91Ó°ÊÓ

The value of good wine increases with age. Thus, if you are a wine dealer, you have the problem of deciding whether to sell your wine now, at a price of \(\$ P\) a bottle, or to sell it later at a higher price. Suppose you know that the amount a wine-drinker is willing to pay for a bottle of this wine \(t\) years from now is \(\$ P(1+20 \sqrt{t}) .\) Assuming continuous compounding and a prevailing interest rate of \(5 \%\) per year, when is the best time to sell your wine?

Explain what is wrong with the statement. A probability density function is always increasing.

Show that the arc length formula for polar coordinates gives the expected answer for the circumference of the circle \(r=a\) for \(0 \leq \theta \leq 2 \pi\)

Set up definite integral(s) to find the volume obtained when the region between \(y=x^{2}\) and \(y=5 x\) is rotated about the given axis. Do not evaluate the integral(s). $The\quad line\quad y=-4$$

Find the area inside the circle \(r=1\) and outside the cardioid \(r=1+\sin \theta\)

Find the volume of a sphere of radius \(r\) by slicing.

Set up definite integral(s) to find the volume obtained when the region between \(y=x^{2}\) and \(y=5 x\) is rotated about the given axis. Do not evaluate the integral(s). $$The\quad line\quad x=-3$$

An oil company discovered an oil reserve of 100 million barrels. For time \(t>0,\) in years, the company's extraction plan is a linear declining function of time as follows: $$ q(t)=a-b t $$ where \(q(t)\) is the rate of extraction of oil in millions of barrels per year at time \(t\) and \(b=0.1\) and \(a=10\) (a) How long does it take to exhaust the entire reserve? (b) The oil price is a constant \(\$ 20\) per barrel, the extraction cost per barrel is a constant \(\$ 10\), and the market interest rate is \(10 \%\) per year, compounded continuously. What is the present value of the company's profit?

Find the center of mass of a cone of height \(5 \mathrm{cm}\) and base diameter \(10 \mathrm{cm}\) with constant density \(\delta \mathrm{gm} / \mathrm{cm}^{3}\).

Give an example of: A density function that is greater than zero on \(0 \leq x \leq\) 20 and zero everywhere else.