Problem 33
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{\ln ^{6} x}{x} d x$$
Problem 33
Guppy Population \(\mathrm{A} 2000\) -gallon tank can support no more than 150 guppies. Six guppies are introduced into the tank. Assume that the rate of growth of the population is \(\frac{d P}{d t}=0.0015 P(150-P)\) where time \(t\) is in weeks. (a) Find a formula for the guppy population in terms of \(t .\) (b) How long will it take for the guppy population to be 100? 125?
Problem 34
Finding Area Find the area of the region enclosed by the \(y\) -axis and the curves \(y=x^{2}\) and \(y=\left(x^{2}+x+1\right) e^{-x}\)
Problem 34
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \tan ^{7} \frac{x}{2} \sec ^{2} \frac{x}{2} d x$$
Problem 34
Gorilla Population A certain wild animal preserve can support no more than 250 lowland gorillas. Twenty-eight gorillas were known to be in the preserve in \(1970 .\) Assume that the rate of growth of the population is \(\frac{d P}{d t}=0.0004 P(250-P)\) where time \(t\) is in years. (a) Find a formula for the gorilla population in terms of \(t\) . (b) How long will it take for the gorilla population to reach the carrying capacity of the preserve?
Problem 35
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int s^{1 / 3} \cos \left(s^{4 / 3}-8\right) d s$$
Problem 35
Logistic Differential Equation Show that the solution of the differential equation \(\frac{d P}{d t}=k P(M-P) \quad\) is \(\quad P=\frac{M}{1+A e^{-M k t}}\) where \(A\) is a constant determined by an appropriate initial condition.
Problem 35
Average Value A retarding force, symbolized by the dashpot in the figure, slows the motion of the weighted spring so that the mass's position at time \(t\) is $$y=2 e^{-t} \cos t, \quad t \geq 0$$ Find the average value of \(y\) over the interval \(0 \leq t \leq 2 \pi\)
Problem 36
Carbon-14 Dating Measurement Sensitivity To see the effect of a relatively small error in the estimate of the amount of carbon-14 in a sample being dated, answer the following questions about this hypothetical situation. (a) A fossilized bone found in central Illinois in the year A.D. 2000 contains 17\(\%\) of its original carbon-14 content. Estimate the year the animal died. (b) Repeat part (a) assuming 18\(\%\) instead of 17\(\%\) . (c) Repeat part (a) assuming 16\(\%\) instead of 17\(\%\) .
Problem 36
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{d x}{\sin ^{2} 3 x}$$
