1 MATH 108 Quiz 3 This quiz is worth 40 points. Questions 1 - 4 are each worth 4 points; do not show your work. Questions 5 – 8 are each worth 6 points; show your work to earn full credit. Write...

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1 MATH 108 Quiz 3 This quiz is worth 40 points. Questions 1 - 4 are each worth 4 points; do not show your work. Questions 5 – 8 are each worth 6 points; show your work to earn full credit. Write answers on the answer sheet that is provided with this quiz. Leave answers in exact form unless otherwise directed to approximate the results. Write all fractions in lowest form. Round decimals to hundredths unless otherwise directed to round. Write answers using positive exponents except when using scientific notation. Simplify all radicals and rationalize the denominators. Write complex numbers in the form a + bi. Applied problems must have the variables identified (Let x =) and an equation for full credit. Reminder: When asked for component form, it will be rectangular (x, y), polar (r, θ), or vector . Select the letter of the one alternative that best completes the statement or answers the question. Check that the answers are all letters (A, B, C, D, or E) and not a value. 1.) Find the rectangular coordinates of (10, ? 3 ). A) (5, −5√3) B) ( √3 2 , 10) C) ( √3 2 , 5) D) (5√3, 5) E) None of these 2.) The letters r and θ represent polar coordinates. Write the equation r – 9cos θ = 7sin θ using rectangular coordinates (x, y). A) ?2 + 9? + ?2 = 2? B) ?2−9? + ?2 − 7? = 0 C) √?2 + ?2 = 16 + ?? D) ?2 + 9? + ?2 + 7? = 0 E) None of these 3.) Find ||3w - v|| if v = -4i + 7j and w = -i - 3j. A) √410 B) √397 C) √257 D) 19 E) None of these 4.) Find the angle in degrees between �⃗� = ⟨ 2, −5⟩ and ? = ⟨1, 4⟩. A) 144.16° B) -18° C) 67.81° D) 125.32° E) None of these The following questions are each worth 6 points. Show your work. 5.) Find the exact polar coordinates of the points of intersection of these two polar equations: r = 1 – 2 sin θ and r = 2. 6.) Find the vector projection of ? onto �⃗⃗� given that ? = 〈2, −3〉 and �⃗⃗� = 〈5, 7〉. 7.) A 350 pound block is hanging on two cables which are attached to the ceiling. The left hand cable makes a 55° angle with the ceiling while the right hand cable makes a 37° angle with the ceiling. What is the tension on each of the cables? 8) Use the polar grid function of the Desmos Graphing Calculator or any other graphing calculator to copy and paste the graph of the polar equation: r = 4 cos 5θ on the interval (-2π, 2π). (3 points) Also include any one point on the graph of the equation. Label the point. (3 points) Hint: select any θ, calculate r, and then, determine (x, y). Next, use the label function to identify the point. (The label should be values, not the letters x and y.) Use the software, do not post a hand-drawn graph and do not label point by hand. MATH 108 Quiz 4 This quiz is worth 40 points. Questions 1 - 4 are each worth 4 points; do not show your work. Questions 5 – 8 are each worth 6 points; show your work to earn full credit. Write answers on the answer sheet that is provided with this quiz. Leave answers in exact form unless otherwise directed to approximate the results. Write all fractions in lowest form. Round decimals to hundredths unless otherwise directed to round. Write answers using positive exponents except when using scientific notation. Simplify all radicals and rationalize the denominators. Write complex numbers in the form a + bi. Applied problems must have the variables identified (Let x =) and an equation for full credit. Select the letter of the one alternative that best completes the statement or answers the question. Check that the answers are all letters (A, B, C, D, or E) and not a value. 1.) Find the equation in standard form for the hyperbola that has center (4, -1), focus (7, -1), and vertex (6, -1). A) (?−1)2 4 − (?+4)2 5 = 1 B) (?−4)2 5 − (?+1)2 4 = 1 C) (?−4)2 4 + (?+1)2 5 = 1 D) (?−4)2 4 − (?+1)2 5 = 1 E) None of these 2.) Graph the equation: (?+1)2 9 + (?+2)2 4 = 1. A) B) C) D) E) None of these 3.) Find the asymptotes of 4?2 − ?2 = 16. A) ? = ± 1 2 ? B) ? = ± 2? C) ? = ±4 ? D) ? = ± 1 4 ? E) None of these 4.) Fill in the blank: ______________________ A sealed-beam spotlight is in the shape of a paraboloid of revolution. The bulb, which is placed at the focus, is 12 centimeters from the vertex. If the depth is to be 20 centimeters, what is the approximate diameter of the headlight at its opening? The following questions are each worth 6 points. Show your work. 5.) An (half) elliptical tunnel arch is constructed which is 20 feet wide at the base and 20 feet tall in the middle. Find the approximate height of the arch exactly 1 foot in from the base of the arch. 6.) Find the equation in standard form for the parabola that has focus (10, 1) directrix is x = 5. 7.) a. Write the equation in standard form of the conic 4?2 + ?2 − 8? − 2? + 1 = 0. (3 points) b. Using the Desmos Graphing Calculator or any other graphing calculator, plot one graph that has the conic in Part a and the center of that graph. Label the point by giving the x and y coordinates. Use appropriate domain and range to include the graph of the conic. Use the software, do not post a hand-drawn graph and do not label point by hand. (3 points) 8.) a. Write the equation in standard form of the conic ?2 − 4? − 2? = 0. (3 points) b. Using the Desmos Graphing Calculator or any other graphing calculator, plot one graph that has the conic in Part a and the vertex and focus of that graph. Label the points by giving the x and y coordinates. Use appropriate domain and range to include the graph of the conic. Use the software, do not post a hand-drawn graph and do not label point by hand. (3 points) MATH 108 Quiz 5 This quiz is worth 40 points. Questions 1 - 4 are each worth 4 points; do not show your work. Questions 5 – 8 are each worth 6 points; show your work to earn full credit. Write answers on the appropriate numbered blank. You can insert additional lines if needed. Leave answers in exact form unless otherwise directed to approximate the results. Write all fractions in lowest form. Round decimals to hundredths unless otherwise directed to round. Write answers using positive exponents except when using scientific notation. Simplify all radicals and rationalize the denominators. Write complex numbers in the form a + bi. Applied problems must have the variables identified (Let x =) and an equation for full credit. Select the letter of the one alternative that best completes the statement or answers the question. Check that the answers are all letters (A, B, C, D, or E) and not a value. 1.) Write the system of equations associated with the augmented matrix. Do not solve. [ 5 5 5 8 3 7 0 −9 4 0 6 8 ] A) { 5? + 5? + 5? = 8 3? + 7? = −9 4? + 6? = 8 B) { 5? + 5? + 5? = 8 3? + 7? = −9 4? + 6? = 8 C) { 5? + 5? + 5? = 8 3? + 7? = −9 4? + 6? =