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
garik1379 [7]
3 years ago
8

An open rectangular box having a volume of 108 in.3 is to be constructed from a tin sheet. Find the dimensions of such a box if

the amount of material used in its construction is to be minimal. Hint: Let the dimensions of the box be x in. by y in. by z in. Then, xyz
Mathematics
1 answer:
murzikaleks [220]3 years ago
6 0

Answer:

6 in x 6 in x 3 in.

Step-by-step explanation:

Given

V = xyz = 108   ⇒   z = 108/(xy)

The amount of the material used is

S = xy + 2yz + 2xz

Put value of z from the volume

S = xy + 2y*108/(xy) + 2x*108/(xy) = xy + 216/x + 216/y

Now, we find the relative minimum of the function S(x,y)

First, we find the critical point. Set Sx = 0  and  Sy = 0

and solve this system:

Sx(x,y) = y - (216/x²) = 0

Sy(x,y) = x - (216/y²) = 0

From the first equation we have

y = 216/x²

Put it in the second equation and find x

x - (216/(216/x²)²) = 0

⇒  x*(1 - (x³/216)) = 0

⇒  x₁ = 0   and  x₂ = 6

Now, we can find y as follows

y₁ = 216/(0)²   which is undefined

y₂ = 216/(6)² = 6

Hence, the only critical point of S is (6, 6). Next, we calculate the second ordered derivatives that we need for the second derivative test:

Sxx(x,y) = 432/x³

Sxy(x,y) = 1

Syy(x,y) = 432/y³

Applying the second derivative test

D(6, 6) = Sxx(6, 6)*Syy(6, 6) - S²xy(6, 6) = 2*2 - 1² = 4 -1 = 3 > 0

Sxx(6, 6) = 2 > 0

Since D(6, 6) > 0   and   Sxx(6, 6) > 0   we can conclude that S has a relative minimum at (6, 6).

z coordinate is:

z = 108/(xy) = 108 / (6*6) = 3

Finally, the dimentions of a box are 6 in x 6 in x 3 in.

You might be interested in
What is the range ? In this picture
aalyn [17]

Answer:

The range is the difference between the highest and lowest number

4 0
3 years ago
Read 2 more answers
8w-15=57 two step equations
insens350 [35]

Answer:

w=9

Step-by-step explanation:

Our equation is: 8w-15=57

Add 15 to both sides

-15 is now gone because we added 15 so it would be zero

and we have 72 because 15 plus 57 is 72

Now we have:

8w=72

Divide by 8 on both sides

8w cancels out to just w

and 72 divided by 8 is 9

And you're left with:

w=9

       

7 0
3 years ago
Help pls it is in the picture
Genrish500 [490]

Answer:

come back

Step-by-step explanation:

5 0
2 years ago
The median of the data set is 81. What number is missing?<br><br> 82, 87, _, 77, 70, 80, 88
mina [271]
Median is the middle number.
Rearrange the numbers to smallest to biggest.
70, 77, 80, _, 82, 87, 88
If the median of the data set is 81 the middle number needs to be 81.
So the missing number is 81.
70, 77, 80, 81, 82, 87, 88
7 0
3 years ago
Read 2 more answers
Can some people answer this for Me please
Alik [6]
Equal to 1, and you'll just add a 4 and that will make it 4/4 and like both are the same numbers it will become 1 whole

7 0
3 years ago
Other questions:
  • T(z)=z-8 <br> Evaluate t(-2)
    6·1 answer
  • What is the variable of 5.76
    10·1 answer
  • Find tha gcf of 20xy and 24xzy
    10·2 answers
  • Using long divison convert the following mixed number into its decimal equivalent -3 2/3 ​
    5·1 answer
  • The diagonals of a square measure 20. what is the length of a side of the square​
    15·1 answer
  • I need help please. Thank you!​
    9·1 answer
  • What is the following sale price for a phone charger with an original price of $1.99 after applying a discount of 20%
    11·1 answer
  • Solve for the unknown in the following diagram. Round the answer to two decimal places.<br>cm​
    11·1 answer
  • Mandy baby sat for 3 1/4 hours on Friday night and 4 5/6 hours on Saturday. How many hours did she babysit this weekend?
    10·1 answer
  • Solve using the quadratic equation -4x² + 7x + 5 = 0​
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!