A container manufacturing company was contracted to design and manufacture cylindrical cans for fruit juice. The volume of each can needs to be 0.946 liters. In order to minimize production costs, the company wishes to design cans that require the smallest amount of material possible. (This means that you need to design a can with volume=.946 liters and the smallest possible surface area. Explain the relationship between the surface area and the amount of material.)

Answers

Answer 1

The volume of the cylinder is the amount of fruit juice it can contain.

The relationship between the volume and the surface area is:

[tex]\mathbf{A = \pi (\sqrt[3]{\frac{V}{2\pi}})^2 + \frac{0.946}{(\sqrt[3]{\frac{V}{2\pi}})}}[/tex]

The given parameter is:

[tex]\mathbf{V = 0.946}[/tex]

The volume of a cylinder is calculated as:

[tex]\mathbf{V = \pi r^2 h}[/tex]

Make h the subject

[tex]\mathbf{h = \frac{V}{ \pi r^2}}[/tex]

The surface area (A) of a cylinder is:

[tex]\mathbf{A = \pi r^2 + \pi rh}[/tex]

Substitute [tex]\mathbf{h = \frac{V}{ \pi r^2}}[/tex]

[tex]\mathbf{A = \pi r^2 + \pi r \times \frac{V}{\pi r^2}}[/tex]

[tex]\mathbf{A = \pi r^2 + \frac{V}{r}}[/tex]

Differentiate

[tex]\mathbf{A' = 2\pi r - Vr^{-2}}[/tex]

Set to 0

[tex]\mathbf{2\pi r - Vr^{-2} = 0}[/tex]

Rewrite as:

[tex]\mathbf{ 2\pi r = Vr^{-2}}[/tex]

Multiply through by r^2

[tex]\mathbf{ 2\pi r^3 = V}[/tex]

Solve for r

[tex]\mathbf{r = \sqrt[3]{\frac{V}{2\pi}}}[/tex]

[tex]\mathbf{A = \pi r^2 + \frac{0.946}{r}}[/tex]

So, we have:

[tex]\mathbf{A = \pi (\sqrt[3]{\frac{V}{2\pi}})^2 + \frac{0.946}{(\sqrt[3]{\frac{V}{2\pi}})}}[/tex]

Hence, the relationship between the volume and the surface area is:

[tex]\mathbf{A = \pi (\sqrt[3]{\frac{V}{2\pi}})^2 + \frac{0.946}{(\sqrt[3]{\frac{V}{2\pi}})}}[/tex]

Read more about surface areas and volumes at:

https://brainly.com/question/3628550


Related Questions

guys help please what is this

Answers

Answer:

A & C

Step-by-step explanation:

You're looking for rational numbers that are also integers, but because (as the figure shows) ALL integers are already rational, you only have to worry about the integer part. An integer is a whole number, positive or negative. So, you need to simplify each element of each set (the numbers inside the curly braces) to see if they're whole.

A - The square root of 100 is just 10 or -10, divided by -2 is -5 or 5. 4, -2, 5, and -5 are all integers.

B - Wrong because 4/3 is not a whole number

C - 16/4 is just 4, and the square root of 49 is 7 or -7. 4, -6, 2, 7, & -7 are all integers.

D - Wrong because 27/2 is not a whole number

E - Wrong because 0.2 is not a whole number

Let me know if you need a more thorough explanation! (:

16% of 95 is what number?

Please answer this fast I'll mark you as brainliest. Thank You.

Answers

Answer:

16% of 95 is 15.2

Step-by-step explanation:

0.16×95= 15.2

16.2


Trust me bro I got you

Direction: the indicated operation reduce the answer to lowest terms whenever possible.

1. 11/12 - 5/12 = ?

2. 6 ½ - 1 9/10 =?

3. 5 ¾ + 9 1/6 =?

4. 7 2/8 - 4 5/6 = ?

5. 5 1/3 - 1 4/5 = ?

6. 7 ¾ - 5 ½ = ?

