------Suppose someone opens the valve on a large water tank so that water drains out. Choose a starting volume of water in the tank (from 100 to 500 gallons), and choose how much water drains out each day (from 2 to 5 gallons). Write the equation that models the relationship between time (x) and water volume (y) in slope-intercept form.
Post your equation. Choose another equation from the discussion board and answer the following.
Write the equation in slope-intercept form, and use the equation to find the amount of water in the tank exactly 2 days after the valve is opened.
Determine the slope of the line. Describe the meaning of the slope in this context.
Rewrite the equation in standard form, and use that equation to find the x- and y-intercepts. Describe the meaning of each intercept.
Do these equations represent an exact or approximate relationship between time and volume? This could be a matter of opinion. What is your opinion? Explain.--------
Let's start by choosing a starting volume of water in the tank of 200 gallons and the amount of water that drains out each day as 3 gallons.
The equation that models the relationship between time (x) and water volume (y) can be written in slope-intercept form as:
y = mx + b
where "m" represents the slope and "b" represents the y-intercept.
In this context, the slope represents the rate at which the water volume is decreasing. The slope can be calculated as the amount of water drained out each day, which is 3 gallons. Therefore, the slope is -3.
The equation in slope-intercept form becomes:
y = -3x + b
To find the value of the y-intercept (b), we can use the starting volume of 200 gallons. Since at the beginning (x = 0), the volume is 200 gallons, we can substitute these values into the equation:
200 = -3(0) + b
200 = b
So, the equation becomes:
y = -3x + 200
To find the amount of water in the tank exactly 2 days after the valve is opened (x = 2), we can substitute this value into the equation:
y = -3(2) + 200
y = -6 + 200
y = 194
Therefore, there would be 194 gallons of water in the tank exactly 2 days after the valve is opened.
Now, let's rewrite the equation in standard form, which is in the form of Ax + By = C:
3x + y = 200
In this form, the x-intercept represents the time it takes for the tank to completely drain. For this equation, the x-intercept is found when y = 0:
3x + 0 = 200
3x = 200
x = 200/3
So, the x-intercept is approximately 66.67 days, which means it would take approximately 67 days for the tank to completely drain.
The y-intercept represents the initial amount of water in the tank at x = 0. In this case, the y-intercept is 200, which corresponds to the initial volume of the tank.
Based on these equations, the relationship between time and volume is approximate since it assumes a constant rate of water draining each day. However, in reality, other factors such as changes in pressure or drainage efficiency could affect the rate of water draining. Therefore, the relationship represented by the equation is an approximation.
To model the relationship between time (x) and water volume (y) in slope-intercept form, we can use the equation:
y = mx + b
Here, m represents the slope of the line and b represents the y-intercept.
Let's choose a starting volume of water in the tank to be 200 gallons and a drainage rate of 4 gallons per day.
Plugging these values into the equation, we get:
y = 4x + 200
If we want to find the amount of water in the tank exactly 2 days after the valve is opened, we can substitute x = 2 into the equation:
y = 4(2) + 200
y = 8 + 200
y = 208 gallons
This means that exactly 2 days after the valve is opened, there will be 208 gallons of water in the tank.
The slope of the line, in this case, is 4. In the context of water draining from a tank, the slope represents the rate at which the water volume decreases per unit of time. In this case, it means that 4 gallons of water are being drained from the tank each day.
To rewrite the equation in standard form, we can rearrange it as follows:
4x - y = -200
The x-intercept can be found by setting y = 0 and solving for x:
4x - 0 = -200
4x = -200
x = -200/4
x = -50
The y-intercept can be found by setting x = 0 and solving for y:
4(0) - y = -200
-y = -200
y = 200
In this context, the x-intercept (-50) represents the number of days it would take for the tank to completely drain if the valve was continuously open. The y-intercept (200) represents the initial volume of water in the tank.
The relationship between time and volume in this equation is an exact relationship because the equation directly calculates the volume of water in the tank based on the given parameters.