You are working to design a skin patch that delivers a drug to the body. Examples of patches on the market today include the Nicoderm patch $($for helping individuals to quit smoking$)$ and patches that contain hormones, including estrogen and testosterone. The skin patch is thin and flat and has a surface area of $1i{n}^2.$

It is placed on the arm with one side up against the skin and the other side exposed to the air. Your intention is to design a patch that delivers a pain medication that might be used after surgery or traumatic injury. To minimize the risk of addiction, the amount of drug delivered decreases as a function of time. You conduct a number of tests, as described below, to help design and characterize the patch. In all tests below, ignore drug and water losses from the edges $($sides$)$ of the patch.

For each test please do the following:

a$)$ Draw the system, boundary, and surroundings.

b$)$ State all reasonable assumptions for the system.

c$)$ Write and simplify the accounting equation for the system.

d$)$ Write the appropriate mass conservation equation for the system.

e$)$ Answer the question asked of each test.

Test A$.$ As noted above, the amount of drug delivered decreases as a function of time. Your research has shown that the rate the drug leaves the patch, $y,$ is described by the following formula:

$y=-1\frac{g}{da{y}^2}t+40\frac{g}{\text{d}\text{a}\text{y}}$

where $t$ is the time. Assume that the drug is delivered from the patch to the skin and that no drug is lost to the air. Assuming that the patch is loaded with $800g$ of drug, how long does it take before the drug is exhausted from the patch?