0. Folded boxes a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way. b. Squares with sides of length x are cut out of each corner of a square piece of cardboard with sides of length /. Find the volume of the largest open box that can be formed in this way.

Box dimensions:

L = 3 ft

D = 6 ft

H = 1 ft

Volume  V = 18 ft³

Step-by-step explanation:

a) Each side of the future box is

L = 5 - 2*x     D = 8 - 2*x  

height of the box is   x

Then the volume of the box as function of x is:

V(x) =  ( 5 - 2*x ) * ( 8  - 2*x ) * x

V(x) = ( 40 - 10*x - 16*x + 4*x² ) * x

V(x) = 40*x - 26*x² + 4*x³

Tacking derivatives on both sides of the equation:

V´(x) = 40 - 52*x + 12x²     or     V´(x) = 20 - 26*x + 6*x²

If V´(x) = 0

6*x² - 26*x + 20  = 0

Solving for x

x₁,₂  = (-b ± √ b² - 4*a*c  ] /2*a

x₁,₂  = ( 26  ±  √ 676  - 480  ) / 12

x₁,₂  =  (26 ± 14 )/ 12

x₁  = 1

x₂ = 3,33        ( we dismiss this solution since is not feasible according to cardboard dimensions )

Then the sides of the box are:

L = 5 -2*x      L  =  5 - 2*1     L = 3 ft

D = 8 - 2*x    D  = 8 - 2*1     D = 6 ft

H = x = 1 ft

And the volume is    V = 6*3*1 = 18 ft³

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3

the correct answer is $7.38

step-by-step explanation: