The slope is the relationship between the cost of the gym and how long you are there for.
the y-intercept is how much it costs to go to the gym for no hours ie the start-up cost.
Answer:
a dog
Step-by-step explanation:
Answer: The amount of salt in the tank after 8 minutes is 36.52 pounds.
Step-by-step explanation:
Salt in the tank is modelled by the Principle of Mass Conservation, which states:
(Salt mass rate per unit time to the tank) - (Salt mass per unit time from the tank) = (Salt accumulation rate of the tank)
Flow is measured as the product of salt concentration and flow. A well stirred mixture means that salt concentrations within tank and in the output mass flow are the same. Inflow salt concentration remains constant. Hence:
![c_{0} \cdot f_{in} - c(t) \cdot f_{out} = \frac{d(V_{tank}(t) \cdot c(t))}{dt}](https://tex.z-dn.net/?f=c_%7B0%7D%20%5Ccdot%20f_%7Bin%7D%20-%20c%28t%29%20%5Ccdot%20f_%7Bout%7D%20%3D%20%5Cfrac%7Bd%28V_%7Btank%7D%28t%29%20%5Ccdot%20c%28t%29%29%7D%7Bdt%7D)
By expanding the previous equation:
![c_{0} \cdot f_{in} - c(t) \cdot f_{out} = V_{tank}(t) \cdot \frac{dc(t)}{dt} + \frac{dV_{tank}(t)}{dt} \cdot c(t)](https://tex.z-dn.net/?f=c_%7B0%7D%20%5Ccdot%20f_%7Bin%7D%20-%20c%28t%29%20%5Ccdot%20f_%7Bout%7D%20%3D%20V_%7Btank%7D%28t%29%20%5Ccdot%20%5Cfrac%7Bdc%28t%29%7D%7Bdt%7D%20%2B%20%5Cfrac%7BdV_%7Btank%7D%28t%29%7D%7Bdt%7D%20%5Ccdot%20c%28t%29)
The tank capacity and capacity rate of change given in gallons and gallons per minute are, respectivelly:
![V_{tank} = 220\\\frac{dV_{tank}(t)}{dt} = 0](https://tex.z-dn.net/?f=V_%7Btank%7D%20%3D%20220%5C%5C%5Cfrac%7BdV_%7Btank%7D%28t%29%7D%7Bdt%7D%20%3D%200)
Since there is no accumulation within the tank, expression is simplified to this:
![c_{0} \cdot f_{in} - c(t) \cdot f_{out} = V_{tank}(t) \cdot \frac{dc(t)}{dt}](https://tex.z-dn.net/?f=c_%7B0%7D%20%5Ccdot%20f_%7Bin%7D%20-%20c%28t%29%20%5Ccdot%20f_%7Bout%7D%20%3D%20V_%7Btank%7D%28t%29%20%5Ccdot%20%5Cfrac%7Bdc%28t%29%7D%7Bdt%7D)
By rearranging the expression, it is noticed the presence of a First-Order Non-Homogeneous Linear Ordinary Differential Equation:
, where
.
![\frac{dc(t)}{dt} + \frac{f_{out}}{V_{tank}} \cdot c(t) = \frac{c_0}{V_{tank}} \cdot f_{in}](https://tex.z-dn.net/?f=%5Cfrac%7Bdc%28t%29%7D%7Bdt%7D%20%2B%20%5Cfrac%7Bf_%7Bout%7D%7D%7BV_%7Btank%7D%7D%20%5Ccdot%20c%28t%29%20%3D%20%5Cfrac%7Bc_0%7D%7BV_%7Btank%7D%7D%20%5Ccdot%20f_%7Bin%7D)
The solution of this equation is:
![c(t) = \frac{c_{0}}{f_{out}} \cdot ({1-e^{-\frac{f_{out}}{V_{tank}}\cdot t }})](https://tex.z-dn.net/?f=c%28t%29%20%3D%20%5Cfrac%7Bc_%7B0%7D%7D%7Bf_%7Bout%7D%7D%20%5Ccdot%20%28%7B1-e%5E%7B-%5Cfrac%7Bf_%7Bout%7D%7D%7BV_%7Btank%7D%7D%5Ccdot%20t%20%7D%7D%29)
The salt concentration after 8 minutes is:
![c(8) = 0.166 \frac{pounds}{gallon}](https://tex.z-dn.net/?f=c%288%29%20%3D%200.166%20%5Cfrac%7Bpounds%7D%7Bgallon%7D)
The instantaneous amount of salt in the tank is: