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
Kobotan [32]
3 years ago
7

A cardboard box without a lid is to have a volume of 16,384 cm3. Find the dimensions that minimize the amount of cardboard used.

(Let x, y, and z be the dimensions of the cardboard box.)
Mathematics
1 answer:
Reika [66]3 years ago
6 0

Answer:

The dimensions that minimize the amount of cardboard used is

x = 31 cm ,  y = 34 cm & Z = 15.54 cm

Step-by-step explanation:

Volume of the cardboard = 16,384 cm^{3}

The function that represents the area of the cardboard without a lid is given by

f (x,y,z) = xy + 2xz + 2yz ------  (1)

Volume of the cardboard with sides x, y & z is

xyz = 16384

z =  \frac{16384}{xy}

Put this value of z in equation (1) we get

f (x,y,z) = xy + 2x(\frac{16384}{xy} ) + 2y(\frac{16384}{xy} )

f (x,y,z) = xy + \frac{32768}{y}  + 2y(\frac{32768}{x} )

Differentiate above equation with respect to x & y we get

f_{x}  = y - \frac{32768}{x^{2} }

f_{y}  = x - \frac{32768}{y^{2} }

Take f_{x}  = 0 \ and  \ f_{y} = 0

y - \frac{32768}{x^{2} } = 0

y = 32768 \ x^{-2}  ------ (2)

x - \frac{32768}{y^{2} } = 0

x = 32768 \ y^{-2}  ------- (3)

By solving equation (2) & (3) we get

x^{3} = 32768

x = 31 cm

From equation 2

y = 32768 \ x^{-2}

y = 32768 (31^{-2})

y = 34 cm

z =  \frac{16384}{xy}

z =  \frac{16384}{(34)(31)}

Z = 15.54 cm

Thus the dimensions that minimize the amount of cardboard used is

x = 31 cm ,  y = 34 cm & Z = 15.54 cm

You might be interested in
Find the area of each parallelogram plz help
xxTIMURxx [149]
Answer:  The area of the parallelogram is:
_______________________________________________________
\frac{9393}{8}  in² = 1174 ⅛  in² = 1174.125 in² .
_______________________________________________________
Explanation:
_______________________________________________________
Area of a parallelogram:
_______________________________________________________
 A = base * height = b * h ;

From the figure (from the actual "question"):
_______________________________________
         b = 50.5 in.

         h = 23.25 in.
____________________________________________________________
Method 1)  A = b * h =
                 
                     =  (50.5 in) * (23.25 in) = 1174.125 in² ; or, write as:  1174 <span>⅛ .
</span>____________________________________________________________
Method 2)  A = b * h =
             
                      =  (50 ½ in) * (23 <span>¼ in) =
   
                      =    (</span>\frac{101}{2} in) * (\frac{93}{4}<span> in) ;
</span>___________________________________________________________
Note:  "50 ½ " = [(50*2) + 1 ] / 2  = \frac{101}{2} ;

Note:  "23 ¼ " = [(23*4) + 1 ] / 4  = \frac{93}{4} ;
____________________________________________________________
→  A =  (\frac{101}{2} in) * (\frac{93}{4} in) ;
 
→  A = \frac{(101*93)}{(2*4)}  in² = \frac{9393}{8}  in² ; 
          
→  A = (9393/8) in² =  
         
→  A  = \frac{9393}{8}  in² = 1174 ⅛  in² = 1174.125 in² .
________________________________________________________
5 0
3 years ago
Helppppp rn please.
scoundrel [369]
1) neither. that’s not one of your answer choices so just pick one (maybe odd ‍♀️)
2) 2 is the answer. Plug in 1 for x, raise everything to its power (which will end up being 1), then use PEMDAS to solve
5 0
3 years ago
Solve the system of equations by elimination.
n200080 [17]

Answer:

Step-by-step explanation:

1. 2x - 9y = 23

2 x = 9 y + 23

y = (2 x)/9 - 23/9

2 x - 9 y - 23 = 0

2. 5x - 3y = -1

5 x + 1 = 3 y

y = (5 x)/3 + 1/3

5 x - 3 y + 1 = 0

I belive that should help you out a bit :D

4 0
2 years ago
Read 2 more answers
How do I solve this?
patriot [66]
You already know that y equals -4. It says that x equals 3, so multiply the -4 by 3 and because it has a negative number and a positive number it will equal -12. Remember anytime you see a letter after a number it means multiply. The complete problem should be -10 because 2 is a whole number you subtract it from -12. Hope that helped you out some.
6 0
2 years ago
2.75 As a fraction in simplest form
marshall27 [118]
2.75=2 3/4 because you would say 2.75 as two and seventy-five hundredths. Seventy-five hundredths is just like 3 quarters out of 4 because 1 quarter=25. 2=50. 3=75. So, 2.75 as a fraction in simplest form is 2 3/4.
Hope I helped:D
7 0
3 years ago
Read 2 more answers
Other questions:
  • Lines l and m are parallel and intersected by transversals t and s as shown in the figure below.
    9·1 answer
  • The button on Preston's jacket has a radius of 4 millimeters. What is the button's area?
    10·1 answer
  • 14 fluid ounces= ___cups
    5·1 answer
  • 3 people ar splitting a 10 ounce bag of pop corn if they share this popcorn equally how many ounces of popcorn will each person
    13·1 answer
  • Will give brainiest help
    11·2 answers
  • HELPPP PLEASEEEEE
    12·1 answer
  • 7) Bob has x Cheetos. Joey has 5 fewer that three times the number of
    10·2 answers
  • Part A: Explain why the x-coordinates of the points where the graphs of the equations y = ^2−x and y = ^4x + 3 intersect are the
    10·1 answer
  • Three ropes of length 2 m 84 cm, 4 m 26 cm and 7 m 10 cm are to be cut into small pieces of equal length. What is the greatest p
    12·2 answers
  • Ab= 6 / a+b = 5 <br><br> find a² + b²
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!