A closed-top cylindrical container is to have a volume of 250 in2. 250 , in squared , . What dimensions (radius and height) will minimize the surface area?

Answers

Answer 1

Answer:

radius r = 3.414 in

height h = 6.8275 in

Step-by-step explanation:

From the information given:

The volume V of a closed cylindrical container with its surface area can be expressed as follows:

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

[tex]S = 2 \pi rh + 2 \pi r^2[/tex]

Given that Volume V = 250 in²

Then;

[tex]\pi r^2h = 250 \\ \\ h = \dfrac{250}{\pi r^2}[/tex]

We also know that the cylinder contains top and bottom circle and the area is equal to πr²,

Hence, if we incorporate these areas in the total area of the cylinder.

Then;

[tex]S = 2\pi r h + 2 \pi r ^2[/tex]

[tex]S = 2\pi r (\dfrac{250}{\pi r^2}) + 2 \pi r ^2[/tex]

[tex]S = \dfrac{500}{r} + 2 \pi r ^2[/tex]

To find the minimum by determining the radius at which the surface by using the first-order derivative.

[tex]S' = 0[/tex]

[tex]- \dfrac{500}{r^2} + 4 \pi r = 0[/tex]

[tex]r^3 = \dfrac{500 }{4 \pi}[/tex]

[tex]r^3 = 39.789[/tex]

[tex]r =\sqrt[3]{39.789}[/tex]

r = 3.414 in

Using the second-order derivative of S to determine the area is maximum or minimum at the radius, we have:

[tex]S'' = - \dfrac{500(-2)}{r^3}+ 4 \pi[/tex]

[tex]S'' = \dfrac{1000}{r^3}+ 4 \pi[/tex]

Thus, the minimum surface area will be used because the second-derivative shows that the area function is higher than zero.

Thus, from [tex]h = \dfrac{250}{\pi r^2}[/tex]

[tex]h = \dfrac{250}{\pi (3.414) ^2}[/tex]

h = 6.8275 in


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Answer:

Part A:  y = 4.5x + 22

Part B:  31 inches

Step-by-step explanation:

Part A:

the y-intercept appear to be at 22

the slope can be found by using the points (0, 22) and (4, 39)

m = 39 - 22

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Step-by-step explanation:

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Answer:

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Step-by-step explanation:

Answer:

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Answers

9514 1404 393

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Step-by-step explanation:

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at a summer camp the campers are divided into groups. each group of 16 campers requires 2 cabins. how many cabins are needed for 112 campers? (please show your work)​

Answers

Answer:

14 cabins

Step-by-step explanation:

16 ÷ 2 = 8 in 1 cabin

112 ÷ 8 = 14 cabins

Please consider marking brainliest :)

Hope this helps

Have a great day!!

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Answers

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Answers

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-2

Step-by-step explanation:

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Answer:

-2

Step-by-step explanation:

[tex]=\frac{-2}{3} \times\frac{3}{1}[/tex]

[tex]=\frac{-2 \times3}{3\times1}[/tex]

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The Perimeter of a rectangle the garden is 43.8 feet it’s length is 12.4 feet what is it width

Answers

12.4 + 12.4 = 24.8


43.8- 24.8 = 19

19/2 = 9.5


I think that is right hope I helped!!

Answer: 9.5 feet

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Answers

Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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Step-by-step explanation:

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Answers

Answer:

117

Step-by-step explanation:

So basically, how I figured it out is 20-15 is 1/4 of 20. So I had to find the 1/4 of 52 which is 13.  But 15 is 3/4 of 20 so then you multiply 13 times 3 which is 39.

10 is half of 20. So you find half of 52. Which is 26.

52+39+26 is your equation.

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I used 20 because in the smaller triangle 20 was the largest side like 52 in the biggest triangle.

Brainliest would be appreciated

First answer

Have a great day!

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Answers

Answer:

slope = 9/5

Step-by-step explanation:

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Answers

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Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:

Given a parallelogram LMNO,

LM = ON and OL = NM

Given;

ON = 9x - 8

LM = 8x + 4

NM = x - 8, and

OL = 3y - 5

Substitute into the expression

LM = ON

8x + 4 = 9x - 8

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