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Hogar Banco de preguntas Matemáticas Precálculo Archive 2025 January 25

Pre-calculus Questions from Jan 25,2025

Browse the Pre-calculus Q&A Archive for Jan 25,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Find the direction angles of the vector. (Round your answers to three decimal places.) \[ \mathbf{u}=-2 \mathbf{i}+7 \mathbf{j}+5 \mathbf{k} \] \( \alpha=\square \mathrm{rad} \) \( \beta=\square \mathrm{rad} \) \( \gamma=\square \mathrm{rad} \) Find \( \mathbf{u} \cdot \mathbf{v}, \mathbf{u} \cdot \mathbf{u},\|\mathbf{u}\|^{2},(\mathbf{u} \cdot \mathbf{v}) \mathbf{v} \), and \( \mathbf{u} \cdot(2 \mathbf{v}) \) \[ \mathbf{u}=\langle 7,11\rangle, \quad \mathbf{v}=\langle-2,3\rangle \] 9. \( f(x)=0.2(5)^{-x} \) Find the unit vector in the direction of \( \mathbf{v} \). \[ \mathbf{v}=-7.1 \mathbf{i}+3.5 \mathbf{j} \] \( \mathbf{u}= \) Find the unit vector in the direction of \( \mathbf{v} \). \[ \mathbf{v}=-7.1 \mathbf{i}+3.5 \mathbf{j} \] The pressure of a fluid flowing through a closed system, \( P \), after \( t \) seconds can be modeled by the function \( P(t)=100+40 \sin \left(\frac{9 \pi}{4} t\right) \). Answer parts (a) through (c) below. (a) In the interval \( [0,1] \), determine the times at which the pressure of the fluid is 100 mmHg . The solution set is \( \{\square \). (Type an integer or decimal rounded to two decimal places as needed. Use a comma to separate answers as needed.) Explain how the graph of the function \( h(x) = -\sqrt{x} \) is affected by the transformation compared to the original \( f(x) = \sqrt{x} \). An airplane is asked to stay within a holding pattern near an airport. The function \( \mathrm{d}(\mathrm{x})=50 \sin (0.88 \mathrm{x})+140 \) represents the distance d , in miles, that the airplane is from the airport at time x , in minutes. Complete parts a through d below. (a) When the plane enters the holding pattern, \( \mathrm{x}=0 \), how far is it from the airport? It is 140 miles from the airport. (b) During the first 15 minutes after the plane enters the holding pattern, at what time x is the plane exactly 110 miles from the airport? The solution set is \( \left\{\begin{array}{l}\text { ? }\end{array}\right. \) (Round the final answer to two decimal places as needed. Round all intermediate values to three decimal places as needed. Use a comma to separate answers as needed.) An airplane is asked to stay within a holding pattern near an airport. The function \( \mathrm{d}(\mathrm{x})=50 \sin (0.88 \mathrm{x})+140 \) represents the distance d , in miles, that the airplane is from the airport at time x , in minutes. Complete parts a through d below. (a) When the plane enters the holding pattern, \( \mathrm{x}=0 \), how far is it from the airport? It is \( \square \) miles from the airport. The pressure of a fluid flowing through a closed system, \( P \), after \( t \) seconds can be modeled by the function \( P(t)=100+40 \sin \left(\frac{12 \pi}{5} t\right) \). Answer parts (a) through (c) below. (a) In the interval \( [0,1] \), determine the times at which the pressure of the fluid is 100 mmHg . The solution set is \( \{0,0.42,0.83\} \). (Type an integer or decimal rounded to two decimal places as needed. Use a comma to separate answers as needed.) (b) In the interval \( [0,1] \), determine the times at which the pressure of the fluid is 140 mmHg . The solution set is \( \{\square\} \). (Type an integer or decimal rounded to two decimal places as needed. Use a comma to separate answers as needed.)
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