16.12 答复 -- -- 第12章:极地坐标和参数等同

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• 选择节答复 -- -- 第12章:极地坐标和参数等同

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  答复 -- -- 第12章:极地坐标和参数等同

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  Section 12.2: Polar Coordinate System
  ::第12.2节:极地坐标系统

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  1. Because a given point may have multiple descriptions
     ::因为给定点可能有多个描述
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  2. If  $\begin{align*}r<0\end{align*}$ , you extend to the left to measure the distance. If  $\begin{align*}\theta > 360\end{align*}$ , you circle around and continue.
     ::如果 r < 0, 则向左延伸以测量距离。 如果 + 360, 则围绕并继续 。
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  3. See point A below.
     ::见下文A点。
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  4. See point B above.
     ::见上文B点。
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  5. See point C above.
     ::见上文C点。
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  6. See point D above.
     ::见上文D点。
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  7. See point E above.
     ::见上文E点。
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  8. (-1.5, -190°) and (1.5, 0.945π)
     :-1.5,-190°)和(1.5, 0.9450°)
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  9. $\begin{align*}\left(5,\tfrac{5\pi }{3}\right)\end{align*}$  and  $\begin{align*}(\text{-}5,\text{-}60^\circ)\end{align*}$  
     :5,53)和(5,60)
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  10. $\begin{align*}(\text{-}3,\text{-}55)\end{align*}$   and  $\begin{align*}(3,1.7\pi) \end{align*}$  
      :3-3-55)和(3,1,7)
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  11. (4, -150°) and (-4,  $\begin{align*}\tfrac{5\pi}{6}\end{align*}$ )
      :4,-150°)和(4,5°6)
  12.  (500, 105°)
  13. 
      
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  14. a.  $\begin{align*}4.189 \ \text{x} \ 10^7\end{align*}$  
      b.  $\begin{align*}3.837 \ \text{x} \ 10^7\end{align*}$  
      c.  $\begin{align*}3.004 \ \text{x} \ 10^7\end{align*}$  
      ::a. 4.189 x 107 b. 3.837 x 107 c. 3.004 x 107
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  15. a.  $\begin{align*}(4.189 \ \text{x} \ 10^7\!, 30^\circ)\end{align*}$  
      b.  $\begin{align*}(3.837 \ \text{x} \ 10^7\!, \text{-}60^\circ)\end{align*}$  
      c.  $\begin{align*}(3.004 \ \text{x} \ 10^7\!, 135^\circ)\end{align*}$  
      ::a. (4.189x107,30)b.(3.837x107,-60)c.(3.004x107,135)
  16. 
      

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  Section 12.3: Polar Equations
  ::第12.3节:极赤道

  1. 
     
  2. 
     
  3. 
     
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  10. 
      
  11. 
      
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  15. 
      
  16. 
      

   

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  Section 12.4: Polar and Cartesian Transformation
  ::第12.4节:极地和笛卡尔转变

  1. (-5, 0)
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  2. (1.5, 2.6) Point plotted below.
     
     :1.5、2.6) 点图如下。
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  3. (-5, -8.66) Point plotted above.
     :5至8.66)以上点图。
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  4. (-15, 0) Point noted above.
     :15,0)上述点。
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  5. $\begin{align*}\left(\sqrt{50}, \text{-}\tfrac{\pi}{4}\right)\end{align*}$  or (7.07, -0.79)
     $\begin{align*}\left(\sqrt{50}, \tfrac{7\pi}{4}\right)\end{align*}$  or (7.07, 5.50)
     ::或(7.07,-0.79)或(7.07,5.50)或(7.07,5.50)
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  6. $\begin{align*}\left(10, \tfrac{\pi}{2}\right)\end{align*}$  or (10, 1.57)
     $\begin{align*}\left(10, \text{-}\tfrac{3\pi}{2}\right)\end{align*}$  or (10, -4.71)
     :10,%2)或(10,1.57) (10,3,%2)或(10,4.71)
  7. (10, 2.50)
     (10, -3.79)
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  8. Approximate equation is  $\begin{align*}y = .33x.\end{align*}$  This is a line with domain $\begin{align*}x\ge0\end{align*}$ . It starts at the origin and has a slope of about 0.325.
     
