1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
scoundrel [369]
3 years ago
13

[10 points] A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimens

ions will produce a box with maximum volume?
Mathematics
1 answer:
MissTica3 years ago
7 0

Answer:

  6 inches square by 3 inches high

Step-by-step explanation:

For a given surface area, the volume of an open-top box is maximized when it has the shape of half a cube. If the area were than of the whole cube, it would be 216 in² = 6×36 in².

That is, the bottom is 6 inches square, and the sides are 3 inches high.

_____

Let x and h represent the base edge length and box height, respectively. Then we have ...

  x² +4xh = 108 . . . . box surface area

Solving for height, we get ...

  h = (108 -x²)/(4x) = 27/x -x/4

The volume is the product of base area and height, so is ...

  V = x²h = x²(27/x -x/4) = 27x -x³/4

We want to maximize the volume, so we want to set its derivative to zero.

  dV/dx = 0 = 27 -(3/4)x²

  x² = (4/3)(27) = 36

  x = 6

  h = 108/x² = 3

The box is 6 inches square and 3 inches high.

_____

<em>Comment on maximum volume, minimum area</em>

In the general case of<em> an open-top box</em>, the volume is maximized when the cost of the bottom and the cost of each pair of opposite sides is the same. Here, the "cost" is simply the area, so the area of the bottom is 1/3 the total area, 36 in².

If the box has a <em>closed top</em>, then each pair of opposite sides will have the same cost for a maximum-volume box. If costs are uniform, the box is a cube.

You might be interested in
Help me please……..<br> Are B,E,and A collinear
Andrew [12]
It can be a yes because the line cross and make a perpendicular
5 0
3 years ago
What number should replace the question mark?
TiliK225 [7]

Answer:

Just a guess but I think 5

Step-by-step explanation:

The top right square on all boxes is the difference of the second box and the first box which equals the third box. This rule is applied to all of the other boxes so if I'm not wrong the question mark should be 5

3 0
3 years ago
Read 2 more answers
What to do from here? (Please don't solve)
Tems11 [23]
You have the answer if you just do a little bit more but other than that you have your answer

5 0
3 years ago
Please help!!! For extra credit.
qaws [65]

Answer:

x=24

Step-by-step explanation:

8x-13 = 7x+11; (because it's a rectangle)

8x - 7x = 11 + 13; (moving xs to one side, the rest to the other)

x = 24

7 0
3 years ago
Most investors in the stock market are trying to buy a stock at a _______
gogolik [260]

Answer:

d

Step-by-step explanation:

7 0
3 years ago
Other questions:
  • Yelp............................
    15·1 answer
  • If f(x) = square root of x+3, what is the equation for f–1(x)? f–1(x) = x2 - 3 f–1(x) = x2 + 3 f–1(x) = (x - 3) 2 f–1(x) = (x +
    8·1 answer
  • 10y - 2x=-6<br> Find the x- intercept and y- intercept
    5·2 answers
  • Plz help I need this to be answered
    12·1 answer
  • What’s the correct answer
    5·2 answers
  • Find the value of x (chords and arcs)
    6·1 answer
  • Negative 6n minus 2n ecualizarán 16
    9·1 answer
  • What’s the measure of angel B
    15·1 answer
  • Write the expression in simpliest form WILL MARK BRAINLIEST<br> 8 ( -3x + 2)
    6·2 answers
  • Valentina planted t fewer trees than Mabel. Mabel planted 91 trees. Write an expression that shows how many trees Valentina plan
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!