If an open box has a square base and a volume of 111 in.3 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. (Round your answers to two decimal places.)

  6.06 inches square by 3.03 inches high

Step-by-step explanation:

An open-top box will have a minimum surface area when its lateral area is double its base area. That is, left-area + right-area = front-area + back area = bottom area.

With that in mind, the simplest way to compute the dimensions is to find the base edge length as the cube root of double the volume.

  b = ∛(2×111 in³) ≈ 6.06 in

  h = b/2 = 3.03 in . . . . . there are 4 faces; each is half the area of the base

The box is 6.06 inches square and 3.03 inches high.

__

Solution using derivatives

Let b represent the edge length of the base. The volume is ...

  V = b^2h

Then the height is ...

  h = V/b^2

  h = 111/b^2

and the total surface area is ...

  A = b^2 + 4bh = b^2 +4·111/b

The area is maximized when the derivative is zero:

  dA/db = 0 = 2b -444/b^2

Solving for b, we get ...

  2b^3 = 444

  b^3 = 222

  b = ∛222 ≈ 6.06 . . . . . . as above

a.   we have two

lines:   y = 4-x   and  

y = 8-x^-1

given two simultaneous equations that are both to be

true, then the solution is the points where the lines cross. the intersection is

where the two equations are equal. therefore the solution that works for both

equations is when

4-x = 8-x^-1

this is where the two lines will cross and that is the

common point that satisfies both equations.

b.   4-x = 8-x^-1

  x           4-x       8-x^-1

-3           7        

8.33

-2           6         8.5

-1           5         9

  0           4        

-

  1           3        

7

  2           2        

7.5

  3           1         7.67

the table shows that none of the x values from -3 to 3 is

the solution because in no case does

4-x = 8-x^-1

to find the solution we need to rearrange the equation to

find for x:

4-x = 8-x^-1

multiply both sides with x:

4x-x^2 = 8x-1

x^2+4x-1=0

x= -4.236, 0.236

therefore there are two points that satisfies the

equation.

find y:  

x=-4.236

y = 4-x   = 4 – (-4.236)

= 8.236

y = 8-x^-1 =   .236)^-1

= 8.236

x=0.236

  y = 4-x   = 4 – (0.236) = 3.764

y = 8-x^-1 =   8-(0.236)^-1

= 3.764

thus the two lines cross at 2 points:

(-4.236, 8.236) & (0.236, 3.764)

c.   to solve

graphically the equation 4-x = 8-x^-1

we would graph both lines: y = 4-x   and   y

= 8-x^-1

the point on the graph where the lines cross is the

solution to the system of equations.

just graph the points on part b on a cartesian coordinate

system and extend the two lines.   the

solution is, as stated, the point where the two lines cross on the graph.

step-by-step explanation:

mark brainliest!

yes, gavin is right.

step-by-step explanation:

gavin is correct. 

this problem is about converting 9% per year to how many percent per quarter. the rate is still the same. it only differed in the time period. so, through dimensional analysis,

9% per year  × (1 year / 4 quarters) = 2.25% per quarter

if he was asked to convert 9% per year to how many percent per quarter  compounded quarterly, then you will not simply divide by four. this is because compounding is not linear.

hope this , have a good day. c;