from a thin piece of cardboard 40 in by 40 in, square corners are cut out so that the sidescan be folded up to make a box. What dimensions will yield a box of maximum volume

20- max volume dimension

Step-by-step explanation:

since we don't know the amount of cardboard cut off, we'll take it as "x"^^

so the bottom of the box have dimensions of 40-2x(since squares are cut from both corners in a side) by 40-2x hence we can take area of base as

(40-2x)^2

since v=base*height

lets take height as x

x(40-2x) (40-2x)=v

(40x-2x^2)(40-2x)=v

4x^3-160x^2+1600x=v

take the derivitive: 12x^2-320x+1600

factor:

4(3x^2-80x +400)=0

4(3x-20)(x-20)=0

12x-80=0

x-20=0

x=20, 6.667(reduced from 6.66666666667)

dilation scale factor 2: starting with δabc, draw the dilation image of the triangle with a center at the origin and a scale factor of two. notice that every coordinate of the original triangle has been multiplied by the scale factor (x2). dilation with scale factor 2, multiply by 2.