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Log(x) / Log(2) = 3^2, solve for x

Log(x) = 9log(2)

Log(x) = log(2^9)

x = 512

....and the second part....y'all must be using the same home schooling programs!

https://web2.0calc.com/questions/part-of-a-two-column-proof-is-shown-with-just-the_1

Here is the first part from yesterday:

https://web2.0calc.com/questions/a-right-triangle-is-shown-below

First let's find out how many CALORIES are required, and then convert to joules , knowing

1 cal = 4.184 j

One calorie will raise ONE GRAM of water ONE DEGREE Celsius

#calories = 7000 gm  x 78 C = 546000  CALORIES

546000 calories x 4.184 j/calorie = 2284.5 kj    (kj = KILOjoule)

2.  First you have to warm the ice up to 0 degrees

     300 g x 1cal/gram-C x  15 C = 4500 cal

     Then you have to add heat to it AT A CONSTANT TEMP to melt the ice @ 80 cal/gm

300gm x 80 cal/gm = 24000 cal

     Then you have to warm the water (melted ice at 0 degrees C) to 20 C

300 gm x 1 cal/gram-C  x 20 C  = 6000 calories

Add them all up  4500+24000+6000 = 34500 cal     =    144.3 kj

Solve for x:

log(log(x)/log(2))/log(3) = 2

Multiply both sides by log(3):

log(log(x)/log(2)) = 2 log(3)

2 log(3) = log(3^2) = log(9):

log(log(x)/log(2)) = log(9)

Cancel logarithms by taking exp of both sides:

log(x)/log(2) = 9

Multiply both sides by log(2):

log(x) = 9 log(2)

9 log(2) = log(2^9) = log(512):

log(x) = log(512)

Cancel logarithms by taking exp of both sides:

x = 512

Thanx for the correction GA !     ~EP

You can also solve it using just one variable:

Let the number of adults = A, then:

The number of children = 292 - A

4A + 1.5[292 - A] = 888

4A + 438 - 1.5A = 888

4A - 1.5A = 888 - 438

2.5A = 450

A =450 / 2.5

A = 180 - Number of adults

292 - 180 = 112 - Number of children.

Let the number of adults = A

Let the number of children = C

A + C =292

4A + 1.5C = 888. solve for A, C

A = 292 - C

4[292 - C] + 1.5C = 888

1,168 - 4C + 1.5C = 888

-2.5C = - 280

C = -280 / -2.5

C = 112 - Number of childern

292 - 112 = 180 - Number of adults  

(n - 2) (n + 4)  < 0

If we solved this

(n - 2) ( n + 4)  = 0

The solutions would be  n  = 2  and n  =  -4

So......the solutions to  the inequality lie on the possible intervals

(-inf, - 4) , ( -4, 2)  ,  (2, inf )

Testing  n  =  0  in the middle interval   will make the inequality true, while the outside two intervals don't 

So...the solution is  (-4, 2)   ....and  5 integers lie in this interval

To convert the standard deviations to percentage of AUC, use a Z-chart or calculator.

Using a Z-chart, find the 1.5 and 0.5 Sdvs area values, then subtract to find the area range.

The area from 0.5 to 1.5 Sdv is 0.9332 of the area (above 1.5) minus 0.6915 of the area (below 0.5) giving 24.2% of the total area.

Here are the percentages for the other deviations.

(A)  1.5 standard deviations above the mean or higher

       (1-0.9332) = 0.0668                          6.7%

(B)   0.5 to 1.5 standard deviations above the mean 

        (0.9332-0.6915) = 0.2417              24.2%

(C)   within 0.5 standard deviation of the mean

       (0.6925 - 0.3085) = 0.384              38.4%

(D)   0.5 to 1.5 standard deviations below the mean

       (0.3085 - 0.0668) = 0.2417           24.2%

(F)   1.5 standard deviations below the mean or lower

        (1-0.9332) = 0.0668                      6.7%

Note: when the standard deviations are below the mean the values are negative, use the “negative” or “left of mean” chart (or adjust by subtracting from 1).  

