hectictar

Point  A  is the current location of the plane.

Point  B  is the point on the ground directly under the plane.

Point  C  is the location of the airport.

m∠BAC   =   90° - 6°   =   84°

a)

plane's ground distance to the airport =  BC

tan 84°  =   BC / AB

tan 84°  =   BC / 3900

3900 tan 84°  =  BC

BC  ≈  37106.021    feet

b)

distance between plane and runway   =   AC

cos 84°   =   AB / AC

cos 84°   =   3900 / AC

AC cos 84°  =  3900

AC  =  3900 / cos 84°

AC  ≈  37310.412    feet

I agree with Melody...I think it's false.

The zeros, or roots, of a polynomial are the values of the independent variable that cause a polynomial to be equal to zero, not the factors.

(f⋅g)(x)

=   f(x) * g(x)

=   (x² + 2x - 6) * (x⁴ + 3)

=   x⁶ + 3x² + 2x⁵ + 6x - 6x⁴ - 18

=   x⁶ + 2x⁵ - 6x⁴ + 3x² + 6x - 18

xy  =  8      Divide both sides of this equation by  y .

x  =  8/y     Use this value for  x  in the second given equation.

x² + y²  =  20

(8/y)² + y²   =   20

64/y² + y²   =   20           Multiply through by  y²

64 + y⁴  =  20y²              Subtract  20y²  from both sides.

y⁴ - 20y² + 64  =  0         We can rewrite this as...

(y²)² - 20y² + 64  =  0     Factor like this:   u² - 20u + 64  =   (u - 4)(u - 16)

(y² - 4)(y² - 16)  =  0       Factor each difference of squares.

(y + 2)(y - 2)(y + 4)(y - 4)  =  0      Set each factor equal to zero.

y + 2  =  0     or     y - 2  =  0     or     y + 4  =  0     or     y - 4  =  0

y  =  -2          or     y  =  2          or     y  =  -4          or     y  =  4

Find  x  at each of these values for  y .

When  y  =  -2  ,     x  =  8 / y  =  8 / -2  =  -4

When  y  =  2  ,     x  =  8 / y  =  8 / 2  =  4

When  y  =  -4  ,     x  =  8 / y  =  8 / -4  =  -2

When  y  =  4  ,     x  =  8 / y  =  8 / 4  =  2

The solution set is   { (-2, -4), (2, 4), (-4, -2), (4, 2) }

x + y  =  5     Subtract  y  from both sides of this equation.

x  =  5 - y      Use this value of  x  in the second equation.

x² + y²   =  13

(5 - y)² + y²   =  13

(5 - y)(5 - y)  +  y²   =  13

25 - 10y + y²  +  y²   =  13

25 - 10y + 2y²   =  13              Subtract  13  from both sides of the equation.

12 - 10y + 2y²   =  0                Rearrange.

2y² - 10y + 12   =  0                Divide through by  2 .

y² - 5y + 6   =   0                     Factor left side.

(y - 3)(y - 2)  =  0                      Set each factor equal to zero.

y - 3  =  0     or     y - 2  =  0

y  =  3          or     y  =  2

Use both of these values of  y  to find  x .

When   y  =  3 ,   x  =  5 - y  =  5 - 3  =  2

When   y  =  2 ,   x  =  5 - y  =  5 - 2  =  3

So...

One solution is   x = 2   and   y = 3

Another solution is   x = 3   and   y = 2

The solution set is  { (2, 3), (3, 2) }

I'm sure there is an easier way to do this, but here is one way to get the answer....

The intersection point of  y = -x - 1  and  x = 2   is   (2, -3)

The intersection point of  y = -x - 1  and  y  =  1/5x + 13/5   is   (-3, 2)

The intersection point of  x = 2  and  y = 1/5x + 13/5   is   (2, 3)

equation of a circle:    (x - h)² + (y - k)²  =  r²

Using the point  (2, -3)  we can make the equation:    (2 - h)² + (-3 - k)²  =  r²

Using the point  (-3, 2)  we can make the equation:    (-3 - h)² + (2 - k)²  =  r²

Using the point  (2,  3)  we can make the equation:    (2 - h)² + (3 - k)²  =  r²

Set the first and third values of  r²  equal to each other.

(2 - h)² + (-3 - k)²   =    (2 - h)² + (3 - k)²

(-3 - k)²   =   (3 - k)²

-3 - k   =   ±(3 - k)

-3 - k   =   3 - k         or         -3 - k   =   -(3 - k)

not a solution                       -3 - k   =   -3 + k

                                             k   =   0

Use this value of  k  and set the first and second values of  r²  equal to each other.

(2 - h)² + (-3 - 0)²   =   (-3 - h)² + (2 - 0)²

(2 - h)² + 9   =   (-3 - h)² + 4

(2 - h)² + 5   =   (-3 - h)²

(2 - h)² - (-3 - h)²  =  -5

(4 - 4h + h²) - (9 - 6h + h²)  =  -5

-5 + 2h  =  -5

2h  =  0

h  =  0

Now we can find  r² .

(2 - 0)² + (-3 - 0)²  =  r²

13  =  r²

So the equation of the circle is...

x² + y²  =  13

(f+g)(x)

=   f(x)  +  g(x)

=   (x³ - x)  +  (x³ + 2x² - 10)

=   x³ - x + x³ + 2x² - 10

=   2x³ + 2x² - x - 10

avg rate of change over interval [7, 9]

=   slope between  ( 7, f(7) )  and  ( 9, f(9) )

=   slope between  ( 7, 1852 )  and  ( 9, 3878 )

=   \(\frac{3878-1852}{9-7}\)

=   \(\frac{2026}2\)

=   1013

avg rate of change over interval [4, 6]

=   slope between  ( 4, f(4) )  and  ( 6, f(6) )

=   slope between  ( 4, 358 )  and  ( 6, 1178 )

=   \(\frac{1178-358}{6-4}\)

=   \(\frac{820}2\)

=    410

How much greater is  1013  than  410   ?

1013 - 410  =  603

1013  is  603  greater than  410 .

your breath?

5.

First let's find what time the rockets are at the same height.

height of Brynn's rocket  =  f(x)

height of Denise's rocket  =  g(x)

height of Brynn's rocket  =  height of Denise's rocket

f(x)  =  g(x)

-4.9x² + 75x   =   -4.9x² + 50x + 38     Solve this equation for  x . Add  4.9x²  to both sides.

75x  =  50x + 38                                   Subtract  50x  from both sides.

25x  =  38                                             Divide both sides by  25.

x   =   1.52

Now we know that after 1.52 seconds, the rockets are at the same height.

To find what this height is, plug in  1.52  for  x  into either function.

f(1.52)  =  -4.9(1.52)² + 75(1.52)   ≈   102.7     meters