91Ó°ÊÓ

Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for \(\$ 2.5\) million. For tax purposes, the value of the machine can be depreciated by \(6.8 \%\) of its current value each year. a. What is the value of the machine after 10 years? b. After how many years is the value of the machine \(10 \%\) of its original value?

Evaluate the following integrals. Include absolute values only when needed. $$\int \frac{e^{2 x}}{4+e^{2 x}} d x$$

Atmospheric pressure The pressure of Earth's atmosphere at sea level is approximately 1000 millibars and decreases exponentially with elevation. At an elevation of \(30,000 \mathrm{ft}\) (approximately the altitude of Mt. Everest), the pressure is one-third the sea-level pressure. At what elevation is the pressure half the sea-level pressure? At what elevation is it \(1 \%\) of the sea-level pressure?

Find the derivatives of the following functions. $$f(v)=\sinh ^{-1} v^{2}$$

Carbon dating The half-life of \(C-14\) is about 5730 yr. a. Archaeologists find a piece of cloth painted with organic dyes. Analysis of the dye in the cloth shows that only \(77 \%\) of the \(C\) -14 originally in the dye remains. When was the cloth painted? b. A well-preserved piece of wood found at an archaeological site has \(6.2 \%\) of the \(\mathrm{C}-14\) that it had when it was alive. Estimate when the wood was cut.

Find the derivatives of the following functions. $$f(x)=\operatorname{csch}^{-1}\left(\frac{2}{x}\right)$$

Evaluate the following integrals. Include absolute values only when needed. $$\int \frac{d x}{x \ln x \ln (\ln x)}$$

Evaluate the following integrals. Include absolute values only when needed. $$\int_{e^{2}}^{e^{3}} \frac{d x}{x \ln x \ln ^{2}(\ln x)}$$

Uranium dating Uranium- 238 (U-238) has a half-life of 4.5 billion years. Geologists find a rock containing a mixture of \(\mathrm{U}-238\) and lead, and they determine that \(85 \%\) of the original \(\mathrm{U}-238\) remains: the other \(15 \%\) has decayed into lead. How old is the rock?

Radioiodine treatment Roughly 12,000 Americans are diagnosed with thyroid cancer every year, which accounts for \(1 \$$ of all cancer cases. It occurs in women three times as frequently as in men. Fortunately, thyroid cancer can be treated successfully in many cases with radioactive iodine, or \)1-131 .\( This unstable form of iodine has a half-life of 8 days and is given in small doses measured in millicuries. a. Suppose a patient is given an initial dose of 100 millicuries. Find the function that gives the amount of \)1-131\( in the body after \)t \geq 0\( days. b. How long does it take the amount of \)1-131\( to reach \)10 \%\( of the initial dose? c. Finding the initial dose to give a particular patient is a critical calculation. How does the time to reach \)10 \%\( of the initial dose change if the initial dose is increased by \)5 \% ?$