91Ó°ÊÓ

Change each equation to its exponential form. $$\ln e^{2}=2$$

Use a calculator to evaluate the exponential function for the given \(x\) -value. Round to the nearest hundredth. $$f(x)=3^{x}, x=-1.5$$

Sodium-24 is a radioactive isotope of sodium that is used to study circulatory dysfunction. Assuming that 4 micrograms of sodium- 24 is injected into a person, the amount \(A\) in micrograms remaining in that person after \(t\) hours is given by the equation \(A=4 e^{-0.046 t}\) a. Graph this equation. b. What amount of sodium- 24 remains after 5 hours? c. What is the half-life of sodium-24? d. In how many hours will the amount of sodium- 24 be 1 microgram?

Polonium ( \(^{210} \mathrm{Po}\) ) has a half-life of 138 days. Find the decay function for the amount of polonium ( \(^{210}\) Po ) that remains in a sample after \(t\) days.

Geologists have determined that Crater Lake in Oregon was formed by a volcanic eruption. Chemical analysis of a wood chip that is assumed to be from a tree that died during the eruption has shown that it contains approximately \(45 \%\) of its original carbon- 14 Determine how long ago the volcanic eruption occurred. Use 5730 years as the half-life of carbon- 14

The Rhind papyrus, named after A. Henry Rhind, contains most of what we know today of ancient Egyptian mathematics. A chemical analysis of a sample from the papyrus has shown that it contains approximately \(75 \%\) of its original carbon-14. What is the age of the Rhind papyrus? Use 5730 years as the halflife of carbon- 14

The current \(I(t)\) (measured in amperes) of a circuit is given by the function \(I(t)=6\left(1-e^{-25 t}\right),\) where \(t\) is the number of seconds after the switch is closed. a. Find the current when \(t=0\) b. Find the current when \(t=0.5\) c. Solve the equation for \(t\) GRAPH CANT COPY

Assuming that air resistance is proportional to velocity, the velocity \(v,\) in feet per second, of a falling object after \(t\) seconds is given by \(v=32\left(1-e^{-t}\right)\) a. Graph this equation for \(t \geq 0\) b. Determine algebraically, to the nearest 0.01 second, when the velocity is 20 feet per second. c. Determine the horizontal asymptote of the graph of \(v\) d. the horizontal asymptote in the context of this application.

Find the domain of the function. Write the domain using interval notation. $$P(x)=\ln \left(x^{2}-4\right)$$

Vinegar has a hydronium-ion concentration of \(1.26 \times 10^{-3}\) mole per liter. Determine the pH of vinegar and state whether it is an acid or a base.