GA

You are correct, g(x)  =  x  / [ x + 2]

So

(g ° h) x  =     g ( h(x))   =   

[ (3x + 7) / (3x + 9) ]  /  [ (3x + 7) / (3x + 9) + 2  ]   =

[ (3x + 7)/ (3x+9) ]  /  [   ( 2 (3x + 9) + (3x + 7) )  /  (3x + 9) ]  =

[ (3x + 7)/ (3x + 9) ]  /  [ (9x + 25) / (3x + 9) ]  =

3x + 7               3x + 9

_______   *      _______

3x + 9               9x + 25

  3x + 7

_______

  9x + 25

 √ [ 2x - 3 ]  + 3  =  x        subtract 3 from both sides

 √ [ 2x - 3 ]  =  x - 3        square both sides

2x - 3   =  x^2 - 6x + 9       subtract/add   2x, 3  to both sides

0  = x^2 - 8x + 12      factor

(x - 6) ( x - 2)  = 0

Setting each factor to 0  and solving for x  gives x  = 6  and x  =  2

Only the first solution solves the original problem

Here's another way to do this

The  new volume will  be   (1/2)^3  =  1/8  of the original volume....so

904.32  * (1/8)   =  113.04  in^3

Third one

We have 

1 + 2 + 4 + .....+ 2^(n - 1)

The sum  is

1 ( 1 - 2^12)  / ( 1 - 2)   =    4095  cars

Last one

The total distance  traveled in the first 5 bounces is

80 + 80  ( 1 - (1/2)^(5 - 1) ) / ( 1 - 1/2)    = 230 ft

And the distance traveled from  the 5th bounce to the next peak is  80 (1/2)^5 =  2.5 ft

So...the total distance traveled is  230  + 2.5  =   232.5 ft

Note that when  all sides = 25.....the perimeter = 100  and the area  =  25^2  =  625 units^2

Note that when one side is 1  and the other is 49....the perimeter is

2(1 + 49)  = 2* 50  = 100

And the area is  1 * 49  =  49 units^2

And this is the least possible area for integer sides with a perimeter of 100

1.  -4 + 8 - 16 + 32 - 64 + 128 - 256   =

(-4 + 8)  + (-16 + 32) + (-64 + 128)  - 256  =

4  +  16  + 64 -  256  =

84  -  256  =

-172

Second one

4 + 12 +  36   +  108  +  .....+  8748

We need to find what term the last is

We can solve this

4*3^(n-1)  =  8748    divide both sides by 4

3^(n - 1)  =   2187      take the log of both sides

log 3^(n- 1)  =  log 2187     and we can write

(n - 1) log 3  = log 2187      divide both sides by log 3

(n - 1)  =   log 2187  / log 3

n - 1  =  7    add 1 to both sides

n  = 8

The sum is given by

4 ( 1 - 3^8) / ( 1 - 3)   = 13,120

Thank you all

Alan is correct....

if x  = 10  ⇒   (10 + 5) (10 - 5)   =  15 * 5   =  75

If x  =  -10  ⇒  (-10 + 5)  (-10 - 5)  =  -5 * -15   =  75

A recursive formula  can be used to find a term based upont the previous term

An algebraic formula can be used to find any term

First one

a_n  = 5  + 5(n - 1)    where n is the nth term

Second one  is a geometric series

a_n  =   3ⁿ     where n is the nth term

Last one

a_n   =       n 

             ____        where n is the nth term

              n + 1

Thanks for your help and explanation! I tried 9 percent for students getting an A, my homework said wrong answer. Also for grade B 24 percent was wrong too. I'm I missing information for this problem?

Correct answers ended up being 6.7% for an A.       100- 93.32= 6.68 round one place 6.7 %

For B it's    93.32 - 69.1463 = 24.17  round one place so   24.2%

(x-5)   *   (x+5)       are the two numbers multiplied together,,...... 

1)  The triangle is a right-angle triangle, therefore:

Cos(60) = The distance to helicopter / distance between her and Jeremiah

1/2         = D / 4,000

D = 4,000 x 1/2

D = 2,000 ft. distance between Amanda and the helicopter.

2)

Cos(30) = Distance from Jeremiah to helicopter / 4,000

0.866     = D / 4,000

D =0.866 x 4,000

D =~ 3,464 ft -distance between Jeremiah and the helicopter.

sorry don't understand "the product of the result is 75".to have a product you need at least 2 numbers

Let x be the number. Then:

(x-5)(x+5) = 75

x² - 25 = 75

x² = 100

x = 10.  

(x = -10 is also a solution)

RE-arranging just a bit results in this:

x^2 - 2x -35 > 0

(x+5)(x-7)    >0      it IS zero between -5  and 7   and greater than 0 outside of this

-5>x>7

Here is the graph: