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LenaWriter [7]
2 years ago
5

If the triangles are similar, what is the value of x?

Mathematics
1 answer:
Yuri [45]2 years ago
6 0

9514 1404 393

Answer:

  1. 5/9
  2. 17/90
  3. 12/45
  4. 19/66

Step-by-step explanation:

You can do these conversions the way shown in your attachment:

2) x = 0.18_8

  10x = 1.88_8

  9x = 1.88_8 -0.18_1 = 1.7

  x = 1.7/9 = 17/90

Or, you can recognize that 0.88_8 is 8/9, but 0.088_8 is that value divided by 10: 8/90. Then 0.18_8 is 0.1 + 0.088_8 = 1/10 + 8/90 = (9+8)/90 = 17/90

__

Here are the others:

1) 0.5_5 = 5/9

3) 0.26_6 = 1/5 + 6/90 = (18 +6)/90 = 24/90 = 12/45

4) 0.287_87 = 1/5 + 87/990 = (198+87)/990 = 19/66

_____

<em>Further development</em>

You have probably figured out that a 1-digit repeat that starts at the decimal point is expressed as a fraction with that digit divided by 9.

  x = 0.dd_d

  10x d.dd_d

  9x = d

  x = d/9 . . . . . . for any single-digit repeat

Now doubt, you're used to seeing 1/3 = 0.33_3 = 3/9, and 2/3 = 0.66_6 = 6/9.

__

Similarly, a 2-digit repeat is the fraction with those two digits divided by 99 (an equal number of 9s). This holds for all n-digit repeats. The "raw" divisor is 10^n -1. In many cases, the fraction can be reduced, as in ...

  0.142857_142857 = 142857/999999 = 1/7

__

When the repeating part of the decimal does not start at the decimal point, then the divisor (some number of 9s) must be adjusted by an appropriate power of 10. The non-repeating part can be expressed as a fraction in the usual way, then added to the value of the repeating part. Or, you can do the several-step process you show. Both work just fine.

_____

<em>About notation</em>

Overbars are difficult to show in plain text. Not all methods of creating overbars are rendered properly in documents that show plain text. In typeset text, they are used to show the repeating digits. So, other methods are sometimes used in plain text to show the repeating digits. Here are a few:

  0.166_6 . . . . shows the 6 repeats

  0.16(6) . . . . . shows the 6 repeats [this notation is also used to indicate error limits for the number, so can be ambiguous]

  0.16... (repeating 6) . . . . . shows the 6 repeats

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How many numbers of solutions are there for the following equation<br> 2(4x+1) −5x +2 = 3x+4
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2(4x+1) −5x +2 = 3x+4

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Find the general expression for the slope of a line tangent to the curve of y=2x^2+4x at the point P(x,y) . Then find the slopes
ArbitrLikvidat [17]

Complete Question

Find the general expression for the slope of a line tangent to the curve of y=2x^2+4x at the point P(x,y) . Then find the slopes for x = 3 and x=0.5. Sketch the curve and the tangent lines. What is the general expression for the slope of a line tangent to the curve of the function y=2x^2+4 at the point P(x,y) ​?

Answer:

The  generally expression for the slope of y  = 2x^2 + 4x is  y' =  4x +4

The graph is shown on the first uploaded image

The  generally expression for the slope of y=2x^2+4 is   y' =  4x

Step-by-step explanation:

From the question we are told that

  The  equation of the curve is y  = 2x^2 + 4x

First we differentiate the equation

So  

     y' =  4x +4

Therefore the generally expression for the slope tangent to the curve y=2x^2+4x is   y' =  4x +4

The  next step is to substitute for x =  3 and  x =  0.5

So  for x_1 =  3

    y' =  4(3) +4

     y' =m_1=  16

And  for  x_2 =  0.5

      y' =  4(0.5) +4

       y' =m_2=  6

Here m_1  and  m_2 are slops of the curve

Next we obtain the coordinates of the tangent lines

So  at x_1 =  3

   y_1  = 2(3)^2 + 4(3)

  y_1  =  21

So the coordinate for the first tangent line is  

    (x_1 , y_1 ) =  (3 ,  21)

At  x_2 = 0.5      

    y_2  = 2(0.5)^2 + 4(0.5)

=>  y_2  = 2.5

So the coordinate for the second  tangent line is  

    (x_2 , y_2 ) =  (0.5 ,  2.5)

Next we obtain the equation for the tangent lines

 So generally the slope is mathematically represented as

        m  =  \frac{y - y_1 }{x-x_1}

For   (x_1 , y_1 ) =  (-3 ,  21) and  y' =m_1=  16

       16 =  \frac{y -21 }{x-3)}

=>    y   = 16x - 27

For  (x_2 , y_2 ) =  (0.5 ,  2.5) and  y' =m_2=  6

       6  =  \frac{y -2.5 }{x-0.5}

       y  = 6x -0.5

Generally the general expression for the slope of a line tangent to the curve of the function y=2x^2+4 at the point P(x,y) is mathematically evaluated by differentiating  y=2x^2+4 as follows

     y' =  4x

     

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3 years ago
Irect Variation
Anit [1.1K]

For this case we have to by definition, if "y" varies directly with "x" then it is true that:

y = kx

Where:

k: Is the constant of proportionality

We find the constant "k" according to the given data:

14 = k (-4)\\k = - \frac {14} {4} = -\frac {7} {2}

Thus, the constant of proportionality is - \frac {7} {2}.

Now, we find the value of "y" when x = -6:

y = - \frac {7} {2} (- 6)\\y = \frac {42} {2}\\y = 21

Thus, whenx = -6, the value of y is 21.

ANswer:

y = 21

4 0
3 years ago
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