MS 101 Alg & Trig test seven • 16 July 2014 Name:

1. Day	Density (kg/m³)
   1	156
   2	160
   3	161
   4	162.5
   5	163
   6	163.5
   7	164
   8	165.5
   9	166
   10	166.5
   11	166.7
   12	167
   13	167.5
   14	169
   15	169.5
   16	170
   17	171

   For the athletes in the MicroGames this summer top performance will depend in part on having the best equipment and gear. Running shoes lose their elastic rebound and cushion with age. The ethylene-vinyl acetate foams used in the midsole break down, crush, and increase in density with use. The data is based on placing running shoes with EVA forumula 147 into an impact machine for 120 minutes a day to simulate two hours of working out. The increase in density each day was recorded. (p140, Verdejo, 2003). Fit a logarithmic trend line to this data.
   
   1. Graph the data, add a LOGARITHMIC trend line and equation to the graph, and then write the logarithmic equation:
   2. ____________ Based on the logarithmic equation in part a, calculate the density for day 36.
   3. ____________ Based on the logarithmic equation in part a, calculate the day when the density would be predicted to be 175 kg/m³
   4. Solve the equation in part a for x, writing out the resulting inverse equation below:
2. Calculate or solve for x the following expressions and equations.
   1. ____________ 14⁰
   2. ____________ log₁₄14
   3. ____________ log 10000
   4. ____________ e^x = 8103.09
   5. ____________ ln x = 2.7726
   6. ____________ log₅x = 2
   7. ____________ 6 ln(6x) = 32.2517
   8. ____________ $e^{\left(\frac{x}{7}\right)}=1096.6332$
   9. ____________ ln(2) + ln (32) = ln(x)
   10. ____________ 2 log(3) − log(x + 144)= −2 log(5)
   11. ____________ ${\mathrm{x}}^{{\log }_{\mathrm{x}}\mathrm{x}}=100$
3. _________ The Pohnpei State track has been redone with a Conica Conipur track surface. Lane six of the newly refurbished and resurfaced track is 109 meters long. What is the radius of lane six in meters?
4. Convert the following angles in radians to degrees
   $\frac{2\mathrm{\pi }}{5}$ = _________° $\frac{\mathrm{\pi }}{5}$ = _________° $\frac{\mathrm{\pi }}{10}$ = _________°
5. Convert 40° to radians expressed as π over a number:
6. Convert 285° to radians expressed as π over a number:
7. The following chart displays three functions. Each function is shown using a marker of a different shape: circles, squares, and triangles.
   
   1. __________ What marker shape is f(x) = e^x?
   2. __________ What marker shape is f(x) = e^−x?
   3. __________ What marker shape is function f(x) = $\frac{24}{(1+e^{-3x})}-12$ ?
   4. ________________ What is the name of the type of function seen in letter c?
   5. ________________ On the 12th of June I introduced the function seen in letter c as being characteristic of the time versus velocity for what sport?
8. Calculate the following values:
   1. __________________ Calculate: 100 sin(72°)
   2. __________________ Calculate: 100 cos(72°)
   3. __________________ Calculate: 100 tan(72°)
   4. __________________ Calculate: 100 sin(36°)
   5. __________________ Calculate: 100 cos(36°)
   6. __________________ Calculate: 100 cos(36°)
   7. __________________ Calculate: 100 sin(18°)
   8. __________________ Calculate: 100 cos(18°)
   9. __________________ Calculate: 100 tan(18°)
   10. __________________ Calculate: 100 sin(0°)
   11. __________________ Calculate: 100 cos(0°)
   12. __________________ Calculate: 100 tan(0°)
   13. __________________ Calculate: 100 tan(90°)
9. Determine the SVG coordinates of the five vertices (corners) of a pentagon inscribed in a circle. The center of the circle is (0, 0). The radius of the circle is 100. Remember that in SVG coordinates +y is down, -y is up.
   
   1. ( ________ , _________ )
   2. ( ________ , _________ )
   3. ( ________ , _________ )
   4. ( ________ , _________ )
   5. ( ________ , _________ )
   Wave equation in terms of wavelength λ and distance x: $y=\mathrm{a}sin\left(\frac{2\mathrm{\pi }x}{\mathrm{\lambda }}\right)$
   Wave equation in terms of period τ and time t: $y=\mathrm{a}sin\left(\frac{2\mathrm{\pi }t}{\mathrm{\tau }}\right)$
   Period τ and frequency f relationship: $\mathrm{\tau }=\frac{1}{f}$
10. A RipStik lays down a wave according to the equation: $y=16sin\left(\frac{2\mathrm{\pi }x}{32}\right)$
    1. a = _______________ Write the amplitude a of the equation.
    2. λ = _______________ Write the wavelength λ of the equation.
11. A spring oscillates vertically according to the equation: $y=40sin\left(\frac{2\mathrm{\pi }t}{2.7}\right)$
    1. a = _______________ Write the amplitude a of the equation.
    2. τ = _______________ Write the period τ of the equation.
    3. f = _______________ Calculate the frequency f of the equation.
12. Sketch a graph of the following function: $y=50cos\left(\frac{2\mathrm{\pi }x}{12}\right)$
    
13. A RipStik was swizzled across a large sheet of paper. The swizzle wave can be seen in the diagram.
    
    1. ________ How many waves are shown in the diagram?
    2. λ = _______________ Write the wavelength λ
    3. a = _______________ Write the amplitude a.
    4. τ = _______________ Write the period τ.
    5. f = _______________ Calculate the frequency f.
    6. y = ______ sin( _________ x). Write out the equation of the wave in terms of the wavelength λ and distance x.
14. Depth charges are explosives that detonate at a specific depth d under the surface of the water. A destroyer armed with depth charges is using a sonar to determine the angle θ and the distance r to a submarine.
    1. Write an equation for the angle θ in terms of r and d:
    2. d = _________ If the angle θ = 40° and the range r is 156 meters, what is the value of the depth d?
15. Solve 5 sin x = 3 for x in degrees
16. Solve 4(tan x)² − 3 = 0 for x in degrees
17. Given the equation for velocity, the launch angle θ, and the distance:
    $\text{distance d}=\frac{2v^2\sin \left(\text{θ}\right)\cos \left(\text{θ}\right)}{\mathrm{9.8}}$
    For a MicroGames javelin thrown at a velocity v = 8 m/s at θ = 40°
    
    1. v_x = ______________ Calculate the horizontal velocity v_x
    2. v_y = ______________ Calculate the vertical velocity v_y
    3. d = _____________Calculate the distance d the javelin will travel.
18. For the vector $\stackrel{\rightharpoonup }{u}$ = 21 i + 20 j
    1. magnitude = __________ Calculate the magnitude.
    2. θ = __________ Calculate the direction angle in degrees.
19. A shotput is thrown with a force of 55 Newtons at a direction angle of 33° to the horizontal. Write the force vector in i, j component form.
    
    $\stackrel{\rightharpoonup }{F}$ = ________ i + ________ j
20. A force F of 6000 Newtons is being exerted at an angle of 20° to the horizontal on an object.
    1. F ⋅ d ___________ Calculate the energy (work) done to move the object 10 meters using the dot product.
    2. F ⨯ d ___________ Calculate the torque on the object if the lever arm distance is 0.3 meters using the cross product.
21. Given a matrix $A=\left[\begin{array}{cc}7& 21\\ 20& 02\end{array}\right]$ and a matrix $B=\left[\begin{array}{cc}7& 20\\ 20& 14\end{array}\right]$
    
    Write the matrix A + B | Write the matrix A − B | Write the matrix 2A