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
jok3333 [9.3K]
3 years ago
14

A worker is to construct an open rectangular box with a square base and a volume of 196196 ft cubedft3. If material for the bott

om costs ​$88​/ft squaredft2 and material for the sides costs ​$77​/ft squaredft2​, what dimensions will result in the least expensive​ box? What is the minimum​ cost? Let x be the length of one side of the base of the box. Express the total cost of the​ materials, C, in terms of x.
Mathematics
1 answer:
Archy [21]3 years ago
4 0

Answer:

Therefore the dimensions of the box is 7 ft by 7 ft by 4 ft.

Therefore the minimum cost to manufacture the box is $12936.

Step-by-step explanation:

Given the volume of the open box is 196 cube ft.

Let the one side of the base of the box be x.

Since the base of the box is square in shape.

The other side of the base is also x.

The area of the base of the box = x² square ft

Let the height of the box be h.

The volume of the box is = Area of the base× height

                                          =(x²×h) cube ft.

                                           =x²h cube ft.

According to the problem,

x²h=196

\Rightarrow h=\frac{196}{x^2}

The material cost for the bottom is $88 per square ft and the material cost for the sides $77  per square ft.

The material cost for the bottom of the box = $(88×x²)

                                                                         =$88x²

The area sides = 2(length+width)height

                       =2(x+x)h

                      =4xh square ft.

The material cost for the sides of the box =$(77×4xh)

                                                                    =$308 xh

Total cost to make the box is =$(88x²+308 xh)

∴C=88x²+308 xh    [ C is in dollar]

Now putting h=\frac{196}{x^2}

C=88x^2+308x\times \frac{196}{x^2}

\Rightarrow C=88x^2+ \frac{60368}{x}

Differentiating with respect to x

C'=176x- \frac{60368}{x^2}

Again differentiating with respect to x

C''=176+ \frac{120736}{x^3}

Now to find the minimum cost, we set C'=0

\therefore 176x- \frac{60368}{x^2}=0

\Rightarrow 176x= \frac{60368}{x^2}

\Rightarrow x^3= \frac{60368}{176}

\Rightarrow x^3=343

\Rightarrow x=7

Now, C''|_{x=7}=176+ \frac{120736}{7^3}>0

Therefore at x=7, the manufacturing cost will be minimum.

The height of the box is h=\frac{196}{7^2} = 4ft

Therefore the dimensions of the box is 7 ft by 7 ft by 4 ft.

The manufacturing cost is C=88.7^2+ \frac{60368}{7}   [ putting x=7]

                                             =$12936

Therefore the minimum cost to manufacture the box is $12936.

You might be interested in
How are polynomials like other number systems such as whole numbers and integers?
Agata [3.3K]
You can add, subtract, and multiply them. These three operations obey the rules for integers. There's a polynomial division algorithm that fills formally the same role as the usual division algorithm for integers. Polynomials added to, subtracted from, or multiplied by other polynomials yield only polynomials. Likewise, integers added to, subtracted from, or multiplied by other integers yield only integers.
8 0
3 years ago
What is the area of this face plz help!!!!
Alexxx [7]

Answer:

The image is blurry on my screen

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
A bag contains colored skittles.
Anarel [89]
50%


Explanation: There are 24 skittles in total, 12 of then at green. 12 is half of 24, so, it would be 50%
5 0
3 years ago
1x+2x = what?<br> Will mark brainiest whatever that mean
MAXImum [283]

Answer:

3x

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
Sal invests $16,000 at age 35. He hopes the investment will be worth $320,000 when he turns 40. If the interest compounds contin
Kaylis [27]

59.9%

Sorry I don’t have time to explain in the middle of an exam

3 0
3 years ago
Other questions:
  • Graph the function y = 2x3 – x2 – 4x + 5. To the nearest tenth, over which interval is the function decreasing?
    12·2 answers
  • A car insurance company suspects that the younger the driver is, the more reckless a driver he/she is. They take a survey and gr
    8·1 answer
  • The one i chose is wrong i think help.
    10·1 answer
  • Which are steps that could be used to solve 0 = 9(x2 + 6x) – 18 by completing the square? Check all that apply.
    9·2 answers
  • The slope-intercept form of a linear equation is y =mx +b, M is the slip of the line and b is where the line crosses the y-axis
    9·1 answer
  • In a town election ,Azeem got 42 percent of the vote cast and Hamza get the remaining votes . If the total number of votes cast
    12·1 answer
  • The percentage of students who attended the talent show for years 2018 to 2013 I shown year | percent of students 2008 | 83% 200
    13·1 answer
  • Write five hundred seven and seven hundredths in a decimal notation
    13·1 answer
  • What is the integration of the function x^2+y^sin(floor(x)))
    11·1 answer
  • Darryl ran 17.3 miles more than Imani last week. Darryl ran 26.3 miles. How many miles did Imani run?
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!