A rectangular box with an open top is to be made from a 14 14-in.-by- 30 30-in. piece of cardboard by removing small squares of equal size from the corners and folding up the remaining flaps. What should be the size of the squares cut from the corners so that the box will have the largest possible volume

Therefore 2.54 in be the size of the square cut from the corner of the cardboard.

Step-by-step explanation:

Given that, a rectangle box is to be made from 14 in by 30 in piece of cardboard with an open top.

The side of the square be x(say) in length.

The length of the rectangular box is =(14 - 2x) in

The width of the rectangular box is = (30-2x) in

The height of the rectangular box is = x in.

The volume of the rectangular box is =(14-2x)(30-2x)x

∴V=(14-2x)(30-2x)x [ where V is in cubic inches]

    =420x- 88x²+4x³

Differentiating with respect to x

V' = 420 -196x+12x²

Again differentiating with respect to x

V''= -196+24x

We set V' = 0 , to find the dimension of the for which the volume the box will be maximum.

∴420 -196x+12x²=0

⇒4(3x²-49x+105)=0

⇒3x²-49x+105=0

Apply the quadratic formula to find the value of x. The formula is , here a= 3, b= -49 , c=105

⇒x=13.80, 2.54

For x= 13.80 , the length of the box will be negative.

∴x=2.54

Therefore when x=2.54 , the volume of the box will be maximum.

Therefore 2.54 in be the size of the square cut from the corner of the cardboard.

ghjmutmum

step-by-step explanation:

answer: e 9, 5.75

step-by-step explanation:

step 1

find the variation ab in the x coordinate

delta x=(x2-x1)=10-> 8 units

find the variation ab  in the y coordinate 

delta y=(y2-y1)=6-> 2 units

step 2

find the coordinates point c

  x coordinate 

cx=ax+delta x*2/> cx=2+8*(2/> cx=3

  y coordinate 

cy=ay+delta y*2/> cy=4+2*(2/> cy=4.25

the coordinates of point c are (3,4.25)

step 3

find the coordinates point d

  x  coordinate 

dx=ax+delta x*4/> dx=2+8*(4/> dx=4

  y  coordinate 

dy=ay+delta y*4/> dy=4+2*(4/> dy=4.5

the coordinates of point d are (4, 4.5)

step 4

find the coordinates point e

  x  coordinate 

ex=ax+delta x*4/> ex=2+8*(6/> ex=5

  y  coordinate 

ey=ay+delta y*4/> ey=4+2*(6/> ey=4.75

the coordinates of point e are (5, 4.75)

step 5

find the coordinates point f

  x  coordinate 

fx=ax+delta x*4/> fx=2+8*(8/> fx=6

  y  coordinate 

fy=ay+delta y*4/> fy=4+2*(8/> fy=5

the coordinates of point f are (6,5)

step 6

find the coordinates point g

  x  coordinate 

gx=ax+delta x*4/> gx=2+8*(10/> gx=7

  y  coordinate 

gy=ay+delta y*4/> gy=4+2*(10/> gy=5.25

the coordinates of point g are (7,5.25)

step 7

find the coordinates point h

  x  coordinate 

hx=ax+delta x*4/> hx=2+8*(12/> hx=8

  y  coordinate 

hy=ay+delta y*4/> hy=4+2*(12/> hy=5.5

the coordinates of point h are (8,5.5)

step 8

find the coordinates point i

  x  coordinate 

ix=ax+delta x*4/> ix=2+8*(14/> ix=9

  y  coordinate 

iy=ay+delta y*4/> iy=4+2*(14/> iy=5.75

the coordinates of point i are (9,5.75)

- r3ktforgood ☕