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Home Question Bank Math Pre Calculus Archive 2025 January 30

Pre-calculus Questions from Jan 30,2025

Browse the Pre-calculus Q&A Archive for Jan 30,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Lamar is walking. \( D(t) \), given below, is his distance in kilometers from Glen City after \( t \) hours of walking. Complete the following statements. Let \( D^{-1} \) be the inverse function of \( D \). Take \( x \) to be an output of the function \( D \). That is, \( x=D(t) \) and \( t=D^{-1}(x) \). (a) Which statement best describes \( D^{-1}(x) \) ? The amount of time he has walked (in hours) when he is \( x \) The reciprocal of his distance from Glen City (in kilometers) after walking \( x \) hours. His distance from Glen City (in kilometers) after he has walked \( x \) hours. The ratio of the amount of time he has walked (in hours) to his distance from Glen City (in kilometers), \( x \). (b) \( D \)-1 ( \( x \) ) = \( \square \) Dean and Becky deposit \( \$ 2,308.00 \) into a savings account which earns \( 4.85 \% \) interest compounded annually. They want to use the money in the account to go on a trip in 3 years. How much will they be able to spend? Use the formula \( A=P\left(1+\frac{r}{n}\right)^{n t} \), where \( A \) is the balance (final amount), \( P \) is the principal (starting amount), \( r \) is the interest rate expressed as a decimal, \( n \) is the number of times per year that the interest is compounded, and \( t \) is the time in years. Round your answer to the nearest cent. Kiara is saving up money to buy a car. Kiara puts \( \$ 5,739.00 \) into an account which earns \( 13.33 \% \) interest, compounded quarterly. How much will she have in the account after 5 years? Use the formula \( A=P\left(1+\frac{r}{n}\right)^{n t} \), where \( A \) is the balance (final amount), \( P \) is the principal (starting amount), \( r \) is the interest rate expressed as a decimal, \( n \) is the number of times year that the interest is compounded, and \( t \) is the time in years. Round your answer to the nearest cent. Ruben opened a savings account and deposited \( \$ 5,415.00 \) as principal. The account earns \( 13 \% \) interest, compounded quarterly. What is the balance after 8 years? Use the formula \( A=P\left(1+\frac{r}{n}\right)^{n t} \), where \( A \) is the balance (final amount), \( P \) is the principal (starting amount), \( r \) is the interest rate expressed as a decimal, \( n \) is the number of times per year that the interest is compounded, and \( t \) is the time in years. Round your answer to the nearest cent. After checking the bacteria population that was assigned to you, you remember that Casey had asked you to provide some analysis for the growth of their bacteria that they are hoping will speed up the production of an important material. Casey has asked you to take the function that models the population size, find the domain, range, and horizontal asymptote, and then plot the results. Casey specifically asked for these values related to the function that fits the data. They also wanted the input value of zero to represent the date that the data was collected. This makes negative input values represent previous days and positive values represent future predictions. After careful analysis, you determine that the function modeling the bacteria population is \( f(x)=4(2)^{x}+6 \). You next carefully find the domain and range in interval notation. To enter \( \infty \) when in text mode, type infinity. To enter \( \infty \) when in symbol mode, choose the \( \infty \) symbol. Domain: Range: asymptote: \( y= \) Write an exponential decay function to model the situation. Compare the average rates of change over the given intervals. initial value: 43 decay factor: 0.9 \( 1 \leq \mathrm{x} \leq 4 \) and \( 5 \leq \mathrm{x} \leq 8 \) The exponential decay function to model the situation is \( f(x)=\square \). \( (x)=\left\{\begin{array}{ll}5 x+6, & \text { for } x \leq-1 \\ x+2, & \text { for }-1<x<2 \\ 4, & \text { for } x \geq 2\end{array}\right. \) 5. The dollar value of a moped is a function of the number of years, \( t \), since the moped was purchased. The function, \( f \), is defined by the equation \( f(t)=2,500 \cdot\left(\frac{1}{2}\right)^{t} \). What is the best choice of domain for the function \( f \) ? A. \( -10 \leq t \leq 10 \) B. \( -10 \leq t \leq 0 \) C. \( 0 \leq t \leq 10 \) D. \( 0 \leq t \leq 100 \) The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at \( 12: 00 \) am and \( 12: 30 \mathrm{pm} \), with a depth of 2.5 m , while high tides occur at \( 6: 15 \) am and \( 6: 45 \mathrm{pm} \), with a depth of 5.5 m . Let \( t=0 \) be \( 12: 00 \mathrm{am} \). Write a cosine model, \( d=a \cos (b t)+k \), for the depth as a function of time. This amplitude is 1.5 \( a=9.5 \) \( \mathrm{X} \Rightarrow-1.5 \) The period is meters. \( b=\square \) hours. 4. 8Cómo se obtienen las gráficas de las siguientes funciones a partir de la gráfica de \( f \) ? \( \begin{array}{lll}\text { (a) } y=-f(x) & \text { (b) } y=2 f(x)-1 & \text { (c) } y=f(x-3)+2\end{array} \) 5. Sin usar una calculadora, haga un diagrama preliminar de la gráfica. \( \begin{array}{lll}\text { (a) } y=x^{3} & \text { (b) } y=(x+1)^{3} & \text { (c) } y=(x-2)^{3}+3 \\ \text { (d) } y=4-x^{2} & \text { (e) } y=\sqrt{x} & \text { (f) } y=2 \sqrt{x} \\ \text { (g) } y=-2^{x} & \text { (h) } y=1+x^{-1} & \end{array} \)
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