     
     ::近似方程式为 Y= 33x。 这是域的直线 x_0。 它从原点开始, 斜度约为 0. 325 。
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  9. $\begin{align*}{x}^{2} + {y}^{2} = 64.\end{align*}$  This is a circle centered at the origin with radius 8.
     
     
     ::x2+y2=64。 这是一个圆, 以原点为中心, 半径为 8 。
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  10. $\begin{align*}y = 7.\end{align*}$  This is a horizontal line through (0, 7).
      
      
      ::y=7. 这是一条通过 0, 7 的水平线 。
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  11. $\begin{align*}x = \text{-}3.\end{align*}$  This is a vertical line through (-3, 0).
      
      
      ::x=-3. 这是一条直径( 3, 0) 的垂直线 。
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  12. $\begin{align*}r = 2\cos{\theta}.\end{align*}$   This is a circle with center at (1, 0) and radius 1.
      
      
      ::r=2cos。 这是一个圆形, 中心在 1, 0 和 半径 1 。
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  13. $\begin{align*}\sin{\theta} =\sqrt{3}\cos{\theta}.\end{align*}$  This is a line with slope equal to the square root of 3.
       
      
      ::这是一条斜坡等于3平方根的直线。
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  14.   $\begin{align*}r\sin{\theta} = \text{-}5.\end{align*}$  This is a horizontal line through (0, -5).
      
      
      ::rsin% - 5。 这是一条通过 0, - 5 的水平线 。
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  15. $\begin{align*}r\cos{\theta }\end{align*}$  ×  $\begin{align*}r\sin{\theta } = 15\end{align*}$  or  $\begin{align*}{r}^{2}\cos{\theta}\sin{\theta} = 15.\end{align*}$  This is a rational function centered at the origin.
      
      
      ::rcos × rsin 15 或 r2cossin 15。 这是一个以源代码为中心的合理函数 。
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  16. $\begin{align*}\sqrt{\tfrac{4,380}{\pi}} = 132 \cos(5\theta)\end{align*}$  
      ::4,380132cos(5__)
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  17. $\begin{align*}l = \tfrac{\sqrt{\tfrac{7,542}{\pi}}}{\cos{5\theta}}\end{align*}$  
      ::7,542cos5
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  18. $\begin{align*}y = \frac{ \tfrac{6}{17}-x \cos{25^\circ} } { \sin{25^\circ} }\end{align*}$  or  $\begin{align*}y \approx \text{-}2.14x + 0.835\end{align*}$  
      ::y=617-xcos25sin25或y-2.14x+0.835

   

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  Section 12.5: Systems of Polar Equations
  ::第12.5节:极赤道系统

