
x = 176
The x-intercept represents the amount of time it takes for the tank to be completely empty. In this context, it means that it will take 176 minutes for the tank to drain completely.
(e) The y-intercept of the linear model is 1584 gallons. In the context of the situation, it means that the tank initially contains 1584 gallons of water when x = 0 minutes.