geno3141

Because there is a 7², we know that it has to be at least 14! and not as big as 21!

Starting at the bottom"

2! = 2¹

3! = 2¹3¹

4! = 2³3¹

5! = 2³3¹5¹

6! = 2⁴3²5¹

7! = 2⁴3²5¹7¹

8! = 2⁷3²5¹7¹ 

9! = 2⁷3⁴5¹7¹ 

10! = 2⁸3⁴5²7¹ 

11! = 2⁸3⁴5²7¹11¹ 

12! = 2¹⁰3⁵5²7¹11¹ 

13! = 2¹⁰3⁵5²7¹11¹13¹ 

14! = 2¹¹3⁵5²7²11¹13¹  

15! = 2¹¹3⁶5³7²11¹13¹  

16! = 2¹⁵3⁶5³7²11¹13¹  

Let  P  be the intersection of AD and BF.

Let  Q  be the intersection of BC and DE.

Triangle(ABP) is congruent to triangle(CDQ).

The area of rectangle(ABC) = 2·3 = 6.

I plan to find the area of triangle(ABP), which is also the area of triangle(CDQ), and subtract these areas

from the area of the rectangle to get the area of the overlapping region.

Let the length of AP = x.

Let the length of PB = y.

By the Pythagorean Theorem, AB² + AP² = PB²   --->   2² + x²  =  y².  --->   y²  =  4 + x²  

Since BP = PD, AP + PD = 3   --->   x + y  =  3     --->     x  =  3 - y

By substitution:  y²  =  4 + (3 - y)²   --->   y²  =  4 + 9 - 6y + y²   --->   6y  =  13   --->   y  =  13/6

Since  x  =  3 - y   --->   x  =  3 - (13/6)  =  5/6

Area of triangle(ABP)  =  ½·2·(5/6)  =  5/6

Area of triangle(CDQ)  =  5/6

Area of the overlapping region  =  6 - (5/6) - (5/6)  =  13/3.

Can anyone find an integer smaller than 345?

Since they lose 72% of their weight, they retain only 28% of their weight.

If x is the total weight of fresh mushrooms:  0.28 · x  =  12.25

     --->     x  =  12.25 / 0.28     --->     x  =  43.75  pounds of fresh mushrooms

Use the quadratic formula to find a and b.

Use these values to find the answer.

Have you found a and b?

The maximum product will occur at one-half the sum.

One-half of 246  =  123:  123 x 123  =  15129.

If you want to graph this, use the equation:  y  =  x(246 - x)

Its vertex occurs at (123, 15129).

2xy + x + 2y  =  899

I re-wrote it:     2xy + x + 2y  =  899     --->     2xy + 2y =  899 - x     --->     (2x + 2)y  =  899 - x

         --->     y  =  (899 - x) / (2x + 2)

From this, I could get these solution;  (11,37)     (19,22)     (35,12)     (59,7)     

                                                            (99,4)       (112,3)     (179,2)     (299,1)

Case 1:  3x - 7 >= 0     and     x + 1 >= 0

                    3x >= 7                      x >= -1

                      x >= 7/3

               Comgining the above:  x >= 7/3

               3x - 7  =  x + 1

                    2x  =  8

                      x  =  4

Case 2:  3x - 7 >= 0     and    x + 1 < 0

                    3x >= 7                    x < -1

                      x >= 7/3

               Combining the above:  impossible

Case 3:  3x - 7 < 0     and     x + 1 >=0

                    3x < 7                     x >= -1

                     x < 7/3

               Combining the above:  -1 <= x < 7/3

                -(3x - 7) = x + 1

                  -3x + 7 = x + 1

                         -4x = -6

                            x = 3/2

Case 4:  3x - 7 < 0     and     x + 1 < 0

                    3x < 7                      x < -1

                     x < 7/3

              Combining the above:  -1 < x < 7/3

              -(3x - 7) = -(x + 1)

                -3x + 7 = -x - 1

                        -4x = -11

                           x = 11/4     (Impossible, not in the required region for x)

The two answers are in bold.

Because she will get 1.5 points for each question left unanswered and she leaves 3 questions unanswered,

she will receive 3 x 1.5 points  =  4.5 points.

To get at least 100 points, she will need 100 - 4.5 points  =  95.5 points on the first 22 questions.

Since she gets 6 points for each correct answer, she will need 95.5 / 6  =  15.92 correct answers --->

she will need 16 correct answers.

16 x 6  =  96 points + 4.5 points  =  100.5 points.

f(x)  =  ln(2x + 5)

                                       y  =  ln(2x + 5)

Interchange x and y:       x  =  ln(2y + 5)

solve for y:                     e^x  =  2y + 5

                                  e^x - 5  =  2y

                           (e^x - 5) / 2  =  y

                                    g(x)  =  (e^x - 5) / 2

                                    g(4)  =  (e⁴ - 5) / 2

Now, it's calculator time!