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  1. They intersect twice. 
     ::它们交叉了两次
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  2. Once in the 1st and once in the 4th quadrant 
     ::一次在第一 一次在第四象限
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  3. They intersect at  $\begin{align*}(1, \frac{\pi }{3})\end{align*}$  and  $\begin{align*}(1, \frac{5\pi }{3}).\end{align*}$  
     ::相交于(1,3,3)和(1,5,3)之间。
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  4. 3 points of intersection
     ::3个交叉点
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  5. Points of intersection are  $\begin{align*}(0, 0), (\frac{1}{2}, \frac{\pi }{2}), \end{align*}$  and  $\begin{align*}(\frac{1}{2}, \frac{5\pi }{3}).\end{align*}$  
     ::交叉点是(0,0,(12,)2)和(12,513)。
  6. $\begin{align*}(2, 0)\end{align*}$  
  7. $\begin{align*}(0, 0)\end{align*}$  
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  8. $\begin{align*}\left(2+\sqrt{2}, \tfrac{3\pi}{4}\right) \ \text{and} \ \left(2-\sqrt{2}, \tfrac{7\pi}{4}\right)\end{align*}$    
     :2,34)和(2,2,74)
  9. $\begin{align*}(\frac{3}{2}, \frac{\pi }{3}), (\frac{3}{2}, \frac{5\pi }{3})\end{align*}$  
  10. $\begin{align*}(\sqrt{2}, \frac{\pi }{4}), (\sqrt{2}, \frac{3\pi }{4})\end{align*}$                                  
  11. $\begin{align*}(0, 0), (1, 0)\end{align*}$                                        
  12. $\begin{align*}(0, 0), (\frac{\sqrt{3}}{2}, \frac{2\pi }{3}), (\frac{\sqrt{3}}{2}, \frac{\pi }{3})\end{align*}$                     
  13. $\begin{align*}(0, 0), (2\sqrt{2}, \frac{5\pi }{4})\end{align*}$                             
  14. (1, 276°), (2.44, 313°)                           
  15. (0, 0), (1.08, 95°), (1,77, 142°), (1.77, 218°), (1.08, 265°) 

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  Section 12.6: Polar Equations of Conics
  ::第12.6节:二次曲线极赤道

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  1. $\begin{align*}8{x}^{2} - 10x + 9{y}^{2} - 25 = 0;\end{align*}$  ellipse
     ::8x2- 10x+9y2- 25=0; 椭圆
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  2. $\begin{align*}3{x}^{2} - 8x + 4{y}^{2} - 16 = 0; \end{align*}$  ellipse
     ::3x2-8x+4y2 - 16=0; 椭圆
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  3. $\begin{align*}3{x}^{2} - 4x + 4{y}^{2} - 4 = 0; \end{align*}$  ellipse
     ::3x2 - 4x+4y2 - 4=0; 椭圆
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  4. $\begin{align*}-12{x}^{2}- 24x + 4{y}^{2} - 9 = 0;\end{align*}$  hyperbola
     ::- 12x2 - 24x+4y2 - 9=0;双波
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  5. $\begin{align*}{x}^{2} - 5x + {y}^{2} = 0;\end{align*}$  circle
     ::x2 - 5x+y2=0; 圆
  6. 
           
  7. 
     
  8.              
     
  9. 
            
  10. 
      
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  11. $\begin{align*}r = \frac{3}{2-\cos{\theta }}\end{align*}$  
      
      ::r=32-cos____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  12. $\begin{align*}r = 10\cos{\theta } - 24\sin{\theta }\end{align*}$  
      
      ::r=10cos24sin
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  13. $\begin{align*}r = \text{-}2\sin{\theta }\end{align*}$  
      ::r=-2sin
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  14. $\begin{align*}r = 2\cos{\theta }\end{align*}$  
      ::r=2cos
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  15. $\begin{align*}r = \frac{1}{1-2\cos{\theta }}\end{align*}$  
      ::r=11-2cos_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  16. $\begin{align*}e=1\end{align*}$ , parabola
      ::e=1, 抛物线
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  17. $\begin{align*}e = \tfrac{1}{2},\end{align*}$  ellipse 
      ::e=12, 椭圆
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  18. $\begin{align*}e=1\end{align*}$ , parabola 
      ::e=1, 抛物线
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  19. $\begin{align*}e=2\end{align*}$ , hyperbola 
      ::e=2,双波
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  20. Answers will vary. One example: On the rectangular grid, you graph points based on  $\begin{align*}x\end{align*}$  and  $\begin{align*}y\end{align*}$  values (how far something is right/left or above/below the origin). On the polar grid, you graph points based on angles and the length of a radius.
      ::答案会有所不同。例如:在矩形网格中,根据 x 和 y 值(东西的右/ 左或上/ 上/ 下) 绘制的图形点。在极地网格中,根据角度和半径长度绘制的图形点。

