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There are 582 such sequences.

By properties of logs, the answer is 2/3.

Lines that are tangent to a circle and meet at a point are congruent, as shown in the graph:

According to the Pythagorean theorem,

\(14^2+(4+x)^2=(x+10)^2\\ 196+16+8x+x^2=x^2+20x+100\\ 112=12x\\ x=\frac{112}{12}=\boxed{\frac{28}{3}}\)

Since the line that cuts through the triangle is parallel to the base, we know that 3 is proportional to x - 3 in the same way 5 is proportional to  2x-10.

Mathematically, we can represent this as: 

\({\frac {5} {2x-10}}={\frac {3} {x-3}}\)

Which can then be solved to get: 

\(5(x-3)=3(2x-10)\)

\(5x-15=6x-30\)

\(x = 15\)

Thus, x = 15.

Solved! :)

To solve this problem, we need to find the maximum of the graph. In this case, the maximum is also the vertex of the graph. So, we just need to find the vertex of the equation. In case you didn't know, the formula for finding the x-coordinate of the vertex is   -b/2a in the form ax^2 + bx + c. 

Plugging in the values(b = 51, a = -1), we get:

x = -(51)/2(-1)

x = -51/-2

x = 51/2 or 25.5

Now, we have the x-coordinate of our vertex(maximum). All we need to do now is plug 51/2 back into the original equation.

y = -(51/2)^2 + 51(51/2) - 201

y = -(650.25) + 1300.5 - 201

y = -650.25 + 1300.5 - 201

y = 449.25

Now, we know the coordinates of the maximum. It is: (25.5, 449.25).

Convert to cash.

tldr: The company must sell their widgets for 25.50 dollars, which leads to a maximum profit of 449.25 dollars. 

To start this problem out, we will find the x and y-intercepts of 2x + 3y = 6. The x-intercept is found when y is 0. So, we plug zero into the equation to get:

2x + 3(0) = 6

2x = 6

x = 3

So now we know that the coordinates of the x-intercept is 3. (this point is b)

The y-intercept is found when x = 0. So, we do the same thing as before, except x = 0.

2(0) + 3y = 6

3y = 6

y = 2

So now we know that the coordinates of the y-intercept is 2. (this point is a)

This is as far as I will go, as the rest of the problem seems a little too complicated and confusing for my 2 little brain cells. Hope this helped though!

The leading term of f(x) is  x^4

The leading term of g(x) is  2x^4

The leading term of their product will be   2x^8

So what is the degree going to be?

Then what would the answer to the second question be?

Thanks!

Nice work, skylersnoop  !!!

The first polygon will have side length will be set to 2x. So, the perimeter of the polygon is 76x. (because 38 * 2x = 76x)

The second polygon will have side length will be set to x, because the second polygon has half the side length of the first polygon. Also, the number of sides that the polygon has will be set to y. So, the perimeter will be xy.

The perimeters of both polygons are equal, so we can set 76x = xy.

Divide both sides by x to get:

76 = y

There are 76 sides in the second polygon. 

Using the Law of Cosines

BC^2  =  4.9^2 + 6.8^2   - 2 ( 4.9 * 6.8)cos (70°)

BC^2  = 47.46

BC  = sqrt ( 47.46)  ≈  6.9 km

After  3 hrs.....Johal  has traveled  3 * 35  =  105 km

And  Julio  has traveled  3 * 40  = 120 km

We  can  use  the Law of Cosines  to  find  the  distance^2, d^2, between them

d^2 =   105^2  + 120^2   -2(105 * 120) cos (63°)

d^2 = 13984

d  = sqrt ( 13984)   ≈   118.25  km

The coordinates of the given points are (-10, 5), and (-5, 6). (you can use the other two points as well). To get from the first coordinate to the second coordinate, you rise by 1 and go over by 5. The formula for slope is rise over run, so the slope of the graph is 1/5. 

Now, we plug that into the slope-intercept equation, to get y = 1/5 x + b.

All that is left to do is plug in any value into our equation to find b. I will choose the point (5, 8).

We get:

8 = 1/5 (5) + b

8 = 1 + b

7 = b

Our complete equation is 1/5 x + 7.

The  46cm chain  forms  the  hypotenuse of a right triangle.....the  height  from  the  ceiling to  the  bottom of  this  chain can  be  found as

sin 60  =  h / 46

46 * sin 60  =  h   ≈  39.84 cm

The longer  chain  will  hang  at  the  same  distance from the ceiling....to find  the  angle it makes with the ceiling  we  can use  the inverse sine  ( arcsin)

arcsin ( 39.84  / 64)   ≈   38.5°

You are welcome Jenny :)

There are 26 ways of placing the balls.

My trip to work is 120 miles. If I go 8 mph faster than my usual speed, I'll get to work 30 minutes earlier. How long does my trip take, in hours, if I go my usual speed?

30 min  = 1/2  hr

Let  your usual  speed =  r

Let  the  greater  speed =  r + 8

Distance / Rate  =  Time

So

Time  at faster speed  + 1/2  hr =  Time  at normal speed.....so....

120 / (r + 8)  +   1/2   =   120 / r  rearrange  as

1/2  =   120/r  -  120/ ( r + 8)        

1/2 =  [ 120 ( r + 8)  - 120 r ]   / ( r * (r + 8) )

1/2  =  [ 120r  + 960 -120r ]  / (r ( r +8))

1/2 =    ( 960 )  / ( r (r + 8))        cross-multiply

r ( r + 8)  =  960 * 2

r^2  + 8r  =  1920

r^2  + 8r  - 1920   =  0

(r  + 48)  ( r - 40) =   0

The  second factor   set to  0  is  what we need

r -40   =   0

r  =  40 mph

Time to work at  normal speed  =   120  / r =   120  /  40 =    3 hrs

Marvin can reach (5,7) in 424 ways.

Total height  given  by

20   ( tan (20°)  +  tan (45°) )   ≈  27.3  m

a)  max height = 20 m

b) 2.5  sec to reach  this height

c )   symmetry  =   2 ( 2.5)  =  5 sec

Thanks!

A three digit number "abc" has the property that the product of a and b is equal to c. The digits a, b, and c are not necessarily distinct. What is the greatest possible value of the three digit number?

919

First  one 

9*15  +  6 * 15 / 2  =

135  +   45  =

180  mm^2

m + b = 4.

second, 114 im rushing