7. 5/8 - 3/8 = ?

8. 7/15 + 8/15 + 2/15 = ?

9. ¾ + ½ + 2/8 = ?

10. 9/10 - 2/5 = ?

PLEASE HELP NOW
MATHEMATICS​

Answers

Answer:

1/24 3/5 14 11/122 5/ 123 8/ 152 1/ 41/ 41 2/ 151 1/ 21/ 2

Can someone tell me the answer to this please?

Answers

Answer:

B

Step-by-step explanation:

HELP, VERY EASY MATHS QUESTION

Answers

Answer:

z=7

Step-by-step explanation:

area = ½ × (a + b) × h.

260 = 37 + 4z × 8 ×0.5

260/0.5 = 520

520/8=65

37 + 4z = 65

65-37 = 28

28/4=7

z=7

Answer:

7

Step-by-step explanation:

1/2 (37 + 4a)  8  = 260       260/8 = 32.5    32.5 * 2 = 65    65-37=28   28/4 = 7  a = 7

How many solutions does the nonlinear system of equations graphed below have?

A. One
B. Four
C. Two
D. Three​

Answers

Answer:

There are three possible types of solutions for a system of nonlinear equations involving a parabola and a line.

Step-by-step explanation:

What is the fraction form of the following decimal expansion?

Answers

Answer: B

Step-by-step explanation:

On a blueprint, the scale says that 2 cm represents 3 m. What is the actual length of a wall if it's measurement on the blueprint is 7 cm?

Answers

Answer:

7!!

Step-by-step explanation:

Whats This Questions Answer Answer It Cuz I Have math Exam After 3Days

Answers

Answer:

bde=30 degrees

Step-by-step explanation:

angle abc = 50 degrees

50+100=150

180-150=30degrees

If x = a sin α, cos β, y = b sin α.sin β and z = c cos α then (x²/a²) + (y²/b²) + (z²/c²) = ?​

Answers

[tex]\large\underline{\sf{Solution-}}[/tex]

Given:

[tex] \rm \longmapsto x = a \sin \alpha \cos \beta [/tex]

[tex] \rm \longmapsto y = b \sin \alpha \sin \beta [/tex]

[tex] \rm \longmapsto z = c\cos \alpha[/tex]

Therefore:

[tex] \rm \longmapsto \dfrac{x}{a} = \sin \alpha \cos \beta [/tex]

[tex] \rm \longmapsto \dfrac{y}{b} = \sin \alpha \sin \beta [/tex]

[tex] \rm \longmapsto \dfrac{z}{c} = \cos \alpha[/tex]

Now:

[tex] \rm = \dfrac{ {x}^{2} }{ {a}^{2}} + \dfrac{ {y}^{2} }{ {b}^{2} } + \dfrac{ {z}^{2} }{ {c}^{2} } [/tex]

[tex] \rm = { \sin}^{2} \alpha \cos^{2} \beta + { \sin}^{2} \alpha \sin^{2} \beta + { \cos}^{2} \alpha [/tex]

[tex] \rm = { \sin}^{2} \alpha (\cos^{2} \beta + \sin^{2} \beta )+ { \cos}^{2} \alpha [/tex]

[tex] \rm = { \sin}^{2} \alpha \cdot1+ { \cos}^{2} \alpha [/tex]

[tex] \rm = { \sin}^{2} \alpha + { \cos}^{2} \alpha [/tex]

[tex] \rm = 1[/tex]

Therefore:

[tex] \rm \longmapsto\dfrac{ {x}^{2} }{ {a}^{2}} + \dfrac{ {y}^{2} }{ {b}^{2} } + \dfrac{ {z}^{2} }{ {c}^{2} } = 1[/tex]

find x if m<2=6x-12
a)12
b)9
c)6
d)-7

Answers

Answer:

12 ,,............,.,...,,,.

The value of x is 12 (Option a) if m∠2 = 6x - 12.

The given figure is an equilateral triangle, it means all three angles are equal. Therefore, we can assume that m∠2 is one of the angles of the equilateral triangle.

