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Home Question Bank Math Pre Calculus Archive 2024 December 04

Pre-calculus Questions from Dec 04,2024

Browse the Pre-calculus Q&A Archive for Dec 04,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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The size of a certain insect population is given by \( \mathrm{P}(\mathrm{t})=400 e^{.03 \mathrm{t}} \), where t is measured in days. At what time will the population equal 1600 ? Determine the implied domain of the following function. Express your answer in interval notation. \[ f(x)=\sqrt{x}-9 \] Answer 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 \( f(x) \). \( f(x)=2 x^{4}+7 x-3 \) The remainder is 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 \( \square \). 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 . Use the graph to evaluate the following expressions. (a) \( (\mathrm{f} \circ \mathrm{g})(1) \) (b) \( (\mathrm{g} \circ \mathrm{f})(0) \) (c) \( (\mathrm{g} \circ \mathrm{g})(0) \) (a) Evaluate \( (\mathrm{f} \circ \mathrm{g})(1) \). ( \( \mathrm{f} \circ \mathrm{g})(1)=1 \) (b) Evaluate \( (\mathrm{g} \circ \mathrm{f})(0) \). ( \( \mathrm{g} \circ \mathrm{f})(0)=0 \) (c) Evaluate \( (\mathrm{g} \circ \mathrm{g})(0) \). ( \( \mathrm{g} \circ \mathrm{g})(0)=0 \) 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.) \( 1 \leftarrow \) Use the formula \( t=\frac{\ln 2}{\mathrm{k}} \) that gives the time for a population, with a growth rate k, to double, to answer the following questions. The growth model \( \mathrm{A}=9 e^{0.005 t} \) describes the population, A , of a country in millions, t years after 2003. a. What is the country's growth rate? \( \square \% \) b. How long will it take the country to double its population? \( \square \) years (Round to the nearest whole number.)
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