CPhill

Let y be the number of large boxes and x be the number of small ones.

The possibilities for this problem occur in the intersection  of the inrequality  y > x with the equation of the line 10x + 8y = 96. as shown in this graph : https://www.desmos.com/calculator/o77ftjozeh

The only integer solutions occur at (0,12) and at (4, 7)

So.....if we require that we send at least one box of both types, the guest's answer of 4 small and 7 large cartons  is correct.

However, if we don't have this requirement......it is possible that we could just ship 12 cartons filled with large boxes only.....

y = 3xcosx

and the slope at any point on this cure is given by the derivative of this function.....so we have....

y ' = 3cosx - 3xsinx  =   3[cosx -xsinx]

So, the slope at (pi, -3pi)  =

y '(pi)  = 3 [ cos (pi)  - (pi) sin (pi)]   =  3 [-1  -  pi*0]  =   3 [-1]  = -3

And the equation  of the tangent line at this point =

y - (-3pi)  = -3(x - pi)

y + 3pi  = -3x + 3pi

y = -3x

Thanks, Rarinstraw and VelvetMusic

Word to "guest".....Rarinstraw's explanation couldn't have been more simple!!!!

150  = .75N    where N  was the totla number of people surveyed   ....divide both sides by .75

150 / .75 = N  = 200 surveyed

Thanks, guest....I made a boo-boo there....let me correct that....!!!

18.  If  BC is parallel to DE, then angle ADE = angle ABC. And by like reasoning, angle AED = angle ACB.  So, by angle-angle congruency, triangle ABC ~ triangle ADE

20. AD = DB   and AE = BC.....no.....for triangle ABC ~ ADE, we would have to have that AD = BC and AE = EC

I assume this is

f(x)= (-5x)/ [ sin(x)+cos(x)  ]   .....if so.....we can write it as

(-5x) [ sin (x) + cos(x)]^(-1)    ......so.....using the product and chain rules, we have.....

f ' (x) =  -5 [ sin (x) + cos(x)]^(-1)   +  (5x)[sin (x) + cos(x)]^(-2)  (cos(x) - sin(x) ]

So    f '  (2pi) =

-5 [ sin (2pi) + cos(2pi)] ^(- 1)  + (5*2pi) [ sin(2pi) + cos(2pi)]^(-2) ( cos(2pi)  - sin (2pi) ]  =

-5  [ 0 + 1]^(-1)  + (5*2pi) [ 0 + 1^(-2) ( 1 - 0)  =

-5 + 5*2pi  =

5 [2pi - 1]  =  about 26.416

Thanks to the guest for pointing out my earlier error....!!!!

Close enough, fellows.......a 5 point bonanza......!!!

Very nice, DS......

I assume that this is :

f(x)= (x+7)/e^x      note that this can be written as :

f (x)   =  ( x  + 7) * e^(-x)     so..........we can use the Product Rule to find the derivative

f ' (x)  = (1) * e^(-x)   + ( x + 7) * ( - 1)e^(-x)  =     e^(-x) [ 1 - ( x + 7) ] =  e^(-x) ( -x - 6)  =

-e^(-x) ( x + 6)

So

f ' (2)  =  -e^(-2)(2 + 6)  =  - [ 8 ] / [ e^2 ]   =  about  - 1.08268