From the given information, we have:

m∠2 = 6x - 12

Since all angles of an equilateral triangle are equal, we can set up the equation:

m∠2 = m∠1 = m∠3

Let's assume m∠1 = m∠2 = m∠3 = y (where y represents the measure of each angle in the equilateral triangle).

We can equate the measures of the angles:

y = 6x - 12

Now, let's solve this equation to find the value of x:

6x - 12 = y

Since y represents the measure of each angle in the equilateral triangle, we can substitute y with 60° (since the sum of the angles in an equilateral triangle is 180°, and each angle is 60°).

6x - 12 = 60

Adding 12 to both sides:

6x = 72

Dividing both sides by 6:

x = 12

Therefore, the value of x is 12.

So, the correct option is: (a) 12

To know more about value, refer here:

https://brainly.com/question/22572776

#SPJ2

what does x equel if there are 9 people 2 run away and the remaining number is x

Answers

Answer:

x=7 people

Step-by-step explanation:

x=9-2

PLS HELP ASAP ILL GIVE BRAINLIEST TYSM

Answers

Answer:

3x+2=16

Step-by-step explanation:

Answer:

the answer is most likely to be rather the first or second choice, but i choose B

Step-by-step explanation:

*Hope this helps*

based on what i understood, 3 green rectangles (or whaever that shape is), the 2 red squares, and the 16 small yellow squares. and the answer choices have

3x - 2 = 16

3x +2 = 16

3x + 4 = 20

3x = 24

and i dont think its 3x + 4 = 20 or 3x = 24 because there isn't any shapes to represent the 4, 20, and 24.

I might be wrong but i hope its right...

George walked 6 miles in 2 hours
What was his average walking rate in miles per hour?
2 miles per hour
3 miles per hour
4 miles per hour
I don't know.

Answers

Answer:

b) 3 miles per hour

Step-by-step explanation:

If George walked 6 miles is 2 hours, his average walking rate would be 3 miles per hour.

6 ÷ 2 = 3.

A mechanic charges a flat fee of $30 plus $25 an hour for labor. Which equation can be used to determine how many hours, 1, the mechanic spends working on a
particular job if the total amount earned is T dollars?

Answers

T = 25x + 30. Where 25 is the charge per hour x is # of hours and 30 is the flat fee only charged 1 time.

(b) It takes 37 pounds of seed to completely plant a 4-acre field. How many pounds of seed are needed per acre?​

Answers

Answer:

9.25 lbs

Step-by-step explanation:

37 is the amount for 4 acres or 4a:

37  = 4a

Isolate and find a by dividing both sides by 4:

4a = 37

4a ÷ 4 = [tex]\frac{37}{4}[/tex]

a = [tex]\frac{37}{4}[/tex] or 9.25 lbs

Hope this helps!

May I please receive help?

Answers

Answer:
I= 61
J=58
K= 61
I have attached the work below.
I know is
l = 61
have a great dayyy

2. Solve x2 + 2x + 7 = 0 using the Quadratic Formula​

Answers

Answer:

-1 ± i√7

Step-by-step explanation:

x² + 2x + 7 = 0D = 2² - 4*7 = 4 - 56 = -52x = (-2 ± √-28)/2 = (-2 ± 2√-7)/2 = -1 ± i√7

Armando wants to raise funds at his school fair. He plans to charge 10 cents to pick marshmallows from his bag without looking. He will give: • 20 cents to anyone who picks a Blue marshmallow. • 50 cents to anyone who picks a Yellow marshmallow. • $1 to anyone who picks a White marshmallow. • Anyone picking a Red or Green marshmallow will lose their money There are as equal number of marshmallows of each color. 3a. Show work and explain why Armando will lose money with this game. Justify using mathematics.

Answers

Since the expected value for this game is negative, Armando will lose money.

The expected value is given by the sum of all outcomes multiplied by it's respective probability.

