The pesticide DDT was widely used in the United States until its ban in 1972. DDT is toxic to a wide range of animals and aquatic life, and is suspected to cause cancer in humans. Scientists and environmentalists worry about such substances because these hazardous materials continue to be dangerous for many years after their disposal. The half-life of DDT is estimated to be 15 years.

1. Write an equation to model the amount of DDT in an object after $t$ years if $500$ mg of the substance is detected.

   Hint (a):

   Since this is a half-life situation, the equation will be of the form: $A(t) = A_0(\frac{1}{2})^{t/b}$

   More Help:

   If you are unsure of the values to input for $A_{0}$ and $b$, use the points $(0, 500)$ and $(15, 250)$ to write and solve a system of equations.

2. A concentration of just $2.73$ mg per pound in a person can cause headaches, nausea, vomiting, confusion, and tremors. How much DDT is this for a person who weighs $130$ pounds? Use your equation to determine how long it would take for the $500$ mg to decay to a safe level for this person.

   Step (b):

   $\frac{2.73 \text{ mg}}{1 \text{ lb}} \cdot 130 \text{ lb} = \:?$

3. Even smaller amounts of DDT can affect small microorganisms. For example, water that contains just $0.0001$ mg of DDT per liter can slow down growth and photosynthesis in green algae. Determine how long it would take for $500$ mg of DDT present in one liter to decay to this level.