For a rectangle with area 100 to have the smallest perimeter, what dimensions should it have?

The following graph shows the approximate value of the U.S. Consumer Price Index (CPI) from September 2004 through November \(2005.32\) The approximating curve shown on the figure is given by $$I(t)=-0.005 t^{3}+0.12 t^{2}-0.01 t+190 \quad(0 \leq t \leq 14)$$ where \(t\) is time in months ( \(t=0\) represents September 2004). a. Use the model to estimate the monthly inflation rate in July \(2005(t=10)\). [Recall that the inflation rate is \(\left.I^{\prime}(t) / I(t) .\right]\) b. Was inflation slowing or speeding up in July \(2005 ?\) c. When was inflation speeding up? When was inflation slowing? HINT [See Example 3.]

If \(f(x)\) is a polynomial of degree 2 or higher, show that between every pair of relative extrema of \(f(x)\) there is a point of inflection of \(f(x)\).

Daily oil imports to the United States from Mexico can be approximated by \(q(t)=-0.015 t^{2}+0.1 t+1.4\) million barrels \((0 \leq t \leq 8)\) where \(t\) is time in years since the start of \(2000 .^{56}\) At the start of 2004 the price of oil was \(\$ 30\) per barrel and increasing at a rate of $\$ 40\( per year. \){ }^{57}$ How fast was (daily) oil expenditure for imports from Mexico changing at that time?

Assume that the demand equation for tuna in a small coastal town is $$p q^{1.5}=50,000$$ where \(q\) is the number of pounds of tuna that can be sold in one month at the price of \(p\) dollars per pound. The town's fishery finds that the demand for tuna is currently 900 pounds per month and is increasing at a rate of 100 pounds per month each month. How fast is the price changing?

A spherical party balloon is being inflated with helium pumped in at a rate of 3 cubic feet per minute. How fast is the radius growing at the instant when the radius has reached 1 foot? (The volume of a sphere of radius \(r\) is \(V=\frac{4}{3} \pi r^{3} .\) HINT [See Example 1.]