In this problem:

There are blue, yellow, white, red and green marshmallows, each with a [tex]\frac{1}{5}[/tex] probability of being picked.If red or green marshmallows are picked, Armando get the charge of $0.1.If someone picks a blue marshmallow, he loses $0.1.If someone picks a yellow marshmallow, he loses $0.4.If someone picks a white marshmallow, he loses $0.9.

Thus, the expected value is:

[tex]E = 0.1\frac{2}{5} - 0.1\frac{1}{5} - 0.4\frac{1}{5} - 0.9\frac{1}{5} = \frac{0.2 - 0.1 - 0.4 - 0.9}{5} = -0.24[/tex]

Since the expected value for this game is negative, Armando will lose money.

A similar problem is given at https://brainly.com/question/24855677

x-2y=-8 slope intercept form
need it asap plz

Answers

Answer:

y = 1/2x + 4

Step-by-step explanation:

hope this helps

Which equation in slope-intercept form has a graph with an x-intercept at 3 and a y-intercept at -2?

Answers

Answer:

a

Step-by-step explanation:

A is the answer, since when you graph it, you are able to visualize y crossing -2 and x crossing 3.

A hardware dealer has 168kg of nails, if one nail weighs 0.25kg, how many nails he have?​

Answers

Answer:

He has 672 nails

Step-by-step explanation:

168/.25

Answer:

He has 672 nails.

Step-by-step explanation:

Weight of all nails=168

Weight of a nail=0 .25

Then,

Number of nails=168/0.25

=672

What is the solution and answer of this question:
3 log 10 + 9 log 10

Answers

Answer:

12

Step-by-step explanation:

the answer is 12 ndndbdbdndnfnnfnfnfnfnfnfnfnfn

12 is the correct answer

Does anyone know this answer for 5 and 6??

Answers

5) direct
6) inverse

Brandon enters bike races 8 1/2 miles every 1/2 hour.

Answers

i have no idea what it is

Show that the expression 1.50(10-r) is equivalent to 15-1.50r

Answers

Answer:

third answer

Step-by-step explanation:

what is the value of x for 10x+1=3(4x+2)-3? ​

Answers

Answer:

x= 7

Step-by-step explanation:

Answer:-1

Step-by-step explanation:

10x+1=3(4x+2)-3

10x+1=12x+6-3

10x+1=12x+3

1=2x+3

-2=2x

1=x

Question 1 (2 points)
What is the common difference d of the sequence?
-15, -4, 7, 18, 29, ...
• 11
•-11
•12
•-12

Answers

The answer is positive 11

Is -3 the same as 0.3

Answers

No they are not same .

Hope it helps

Answer:

Nope! They are not the same. This is because -3 is a negative whole number. On the other hand, 0.3 converts to 3/10 which is a positive decimal number. Hope this helps. I have attached below a number line showing the two different points (with the given number on each end).

Hope this helps and be sure to have a wonderful time ahead at Brainly! :D

Activity Give the exponential model for the following situation​

Answers

Answer:

Suppose that a couple invested $50,000 in an account when their child was born, to prepare for the child's college education. If the average interest rate is 4.4% compounded annually, ( A ) Give an exponential model for the situation, and ( B ) Will the money be doubled by the time the child turns 18 years old?

( A ) First picture signifies the growth of money per year.

( B ) Yes, the money will be doubled as it's maturity would be $108,537.29.

a = p(1 + \frac{r}{n} ) {}^{nt}a=p(1+

n

r

)

nt

a = 50.000.00(1 + \frac{0.044}{1} ) {}^{(1)(18)}a=50.000.00(1+

1

0.044

)

(1)(18)

a = 50.000.00(1 + 0.044) {}^{(1)(18)}a=50.000.00(1+0.044)

(1)(18)

a = 50.000.00(1.044) {}^{(18)}a=50.000.00(1.044)

(18)

50,000.00 ( 2.17074583287910578440507440 it did not round off as the exact decimal is needed.

a = 108.537.29a=108.537.29

Step-by-step explanation:

Hope This Help you!!

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