geno3141

If you understand the answer to solving the problem given the 16^th and 17^th terms, work this problem in a similar fashion.

The equation of the parabola is:  y  =  -2x² + 12x - 8

a = -2     b = 12     c = -8

The formula is:  -b/(2a)  =  -12 / (2·-2)  =  -12 / -4  =  3

To find the value of y, substitute 3 into the equation of the parabola

The graphing formula for a parabola is:  y - k  =  a(x - h)²  where (h, k) = (3, 2)

--->                                                          y - 2  =  a(x - 3)²

Since (1, 0) is on the parabola   --->       0 - 2  =  a(1 - 3)²

                                                                    -2  =  a(-2)²   

                                                                    -2  =  a(4)

                                                                    -½  =  a

--->         y - 2  =  -½(x - 3)²

--->    -2(y - 2)  =  (x - 3)² 

--->     -2y + 4  =  x² - 6x + 9

--->           -2y  =  x² - 6x + 5

--->               y  =  -½x² + 3x - 5/2

The common difference is 2.

The formula for the n^th term of an arithmetic sequence:  t_n  =  t₁ + (n - 1)d

The 16^th term is 8   --->   n = 16   --->   8  =  t₁ + (16 -1)(2)

                                                              8  =  t₁ + (15)(2)

                                                               8  =  t₁ + 30

--->   t₁  =  -22

--->   t₂  =  -22 + 2  =  -20

You can also use the theorem that says that  MA·AT  =  EA·AD

--->   9·9  =  4·AD

--->   AD  =  81/4   

--->   AD  =  20.25

Find the diameter and divide by 2.

Call BE = x         then AC = x

Call DE = y

Call EC = z

Triangle(ACB) is similar to triangle(DEC).

--->   y / x  =  x / (x + z)

Right triangle   --->   y  =  sqrt(1 - x²)

x + y + z  =  2   --->   z  = 2 - x - y   --->   z  =  2 - x - sqrt(1 - x²)

y / x  =  x / (x + z)   --->   sqrt(1 - x²) / x  =  x / [ x + (2 - x - sqrt(1 - x²) ) ]

--->   sqrt(1 - x²) / x  =  x / [ 2 - sqrt(1 - x²) ]

Cross-multiplying:

--->   x²  =  2sqrt(1 - x²) - (1 - x²)

--->   x²  =  2sqrt(1 - x²) - 1 + x²

--->     0  =  2sqrt(1 - x²) - 1

--->    1  =  2sqrt(1 - x²)

--->   ½  =  sqrt(1 - x²)

--->   ¼  =  1 - x²

--->   x²  =  3/4

--->     x  =  sqrt(3)/2       [the negative answer can be ignored]

cos(ABC)  =  [ sqrt(3)/2  ] / 1

angle(ABC)  =  cos^-1[ sqrt(3)/2 ]

angle(ABC)  =  30^o 

The time 7:30 represents 7 hours and 30 minutes.

To this, we need to add:  2 hours, 20 minutes, 2 hours, 45 minutes, and 3 hours.

Hours: 5 mi 2 + 2 + 3  =  7 hours.

Minutes:  20 + 45  =  65 minutes.

      65 minutes = 1 hour, 5 minues.

We will need to add 7 hours + 1 hour + 5 minutes  =  8 hours 5 minutes

To 7:30, which is 7 hours and 30 minutes, we add 8 hours and 5 minutes, to get 15 hours and 35 minutes.

15 hours is 3 in the afternoon, so Randy leaves at 3:35

From 6:00 am to 2:00 pm is 8 hours.

Dropping 3^o per hour for 8 hours will result in a trop of 8 x 3^o  =  24^o.

Since the temperature started at 36^oC, the new temperature will be 36^o - 24^o  =  12^oC.

A diagonal from one vertex to its opposite vertex (through the middle of the box) can be found by using the 

"three-dimensional Pythagorean Theorem":

         Distance² =  Height² + Length² + Depth² 

--->   Distance²  =  8² + 16² + 13²

--->   Distance²  =  489

--->   Distance   =  sqrt(489)  cubic cm

1)  A cube with side length = 4 has an area of 16 for each of its sides; 6 x 16  =  96  for its total area.

     A cube with side length = 5, has an area of 25 for each of its sides; 6 x 25  =  150 for its total area.

    The total area for both is  96 + 150  =  246

    If you glue one of the sides of the smaller cube to one of the sides of the larger cube, you will remove 16 

    from each of the two cubes:  246 - 16 - 16  =  214

2)  length  =  L

     width   =  W

     height  =  H

     L · W  =  6                    --->   W  =  6 / L

     W · H  =  15                 --->   W  =  15 / H

     L · H  =  40

     Since  W  =  6 / L   and   W  =  15 / H     --->     6 / L.  =  15 / H     --->     6H  =  15L     --->     H  =  15L / 6

     Since  L · H  =  40     --->     L · ( 15L / 6 )  =  40     --->     15L²  =  240     --->     L²  =  16     --->     L  =  4

     Since  L  =  4     --->     L · W  =  6     --->     4 · W  =  6     --->     W  =  1.5

     Since  L · H  =  40     --->     4 · H  =  40     --->     H  =  10