CPhill

Note something that we need to be aware of, helperid

If  x  =  -1/2....we will have negatives  under  both radicals....but each side will evaluate to the complex answer, "i"

Thanks, Hectictar.....!!!!

Here it is

The perimeter is given by

P  =  2 ( W + L)

P / 2  =   W + L    ⇒   P/2 - W  =  L     (1)

And the area is

A =  L  * W        sub  (1)  for L

A =  (P/2 - W) * W        simplify

(P/2) W  - W^2     rearrange

-1W^2  + (P/2)W

a =  -1    b  =  (P/2)

Again....the  "W"  that maximizes  the area is given by     -b / [ 2a]  =

- ( P/2) / [2 ( -1) ]  =  - P/ -4  =  P /4

And when  W  =  P/4       L  =  P/2 - P/4  =  P/4

So.....  W = L   =   P/4  =  100 / 4  =   25  ft

This will always be true, AT......for any perimeter......area is maximized  when  W = L  = P/4

I'm a little tired......but the area will be maximized when the rectangle is a square with a side of 25

I'll go through it, AT, if you want me to........

Lateral surface area  =

pi * ( sum of the radii) * sqrt [  difference of radii^2  + height^2 ]  =

pi *  ( 6 + 3) * sqrt [ ( 6 - 3)^2  + 4^2 ]  =

pi * ( 9 ) * sqrt [ 3^2 + 4^2 ]  =

pi * 9 * sqrt [ 25]  =

pi * 9 * 5  = 

45pi   in^2

See your other post for my explanation.....

Sorry.....I should have explained that....!!!!

In a quadratic that has the form  ax^2 + bx + c, the  maximum  [ or minimum]  is produced at the x value  given by :

-b  / [  2a]

So  ......using our function    -1000x^2 + 20000x + 1500000

a =  -1000      and b  =  20000

So....the x value [ additional employees hired ]  that produces the max revenue  is

-20000 / [ 2 ( -10000) ]  =    -20000 / -2000  = 10

Does that help ???

I just answered....see your other post, AT

These are always a little difficult for me to think of the correct function!!!

a. Call x the number  of additional people hired ........  so we have

Sales =  (30 + x) (50000 - 1000x)  =   -1000x^2 + 20000x + 1500000

b. To find the number of additional hires that will maximize revenue......we can use

x =  -20000 / [ -2 * 1000]  =  -20000 / -2000  =  10 additional people hired will maximize the revenue

The max sales  will be   (30 + 10) (50000 - 1000 (10) )  =  40 * 40000  = $ 1,600,000

The 121st element will be the first number in the 16th row = 1 ...  [ I am letting the "1"  at the top of the triangle be the first row ]

The sum of the entries in any row will be  2^(r - 1)   where r is the row number

So....the sum of the first 121 elements will be 

16

Σ  2^(r - 1)  +  1   =   [ 65535 + 1 ]   =  65536

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