   

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  Section 12.7: Polar Form of Complex Numbers
  ::第12.7节:复杂数字的极表形式

  

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  1. See graph above.  $\begin{align*}\left(\sqrt{2}, 45^\circ\right)\end{align*}$
     ::见上文图表。 (第2,45__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  2. See graph above.  $\begin{align*}\left(\sqrt{2}, 135^\circ\right)\end{align*}$  
     ::见上文图表。 (第2,135__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  3. See graph above.  $\begin{align*}\left(6, 270^\circ\right)\end{align*}$  
     ::见上文图表(6,270)
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  4. See graph above.  $\begin{align*}\left(\sqrt{2}, 45^\circ\right) \end{align*}$  
     ::见上文图表。 (第2,45__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  5. See graph above.  $\begin{align*}\left(\sqrt{2}, \text{-}45^\circ\right)\end{align*}$  
     ::见上文图。 (%2,-45_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  6. See graph above.  $\begin{align*}\left(2, 0^\circ\right)\end{align*}$
     
     ::见上文图表(2,0)
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  7. See graph above.  $\begin{align*}\left(2, 60^\circ\right)\end{align*}$       
     ::见上文图表(2,60)
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  8. See graph above.  $\begin{align*}\left(2, \text{-}30^\circ\right)\end{align*}$
     
     ::见上文图表(2,30)
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  9. See graph above.  $\begin{align*}\left(4, 30^\circ\right)\end{align*}$  
     ::见上文图表(4,30)
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  10. $\begin{align*}4.25 + 4.25i\end{align*}$  
      ::4.25+4.25i
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  11. $\begin{align*}\text{-}7.5 + 13i\end{align*}$  
      ::-7.5+13i
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  12. $\begin{align*}6 + 10.4i\end{align*}$  
      ::6+10.4i 6+10.4i
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  13. $\begin{align*}\left(\sqrt{{x}^{2} + {y}^{2}}, \frac{1}{sin}\left(\tfrac{y^2}{r^2}\right) \right)\end{align*}$    
      $\begin{align*}(\cos{\theta } + i\sin{\theta })\end{align*}$  using $\begin{align*}\theta \end{align*}$  from above
      :x2+y2,1sin(y2r2))(cosisin) 使用上面的
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  14. Zeros: 2 + 2i, 2 – 2i
      
       
      ::零:2+2i,2-2i
  15. (0, 0°)
  16. $\begin{align*}\left(\tfrac{\sqrt{3}-1}{2}, 0^\circ\right)\end{align*}$  
  17. $\begin{align*}\left(\sqrt{13}, 213.69^\circ\right)\end{align*}$  
  18. (4, -30°)

     

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  Section 12.8: Product and Quotient Theorems
  ::第12.8节:产品和引号理论

