A 24 inch wire is cut in two and shaped into a square and a regular octagon . What is the minimum possible sum of the two areas?

A(t) = 41,47 in²

Step-by-step explanation:

Let´s call "x" the cut of point to get to pieces of wire, we make a square from x and the regular octagon will be shaped with 24-x

Then Area of the square  A(s)  =  x²

Area of the octagon is  A(o)  = 1/2*p*length of apothem (d)

p = ( 24 - x )

length of apothem (d) :

The side of the octagon is equal to ( 24 - x ) / 8 half the side is  

( 24  -  x ) / 16

tan α  = ( 24- x ) 16 / d      since ∡s in octagon are 360 / 8  = 45°

α ( ∡ between apothem and one of the interiors ∡ of the octagon )half of 45 is  α = 22,5°  

tanα = 0,41

d = (24 - x ) / 16*0,41        d = ( 24 - x ) / 6,56

Then

A(t) = A(s) + A(o)

A(t) =  x²  + (1/2)* ( 24 - x ) ( 24 - x ) / 6,56

Note  A(t) = A(x)

A(x) =  x²  + (1/2) * (24 - x )²/ 6,56

A(x) = x² + ( 1/ 2*6,56) * ( (24)² -48*x + x² )

Taking derivatives on both sides of the equation

A´(x) = 2*x + ( 1/13,12)* ( - 48 + 2x )

A´(x) = 2*x - 48/ 13,12 + 2*x

A´(x) = 4*x - 3,66

A´(x) = 0     4x = 3,66        x  = 0,91 in    and  d =( 24 - x ) / 6,56

d = ( 24 - 0,91 ) / 6,56        d = 3,52

Then  A(s) = (0,91)²      A(s) = 0,83 in²

A(o) = 1/2 * ( 24 - 0,91 )* 3,52

A(o) = 40,63 in²

A(t) = 40,63 + 0,83

A(t) = 41,47 in²

stew reads at the fastest rate of 265 words per minute.

step-by-step explanation:

it's a given that felicia reads at 260 word per minute and on connie's graph we can see that she reads 200 words per minute. since stew's table tells us that he read 795 in 3 minutes, we can divide 795 by 3 to find out how many words he reads per minute.

795/3 = 265

stew reads 265 words per minute - he reads at the fastest rate since felecia reads at 260 words per minute and connie reads at 200 words per minute.

stew (265) > felecia (260) > connie (200)