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Thanks so much! You're a life-saver!

∠DCB  and  ∠CBA  are alternate interior angles, so they have the same measure.

∠CBA  and  ∠BAC  are base angles of isosceles triangle ABC, so they have the same measure.

So..

m∠DCB   =   m∠CBA   =   m∠BAC   =   40°

And since there are  180°  in triangle  ABC,

m∠ACB   =   180° - 40° - 40°   =   100°

And...

m∠ACB  +  m∠DCB  +  m∠ECD   =   180°

100°  +  40°  +  m∠ECD   =   180°

m∠ECD   =   180°  -  40°  -  100°   =   40°

4)The distance from Capital City to Little Village is 660 miles. From Capital City to Mytown is 310 miles, from Mytown to Yourtown is 200 miles, and from Yourtown to Little Village is 150 miles. How far is it from Mytown to Little Village?

Here's the layout [  assuning that all the cities are collinear ]

C                  660                                              L

C       310             M

                            M     200       Y 

                                                  Y      150       L

Mytown to Little Village  =  200 + 150    =  350 miles

3)A triangle with integer sides has perimeter 12. How many such non-congruent triangles are there? (A 3-4-5 triangle is considered congruent to a 3-5-4 triangle because we can reflect and rotate the triangles until they match up.)

Possibilities :

3 - 4 - 5

4 - 4 - 4

5 - 5 - 2

1). CPhill: Did you make a mistake in calculating the sides of the isosceles triangle? 9 + 9 + 20 =38??

Is it 5: 5: 2, or 5/12 x 36 =15, 15, 6 =36 cm??

2.     Assuming that     g(x)   =   20(1.5)^x

the y-intercept of g(x)  =   g(0)   =   20(1.5)⁰   =   20(1)   =   20

the y-intercept of f(x)  =  f(0)   =   -40

The y-intercept of  g(x)  is greater than the y-intercept of f(x) , so  a)  is true.

From  -5  to  -3 ,  f(x)  goes from  -45  to  -49 ,  and   (-49) - (-45)   =   -4

From  -5  to  -3 ,  g(x)  goes from  \(\frac{640}{243}\)  to  \(\frac{160}{27}\) , and  \(\frac{160}{27}\)  -  \(\frac{640}{243}\)   =   \(\frac{800}{243}\)

f(x)  is decreasing on the interval  (-5, -3)  because as  x  gets larger, f(x) gets smaller.

\(\frac{800}{243}\)  >  -4 ,  so  g(x)  is increasing faster than  f(x)  on the interval  (-5, -3) .  b)  is false.

f(1)   =   -33

g(-1)   =   20(1.5)^-1   =   \(\frac{20}{1.5}\)   =   \(\frac{40}{3}\)   ≈   13.3

Is  f(1)  less than  g(-1) ?   Is  -33  less than  13.3 ?  Yes, so  c)  is true.

We already found that  f(x)  is decreasing on the interval  (-5, -3)  , so

f(x)  can't be increasing on the interval  (−∞, ∞) .  d)  is false.

2)In how many ways can we choose one number from the set {1,2,3}, one number from the set {4,5,6}, and one number from the set {7,8,9} such that the three could be the sides of a nondegenerate triangle?

We can  eliminate 1 as one of the sides because  1 + 6  = 7 

But this sum must exceed 7

So...the possibilities are

{2, 6, 7 }

{3, 5, 7 }

{3, 6, 7}

{3, 6, 8}

1)The perimeter of an isosceles triangle is 36 centimeters and two sides of the triangle are in the ratio 2:5. What is the number of centimeters in the length of the longest side?

The sides must  be  in the ratio   2 : 5 : 5

So......each equal part must be   3 cm

So....the number of cm in the longest side =

5 / 12  *  36  = 

5 * (36/12) =

5 * 3  =

15 cm

The shortest side  is  (2/12) (36)  =  6

So   15 + 15 + 6  = 36

EDit to correct a previous error....

Here's one way to do this...but maybe not the most efficient

Move  "B"  to the origin   ....so   ( 1 - 1, - 4 + 4)  = (0, 0)

Apply the same transformation to "A"  and we have   

(5 -1, 3 + 4)  =  ( 4, 7)

Rotating this point 90° counter-clockwise produces (-7, 4)

Now.......reverse the original transformation  and we have

(-7 + 1, 4 - 4)   =    (-6, 0)  = A'

(x , y)  reflected in the x axis ⇒  (x, - y)          

(x, - y) rotated 90°  clockwise ⇒ ( -y, - x)

(-y, - x)  reflected in the y axis ⇒  (y, -x)

Thanks, EP.....to see this more clearly.....note that  the cube that is removed had three exposed sides contributing to the original surface area of the original cube

But...note when the smaller cube is removed....the same surface areas remain on the  altered cube

(3a²b^-1)(4a^-3b)

We   are using   a^m * aⁿ  =  a^{(m + n)}

3*4 * a²* a^-3 * b^-1 * b¹

12 a^-1 b⁰

12 / a

Original surface area    21 x 21 x 6 sides =2646 cm^2

New surface area    21 x 21 x 3 sides    +  15x15x3   +  (21 x 21 - (15x15) ) x 3

                                   1323                +          675            +     648

                                     2646 cm^2

No change in surface area

The amplitude  is 6

We can use the sine here...I'm assuming that the period is 4 seconds.....so each second is 

2pi / 4   =   pi/2  rads

So...we have this function

y = 6sin ( pi * t / 2)

Here's the graph :  https://www.desmos.com/calculator/scp4kbo1z4

X^2 -10x   + 25   +  y^2 + 6y   +   9   = -c   + 25 + 9

(x-5)^2  +  (y+3)^2  = -c  + 34           

-c+34 = 1

-c=-33

c = 33

(x) (x^-3)   =

(x¹) (x^-3)

Using    (a^m) (aⁿ)  = a^{(m + n)}

So  

(x¹) (x^-3)  =  ( x ) ^{(1 - 3)}  =  (x)^-2  =   1 / x²

The center is  ( -1, 1)

And the radius  is   √5

So....the equation is

(x + 1)^2  + (y - 1)^2  = 5        simplify

x^2 + 2x + 1  + y^2 - 2y + 1  =  5

x^2 + 1y^2 + 2x - 2y - 3  =  0

So

A + B + C + D  =

1 + 2 -  2   - 3  =     -2 

By brute force  :

Gil's current age  =  36

Sheila's current age  = 36 - 17  =  19

Perfect square  =  1936   =  44^2

In 13 years

Gil  =  49

Sheila  = 32

Perfect square  = 3249   =  57^2

Amazing, hectictar!

f(x)  =     1000 /  [ 1 + 9e^(-.4x) ] 

Note that......at the start x   = 0....so  e^(-.4(0))  = e^0   =   1

So....at the start we have

1000 / [  1 + 9(1)]   =    1000 / [ 1 + 9 ]   =  1000 / 10 =  100 flowers

So....the first statement is true

To   answer the next two...look at the calculated results from WolframAlpha :

https://www.wolframalpha.com/input/?i=1000+%2F%C2%A0+%5B+1+%2B+9e%5E(-.4x)+%5D,+x+%3D+10,+11

10th year  ≈   858         11th year ≈ 900

So    900 - 858  =  42    .....the second answer is good, too

For the 3rd one......let x  =  15

WolframAlpha  calculates   ≈  978....so...the third statement isn't true

The last one is a little tricky

9th year  ≈ 803

10th year ≈  858     increase  =  55

10th ≈  858

11th ≈ 900    increase =  42

11th ≈ 900

12th ≈ 931   increase  =  31

So....the last statement seems to be true, as well