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  1. $\begin{align*}4\sqrt{2}\left(\cos{15^\circ}+i\sin{15^\circ}\right)\end{align*}$
     ::42(cos15isin15)
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  2. $\begin{align*}8cis(60\end{align*}$ °)
     ::8cis( 60°)
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  3. $\begin{align*}\frac{1}{3}cis(\text{-}120^\circ)\end{align*}$
     ::13CIS(-120)
  4. 60°
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  5. $\begin{align*}33.81 + 9.06i\end{align*}$  
     ::33.81+9.06i
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  6. $\begin{align*}.36+1.35i\end{align*}$  
     ::36+1.35i
  7. $\begin{align*}\frac{3}{16}\end{align*}$  
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  8. $\begin{align*}0 + 40i\end{align*}$  
     ::0+40i
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  9. $\begin{align*}1.4 + \frac{4}{5}i\end{align*}$  
     ::1.4+45i
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  10. $\begin{align*}\frac{1}{2} + \frac{5}{16}i\end{align*}$  
      ::12+516i
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  11. $\begin{align*}\text{-}32+32i\sqrt{3}\end{align*}$  
      ::-32+32i%3
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  12. $\begin{align*}0 + 125i\end{align*}$  
      ::0+125i
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  13.   $\begin{align*}2\sqrt{3}+2 + (2\sqrt{3}-2)i\end{align*}$   
      ::3+2+2+(2+3-2)i
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  14. $\begin{align*}4 + 4\sqrt{3}i\end{align*}$  
      ::4+43i
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  15. $\begin{align*}3 + 2.6i\end{align*}$  
      ::3+2.6i 3+2.6i
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  16. $\begin{align*}\text{-}\tfrac{1}{6} - \tfrac{2}{7}i\end{align*}$  
      ::-16-27i
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  17. $\begin{align*}\text{-}\tfrac{23}{40}- \tfrac{27}{56}i\end{align*}$    
      ::-2340-22756i
  18. $\begin{align*}\left(\tfrac{32}{7}, 30^\circ\right)\end{align*}$  
  19. $\begin{align*}\left(84, 37^\circ\right)\end{align*}$  

   

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  Section 12.9: Powers and Roots of Complex Numbers
  ::第12.9节:复杂数字的权力和根源

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  1. $\begin{align*}\frac{\text{-}1}{2} + \frac{5}{2}i\end{align*}$  
     
     ::-12+52i
  2. $\begin{align*}37\end{align*}$  
     
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  3. $\begin{align*}(\frac{1}{2} - \frac{i\sqrt{3}}{2})\end{align*}$  
     
     :12-i32)
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  4. $\begin{align*}4\sqrt{2}(\cos{15^\circ}+i\sin{15^\circ} )\end{align*}$
     
     ::42(cos15isin15)
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  5. $\begin{align*}8cis(60^\circ)\end{align*}$
     
     ::8CIS( 60)
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  6. $\begin{align*}4cis(\frac{9\pi }{40})\end{align*}$  
     ::4Cisc(940)
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  7. $\begin{align*}\tfrac{1}{3}cis\left(\text{-}120^\circ\right)\end{align*}$  
     
     ::13CIS(-120)
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  8. $\begin{align*}\tfrac{3}{4}cis\left(\text{-}140^\circ\right)\end{align*}$  
     
     ::34Cis(-140)
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  9. $\begin{align*}\text{-}\frac{27}{2} - \frac{27\sqrt{3}}{2}i\end{align*}$  
     
     ::- 272-2732i
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  10. $\begin{align*}\text{-}2\sqrt{2} - 2\sqrt{2}i\end{align*}$  
      
      ::-2-2-22i
  11. $\begin{align*}\text{-}64\end{align*}$  
      
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  12. $\begin{align*}(\sqrt[6]{2})cis15\end{align*}$ °,  $\begin{align*}\sqrt[6]{2}cis135\end{align*}$ °,   $\begin{align*}\sqrt[6]{2}cis255\end{align*}$ ° 
      
      :6_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  13. $\begin{align*}2cis67.5\end{align*}$ °,  $\begin{align*}2cis157.5\end{align*}$ °,   $\begin{align*}2cis247.5\end{align*}$ °, $\begin{align*}2cis337.5\end{align*}$ °
      ::2C67.5°, 2C157.5°, 2C247.5°, 2C337.5°
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  14. $\begin{align*}cis18\end{align*}$ °, $\begin{align*}cis90\end{align*}$ °,  $\begin{align*}cis162\end{align*}$ °,  $\begin{align*}cis234\end{align*}$ °, $\begin{align*}cis306\end{align*}$ °        
      
      ::cis18°, cis90°, cis162°, cis234°, cis306°

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  Section 12.10: Parameters and Parameter Elimination
  ::第12.10节:参数和参数去除

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  1. $\begin{align*}y = \tfrac{4}{9}{x}^{2} + \tfrac{2}{9}x - \tfrac{2}{9}\end{align*}$  
     ::y=49x2+29x-29
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  2. $\begin{align*}x = \tfrac{3}{4}{y}^{2} + \tfrac{9}{2}y + \tfrac{15}{4}\end{align*}$  
     ::x=34y2+92y+154
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  3. $\begin{align*}y = {x}^{2}\end{align*}$  
     ::y=x2 y=x2
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  4. $\begin{align*}y = {x}^{3} + 15{x}^{2} + 75x + 126\end{align*}$  
     ::y=x3+15x2+75x+126
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  5. $\begin{align*}y = {x}^{2} - 8x + 11\end{align*}$  
     ::y=x2 - 8x+11 y=x2 - 8x+11
  6. 
     
  7. 
     
  8. 
     
  9. 
     
  10. 
      
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  11. $\begin{align*}{x}^{2} + \left({\frac{y + 4}{3}}\right)^{2} = 1\end{align*}$  
      
      
      ::x2+(y+43)2=1
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  12. $\begin{align*}{(x-1)}^{2} + {(y - 1)}^{2} = 4\end{align*}$  
      
      
      :x-1)2+(y-1)2=4
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  13. $\begin{align*}x = 2 + 3\cos{t}; \ y = 4 + 3\sin{t}\end{align*}$  
      ::x=2+3cost;y=4+3sint
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  14. $\begin{align*}x = 2\cos{t}; \ y = 5\sin{t}\end{align*}$  
      ::x=2 成本; y=5sint
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  15. $\begin{align*}x = 3\cos{t} + 4; \ y = 6\sin{t} - 1\end{align*}$  
      ::x=3cost+4;y=6sint-1

   

  播放段落

  Section 12.11: Parametric Inverses
  ::第12.11节:参数反数

  播放段落
  1. $\begin{align*}x = {t}^{2} + 2; \ y = t - 4\end{align*}$  
     ::x=t2+2; y=t- 4
  播放段落
  2. Function
     ::函数职能职能职能职能职能职能职能职能
  播放段落
  3. Neither
     ::中 无
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  4. $\begin{align*}x = 4 - t; \ y = {t}^{2}\end{align*}$  
     ::x=4 - t; y=t2 x=4 - t; y=t2
  播放段落
  5. Inverse
     ::反逆
  播放段落
  6. Relation
     ::关系关系
  播放段落
  7. $\begin{align*}x = {t}^{2} - 3; \ y = 2t - 1\end{align*}$  
     ::x=t2 - 3; y=2t - 1
  播放段落
  8. Inverse
     ::反逆
  播放段落
  9. Function
     ::函数职能职能职能职能职能职能职能职能
  播放段落
  10. $\begin{align*}x = {t}^{2}-2t; \ y = 3t+14\end{align*}$  
      ::x=t2- 2t; y=3t+14
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  11. When  $\begin{align*}t = \text{-}2\end{align*}$  at  $\begin{align*}(8, 8)\end{align*}$  and when  $\begin{align*}t = 7\end{align*}$  at  $\begin{align*}(35, 35)\end{align*}$
      ::当 t = 2 时( 8, 8) , 当 t = 7 时( 35, 35)
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  12. $\begin{align*}x = 4t-4; \ y = {t}^{2}\end{align*}$  
      ::x=4t- 4; y=t2
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  13. When $\begin{align*}t=2\end{align*}$  at  $\begin{align*}(4, 4)\end{align*}$
      ::当 t=2 时( 4, 4)
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  14. $\begin{align*}x = t; \ y = t^2 + t - 6\end{align*}$  
      
      
      ::x=t; y=t2+t- 6 x=t; y=t2+t- 6
  播放段落
  15. $\begin{align*}x = t; \ y = t^2 + 3t + 2\end{align*}$  
      
      
      ::x=t; y=t2+3t+2

   

  播放段落

   Section 12.12: Applications of Parametric Equations
  ::第12.12节:参数等量的应用

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  1.   $\begin{align*}x = \text{-}50\sin{\frac{2\pi }{5}t}; \end{align*}$   $\begin{align*}y = \text{-}50\cos({\frac{2\pi }{5}t}) + 53\end{align*}$  
     ::x=- 50sin2°5t; y=- 50cos(2°5t)+53
  2. $\begin{align*}(\text{-}29.4, 93.1)\end{align*}$  
  3. $\begin{align*}(47.55, 37.55)\end{align*}$  
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  4. $\begin{align*}x = \text{-}40\sin({\frac{\pi }{3}(t + 1)}); \end{align*}$   $\begin{align*}y = \text{-}40\cos({\frac{\pi }{3}}(t + 1)) + 43\end{align*}$  
     ::x=- 40sin( 3( t+1); y=- 40cos( 3 ( t+1))+43
  5. $\begin{align*}(0, 83)\end{align*}$  
  6. $\begin{align*}(34.6, 23)\end{align*}$  
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  7. $\begin{align*}x = t \cdot 73.33 \cdot \cos{\frac{\pi }{4}}; \end{align*}$   $\begin{align*}y = \frac{1}{2} \cdot (\text{-}32) \cdot {t}^{2} + t \cdot 73.33 \cdot \sin({\frac{\pi }{4}}) + 5\end{align*}$  
     ::x=t73.33cos4;y=12(- 32)\t2+t73.33sin(- 4)+5
  8. $\begin{align*}(103.7, 44.7)\end{align*}$  
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  9. $\begin{align*}172.7\end{align*}$  feet in about  $\begin{align*}3.33\end{align*}$  seconds
     ::大约3.33秒内172.7英尺
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  10. $\begin{align*}x = t \cdot 102.67 \cdot \cos({\frac{\pi }{3}}) + 8.8t; \end{align*}$   $\begin{align*}y = \frac{1}{2} \cdot (\text{-}32) \cdot {t}^{2} + t \cdot 102.67 \cdot \sin({\frac{\pi }{3}}) + 7\end{align*}$  
      ::x=t102.67cos(3)+8.8t;y=12(-32)t2+t102.67sin(3)+7
  11. $\begin{align*}(120.27, 120.83)\end{align*}$  
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  12. $\begin{align*}338.56 \end{align*}$  feet in about  $\begin{align*}5.63\end{align*}$  seconds
      ::338.56英尺,约5.63秒
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  13. $\begin{align*}x\end{align*}$ ₁  =  $\begin{align*}\text{-}t \cdot 105.6 \cdot \cos({\frac{\pi }{3}}) + 250 + 8.8t; \end{align*}$     $\begin{align*}y\end{align*}$ ₁  =  $\begin{align*}\frac{1}{2} \cdot (\text{-}32) \cdot {t}^{2} + t \cdot 105.6 \cdot \sin({\frac{\pi }{3}})\end{align*}$  
      ::x1 = - t105.6.6cos 3+250+8. 8t; y1 = 12( 32) t2+t105. 6.6sin 3)
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  14. $\begin{align*}x\end{align*}$ ₂ =  $\begin{align*}t \cdot 95.33 \cdot \cos({\frac{\pi }{4}}) + 8.8t; \end{align*}$   $\begin{align*}y \end{align*}$ ₂  =  $\begin{align*}\frac{1}{2} \cdot (\text{-}32) \cdot {t}^{2} + t \cdot 95.33 \cdot \sin({\frac{\pi }{4}})\end{align*}$  
      ::x2 = t95.33cos( 4) + 8.8t; y2 = 12( 32) t2+ t95.33sin( 4)
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  15. While the graphs intersect, each ball passes through the point of intersection at a different time.
      
      
      ::当图形交叉时,每个球会在不同的时间穿过交叉点。