42. Air Traffic Control Suppose you are an air traffic controller directing the pilot of a plane on a hyperbolic flight path. You and another air traffic controller from a different airport send radio signals to the pilot simultaneously. The two airports are 48 km apart. The pilot's instrument panel tells him that the signal from your airport always arrives 100   μ s 100 mu s  (microseconds) before the signal from the other airport.
    1. Which airport is the plane closer to?
    2. If the signals travel at a rate of 300  m / μ s , 300 , m slash mu s comma  what is the difference in distances from the plane to the two airports?
    3. Write the equation of the flight path. ( Hint : k = 2 a ) open , cap hint , colon k equals 2 eh close
    4. Draw the hyperbola. Which branch represents the flight path?

Standardized Test Prep

SAT/ACT

43. The graph of x 2 16 − y 2 4 = 1 fraction x squared , over 16 end fraction , minus , fraction y squared , over 4 end fraction , equals 1  is a hyperbola. Which set of equations represents the asymptotes of the hyperbola's graph?
    1. y = 1 2 x , y = − 1 2 x y equals , 1 half , x comma y equals negative , 1 half , x
    2. y = 2 x , y = − 2 x y equals 2 x comma y equals negative 2 x
    3. x = 1 2 y , x = − 1 2 y x equals , 1 half , y comma x equals negative , 1 half , y
    4. y = 1 4 x , y = − 1 4 x y equals , 1 fourth , x comma y equals negative , 1 fourth , x

44. Simplify 1 y − 1 x 1 x y − 1 . fraction 1 over y , minus , 1 over x , over fraction 1 , over x y end fraction , minus 1 end fraction . .

    6. − y − x x y − 1 negative . fraction y minus x , over x y minus 1 end fraction
    7. x − y 1 − x y fraction x minus y , over 1 minus x y end fraction
    8. x + y 1 + x y fraction x plus y , over 1 plus x y end fraction
    9. x + y
45. How is the graph of y = 4 · ( 1 2 ) x − 3 y equals 4 middle dot . open , 1 half , close super x minus 3 end super  translated from the graph of y = 4 · ( 1 2 ) x ? y equals 4 middle dot . open , 1 half , close to the x . question mark
    1. 3 units right
    2. 3units left
    3. 3 units down
    4. 3 units up

Short Response

46. Using sigma notation, what is an expression for the sum of a 6-term arithmetic sequence with first term of 3 and a common difference of 4? What is the sum?

Mixed Review

See Lesson 10-4.

Find the foci for each equation of an ellipse. Then graph the ellipse.

47. x 2 34 + y 2 25 = 1 fraction x squared , over 34 end fraction , plus , fraction y squared , over 25 end fraction , equals 1
48. 3 x 2 + 2 y 2 = 6 3 x squared , plus 2 , y squared , equals 6
49. 25 x 2 + 16 y 2 = 1600 25 x squared , plus 16 , y squared , equals , 1600

See Lesson 7-5.

Solve each equation. Check your answers.

50. 8 2 x = 4 8 super 2 x end super , equals 4
51. log 8 x = 3 log 8 x equals 3
52. 2 log 3 x − log 3 4 = 2 2 , log base 3 , x minus , log base 3 , 4 equals 2

Get Ready! To prepare for Lesson 10-6, do Exercises 53–55.

See Lesson 4-2.

Rewrite each function in vertex form.

53. y = x 2 − 6 x + 1 y equals , x squared , minus 6 x plus 1
54. y = 2 x 2 + 12 x y equals , 2 x squared , plus 12 x
55. y = 3 x 2 + 24 x − 2 y equals , 3 x squared , plus 24 x minus 2

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