91Ó°ÊÓ

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$f(x)=3 x^{4}+2 x^{3}-x^{2}-x-6 ; \quad p(x)=x^{2}+1$$

The ideal gas law The ideal gas law states that the volume \(V\) that a gas occupies is directly proportional to the product of the number \(n\) of moles of gas and the temperature \(T\) (in K) and is inversely proportional to the pressure \(P\) (in atmospheres). (a) Express \(V\) in terms of \(n, T, P,\) and a constant of proportionality \(k\) What is the effect on the volume if the number of moles is doubled and both the temperature and the pressure are reduced by a factor of one-half?

Poiseuille's law Poiseuille's law states that the blood flow rate \(F\) (in \(\mathrm{L} / \mathrm{min}\) ) through a major artery is directly proportional to the product of the fourth power of the radius \(r\) of the artery and the blood pressure \(P\). (a) Express \(F\) in terms of \(P, r,\) and a constant of proportionality \(k\) (b) During heavy exercise, normal blood flow rates sometimes triple. If the radius of a major artery increases by \(10 \%,\) approximately how much harder must the heart pump?

Find the remainder if the polynomial $$3 x^{100}+5 x^{35}-4 x^{36}+2 x^{17}-6$$ is divided by \(x+1\)

For a particular salmon population, the relationship between the number \(S\) of spawners and the number \(R\) of offspring that survive to maturity is given by the formula $$R=\frac{4500 \mathrm{s}}{s+500}$$ (a) Under what conditions is \(R>S ?\) (b) Find the number of spawners that would yield \(90 \%\) of the greatest possible number of offspring that survive to maturity. (c) Work part (b) with 80 \% replacing 90 \(\$ 6 dollar. (d) Compare the results for \)S\( and \)R$ (in terms of percentage increases) from parts (b) and (c).