Questions   
Sort: 
 #1
avatar+130511 
+10

3^x  = x ^5       take the log of both sides

log 3^x   = logx^5     and we can write

x log3  = 5 log x

Rearrange

x / log x  = 5/log 3

 

Mmmm....that doesn't look too promising, does it???

Let's just graph the original functions and see where they meet (if they do....!!)

Here's a partial graph......https://www.desmos.com/calculator/roelalpkth

The graph shows that one solution occurs at about (1.343, 4.375)

Another solution occurs at about (10.351, 150,462).....shown here....https://www.desmos.com/calculator/ujgxybuyeq

 

This happens because from x values (-∞, 1.343), the graph of 3^x > x^5. Then  from (1.343, 10.351), the graph of x^5 > 3^x.  Finally, the graph of 3^x "overtakes" the graph of x^5 at about x = 10.351, and is always greater at any comparavle x value after that - as we would expect....!!!!

 

 

  

May 2, 2015
 #3
avatar+130511 
+10

I get a different answer from Melody on this one.....

 

Let x  be the number of 50 cent coins in the beginning

Let y be the number of 20 cent coins in the beginning

Let x + 24 be the number of 10 cent coins in the beginning

 

And after the spree we have

x - 18   = the number of 50 cent coins

(2/5)(x + 24) = the number of 10 cent coins = the number of 20 cent coins

 

So the total coins left after the spree = 2(2/5)(x + 24) + ( x -18)  = (4/5)(x + 24) + (x - 18)

And 40% of these = the number of 50 cent coins left   ....  so we have

.40[ (4/5)(x + 24) + (x - 18) ] = (x - 18)

(2/5)[ (4/5 (x + 24) + (x - 18) ] = (x - 18)

[ (4/5 (x + 24) + (x - 18) ]  = (5/2)(x - 18)   multiply through by 10

8(x + 24) + 10( x - 18) = 25(x - 18)

8x + 192 + 10x - 180  = 25 x - 450   simplify

18x + 12  = 25x - 450

462 = 7x    divide by 7 on each side

x = 66  this is the number of 50 cent coins in the beginning

And x + 24 = (66 + 24) = 90 were 10 cent coins in the beginning

And notice that (3/4)y = (2/5) (x + 24)...  so y = (8/15)(66 + 24) = (8/15)(90) = 48 coins 

So....there were 48 20 cent coins originally

 

Now.....let's check the math....

The total number of coins after the spree =

[(4/5)(66 + 24) + (66 - 18) ]  = 120

And (66 - 18) = 48 of these were 50 cent coins

So..... 48 / 120 = 40% were 50 cent coins

And after the spree....there were an equal number of 10 and 20 cent coins.....check ...(2/5)(90) = (3/4)(48) = 36 each

 

So the total spent  = (3/5)(90)(.10) + (1/4)(48)(.20) + (18)(.50)  = $16.80 

 

  

 

I believe Melody has a slight error

She gets 210 10 cent  coins at the start    .....and 2/5 of these are left = 84 left   

And this is equal  to the number of 20 cent coins left

And she calculates the number of 50 cent coins left = 210 - 42 =  168

So....the total coins left = [ 84 + 84 + 168]  = 336 coins left

But 168 / 336 = 50% of the coins left are 50 cent coins......and this is too much

 

Can anyone reconcile these differences ???

May 2, 2015

0 Online Users