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Alex_Xolod [135]
3 years ago
12

A rectangular box with a volume of 684 ftcubed is to be constructed with a square base and top. The cost per square foot for the

bottom is 20cents​, for the top is 10cents​, and for the sides is 2.5cents. What dimensions will minimize the​ cost?
Mathematics
1 answer:
umka21 [38]3 years ago
3 0

Answer:

The dimensions of the rectangular box is 29.08 ft×29.08 ft×4.85 ft.

Minimum cost= 26,779.77 cents.

Step-by-step explanation:

Given that a rectangular box with a volume of 684 ft³.

The base and the top of the rectangular box is square in shape.

Let the length and width of the rectangular box be x.

[since the base is square in shape,  length=width]

and the height of the rectangular box be h.

The volume of rectangular box is = Length ×width × height

                                                       =(x²h) ft³

According to the problem,

x^2h=684

\Rightarrow h=\frac{684}{x^2}.....(1)

The area of the base and top of rectangular box is = x² ft²

The surface area of the sides= 2(length+width) height

                                                =2(x+x)h

                                                =4xh ft²

The total cost to construct the rectangular box is

=[(x²×20)+(x²×10)+(4xh×2.5)] cents

=(20x²+10x²+10xh) cents

=(30x²+10xh) cents

Total cost= C(x).

C(x) is in cents.

∴C(x)=30x²+10xh

Putting h=\frac{684}{x^2}

C(x)=30x^2+10x\times\frac{684}{x^2}

\Rightarrow C(x)=30x^2+\frac{6840}{x}

Differentiating with respect to x

C'(x)=60x-\frac{6840}{x^2}

To find minimum cost, we set C'(x)=0

\therefore60x-\frac{6840}{x^2}=0

\Rightarrow60x=\frac{6840}{x^2}

\Rightarrow x^3=\frac{6840}{60}

\Rightarrow x\approx 4.85 ft.

Putting the value x in equation (1) we get

h=\frac{684}{(4.85)^2}

  ≈29.08 ft.

The dimensions of the rectangular box is 29.08 ft×29.08 ft×4.85 ft.

Minimum cost C(x)=[30(29.08)²+10(29.08)(4.85)] cents

                                =29,779.77 cents

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(x+4)/(3x+1) = 3/4

Cross multiply:

4(x+4) = 3(3x+1)
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Which inequality represents the phrase "paid more than $2.00"?
Nastasia [14]
Is greater than or equal to $.00
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2 years ago
Cw 10.5 10.5 V. W<br> A great
Semmy [17]

Answer:

Refer to step-by-step.

Step-by-step explanation:

12.  x = 28

BC = x

AB = 96

AC = 100

We use the Pythagorean theorem to find the value of x.

a² + b² = c²

x² + 96² = 100²

x² + 9216 = 10000

x² = 10000-9216

x² = 784

√x² = √784

x = 28

13. x = 64

a² + b² = c²

48² + x² = 80²

2304 + x² = 6400

x² = 6400 - 2304

x² = 4096

√x² = √4096

x = 64

14. YES and 25 = 25

a² + b² = c²

3² + 4² = 5²

9 + 16 = 25

25 = 25

So this means that AB is tangent to the circle.

15. NO and 45 ≠ 49

a² + b² = c²

3² + 6² = 7²

9 + 36 = 49

45 = 49

So this means the AB is not tangent to the circle.

16. x = 4.5 and P = 52

To find the value of x, we need to determine the value of our hypotenuse.

QU is congruent to QT, therefore, QT = 4

UR is congruent with SR, therefore, UR = 13

PS is congruent to AB, therefore:

2x = 9

Divide both sides by 2

x = 4.5

The perimeter of a triangle is:

P = a + b + c

a = 9 + 4 or 13

b = 2(4.5) + 13 or 22

c = 4 + 13 or 17

P = 13 + 22 + 17

P = 52

17. x = 13 and P = 72

TJ is congruent to UJ, therefore, TJ = 13

x = 13

The perimeter of a parallelogram is:

P = 2(a+b)

a = HR + RK

b = KU + UJ

a = 13 + 5 or 18

b = 5 + 13 or 18

P = 2(18 + 18)

P = 2(36)

P = 72

18. x = 8 and P = 80

We know that part of the whole of 26 is 18 because one side is congruent to 18.

to find the value of the other half, we simply subtract 18 from 26.

26 - 18 = 8

x is congruent to 8, therefore x = 8

The line segment 14 is congruent to the opposite segment of x, therefore making the value 14.

So then we have:

a = 8 + 14 or 22

b = 26

c = 18 + 14 or 32

P = a + b + c

P = 22 + 26 + 32

P = 80

19. x = 6 and P = 52

Now we have the case of x + 2 is congruent to 8.

x + 2 = 8

Combine like terms.

x = 8 - 2

x = 6

Now that we have the value of x, we can simply look for the value of the line segment attached to it.

4 is congruent to the line segment attached to x + 2, therefore the value is 4.

To find the perimeter we have to add all sides together.

P = (8+5)+(5+9)+(9+4)+(8+4)

P = 52

20. x = 5, y = 2, z =10, and P = 68

Let's take this one step at a time.

First we look for x.

2x + 2 = 3x - 3

Combine like terms.

2x - 3x = -3 - 2

-x = -5

Divide both sides by -1.

x = 5

Now let's get the value of y.

5y - 2 = 3y + 2

Combine like terms.

5y - 3y = 2 + 2

2y = 4

Divide both sides by 2.

y = 2

Now let's look for z.

34 - 2z = z + 4

Combine like terms.

-2z - z = 4 - 34

-3z = -30

Divide both sides by -3.

z = 10

Now that we have the values of x, y, and z. We can substitute them to find the values of our segments.

2x + 2

2(5) + 2 = 12

3x - 3

3(5)-3 = 12

5y - 2

5(2) - 2 = 8

3y + 2

3(2) + 2 = 8

z + 4

10 + 4 = 14

34 - 2z

34 - 2(10)

34 - 20 = 14

Now that we have our values let's look for our perimeter.

P = a + b + c

P = (12 + 8) + (12 + 14) + (14 + 8)

P = 20 + 26 + 22

P = 68

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3 years ago
The ratio of boys to girls in the Science Club is 3.5. If there are 60 girls, how many boys are there?
liubo4ka [24]

Answer:

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Step-by-step explanation:

                                                            3

Please write the ratio as 3:5 or as  --------

                                                             5

Note that 3 + 5 = 8, so that the fraction of boys in the club is 3/8 and that of girls is 5/8.  

How many are in the club altogether?  Let this count be represented by c.  Then (5/8)c = 60.

Multiplying both sides of this equation by (8/5), to isolate c, we get:

c = (8/5)(60) = 96.  Thus, there are 96 Science students altogether.

3/8 of this number are boys.  This comes out to (3/8)(96) = 36 boys.

We could also solve this problem in the following manner:

 3          b

------ = ------- , where b represents the number of boys.

  5        60

Then 5b = 180.  Dividing both sides by 5, we get b = 180/5 = 36 boys

3 0
3 years ago
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