CPhill

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

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) (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