Logistic Growth Animal populations are not capable of unrestricted growth because of limited habitat and food sup- plies. Under such conditions the population follows a logistic growth model: \( P ( t ) = \frac { d } { 1 + k e ^ { - c t } } \) where \( c , d \) , and \( k \) are positive constants. For a certain fish population in a small pond \( d = 1200 , k = 11 , c = 0.2 \) and \( t \) is measured in years. The fish were introduced into the pond at time \( t = 0 \) . (a) How many fish were originally put in the pond? (b) Find the population after \( 10,20 \) , and \( 30 \) years. (c) Evaluate \( P ( t ) \) for large values of \( t \) . What value does the population approach as \( t \rightarrow \infty \) ? Does the graph shown confirm your calculations? 

Use transformations of the graph of \( y=x^{5} \) to graph the function. \( f(x)=(x-8)^{5}+6 \) Which tranformations are needed to graph the function? A. The graph of \( y=x^{5} \) should be horizontally shifted to the left by 6 units and shifted vertically down by 8 . B. The graph of \( y=x^{5} \) should be horizontally shifted to the right by 6 units and shifted vertically up by 8 . The graph of \( y=x^{5} \) should be horizontally shifted the the left by 8 units and shifted vertically down by 6 . D. The graph of \( y=x^{5} \) should be horizontally shifted to the right by 8 units and shifted vertically up by 6 .

Find the maximum \( y \)-value on the graph of \( y=f(x) \). \[ f(x)=-4 x^{2}+x-7 \] The maximum \( y \)-value is \( y=\square \). (Type an integer or a fraction.)

The function \( f(t)=60,000(2)^{\frac{1}{40}} \) gives the number of bacteria in a population \( t \) minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?

The number of U.S. travelers to other countries during the period from 1990 through 2009 can be modeled by the polynomial function \( P(x)=-0.00530 x^{3}+0.1366 x^{2}+1.250 x+46.49 \) where \( x=0 \) represents \( 1990, x=1 \) represents 1991 , and so on and \( P(x) \) is in millions. Use this function to approximate the number of \( U \).S. travelers to other countries in each given year. (a) 1990 (b) 2000 (c) 2009 (a) In 1990 , the number of U.S. travelers to other countries was approximately \( \square \) million. (Simplify your answer. Type an integer or decimal rounded to the nearest tenth as needed.) (b) In 2000, the number of U.S. travelers to other countries was approximately \( \square \) million. (Simplify your answer. Type an integer or decimal rounded to the nearest tenth as needed.) (c) In 2009, the number of U.S. travelers to other countries was approximately \( \square \) million. (Simplify your answer. Type an integer or decimal rounded to the nearest tenth as needed.)

Use the remainder theorem to find the remainder when \( f(x) \) is divided by \( x-1 \). Then use the factor theorem to determine whether \( x-1 \) is a factor of \( f(x) \). \( f(x)=3 x^{3}-2 x^{2}+7 x-2 \) The remainder is \( \square \).