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
Dominik [7]
3 years ago
7

A metal cylinder can with an open top and closed bottom is to have volume 4 cubic feet. Approximate the dimensions that require

the least amount of material. What would this amount be?
Mathematics
1 answer:
Aleksandr-060686 [28]3 years ago
8 0

Answer:

r\approx 1.084\ feet

h\approx 1.084\ feet

\displaystyle A=11.07\ ft^2

Step-by-step explanation:

<u>Optimizing With Derivatives </u>

The procedure to optimize a function (find its maximum or minimum) consists in :

  •  Produce a function which depends on only one variable
  •  Compute the first derivative and set it equal to 0
  •  Find the values for the variable, called critical points
  •  Compute the second derivative
  •  Evaluate the second derivative in the critical points. If it results positive, the critical point is a minimum, if it's negative, the critical point is a maximum

We know a cylinder has a volume of 4 ft^3. The volume of a cylinder is given by

\displaystyle V=\pi r^2h

Equating it to 4

\displaystyle \pi r^2h=4

Let's solve for h

\displaystyle h=\frac{4}{\pi r^2}

A cylinder with an open-top has only one circle as the shape of the lid and has a lateral area computed as a rectangle of height h and base equal to the length of a circle. Thus, the total area of the material to make the cylinder is

\displaystyle A=\pi r^2+2\pi rh

Replacing the formula of h

\displaystyle A=\pi r^2+2\pi r \left (\frac{4}{\pi r^2}\right )

Simplifying

\displaystyle A=\pi r^2+\frac{8}{r}

We have the function of the area in terms of one variable. Now we compute the first derivative and equal it to zero

\displaystyle A'=2\pi r-\frac{8}{r^2}=0

Rearranging

\displaystyle 2\pi r=\frac{8}{r^2}

Solving for r

\displaystyle r^3=\frac{4}{\pi }

\displaystyle r=\sqrt[3]{\frac{4}{\pi }}\approx 1.084\ feet

Computing h

\displaystyle h=\frac{4}{\pi \ r^2}\approx 1.084\ feet

We can see the height and the radius are of the same size. We check if the critical point is a maximum or a minimum by computing the second derivative

\displaystyle A''=2\pi+\frac{16}{r^3}

We can see it will be always positive regardless of the value of r (assumed positive too), so the critical point is a minimum.

The minimum area is

\displaystyle A=\pi(1.084)^2+\frac{8}{1.084}

\boxed{ A=11.07\ ft^2}

You might be interested in
I really need help...
lianna [129]
It could not be 73º because if you add that to 120 then it will be more than 180 degrees, which is the total number of degrees in a triangle. Hope this helps!
8 0
3 years ago
Read 2 more answers
8s in 5,188 what is the answer
Karo-lina-s [1.5K]

Answer:

648.5

Step-by-step explanation:

5188/8

648.5

Good luck!

5 0
3 years ago
Help me and u will get 100 points at the end
Lera25 [3.4K]

Answer:

x > 0 and y < 2

Step-by-step explanation:

By definition the inverse of a function is simply, when the domain and range are swapped. So if the domain of g(x) is x < 2 and y > 0, then the inverse will have these swapped. So x > 0 and y < 2

3 0
1 year ago
Read 2 more answers
Please help me, question is the picture
Hitman42 [59]
The answer is 425 miles.

Both of the companies can be represented by an equation. The first company being f(a) and the second company bring f(b).
f(a)=$0.06x+65
f(b)=$0.10x+48
We want to find where they are equal, so we can set the equations equal to each other.
$0.06x+65=$0.10x+48
From here we can simplify the equation.
17=$0.04x
425=x
8 0
3 years ago
Read 2 more answers
Does the equation 6x + 35 = -6x - 35 have no, one, or infinite solutions?
fenix001 [56]

Answer:

The equation has one solution.

Step-by-step explanation:

6x + 35 = -6x - 35

subtract 6x from both sides of the equation

35 = -35 - 12x

add 35 to both sides of the equation

70 = -12x

divide -12x on both sides of the equation

x = 5.833333333

6 0
2 years ago
Other questions:
  • a truck and its load weigh 22,800 pound. if the load weight 12,400 pounds more than the truck, how many does the truck weight?
    12·1 answer
  • One day in the school cafeteria the number of children who bought white milk was twice the number who bought chocolate milk how
    14·2 answers
  • Find the hypo when the opposite is 36 and the adjacent is 27​
    13·2 answers
  • Given a bottle with a height vs. volume graph that is created as the bottle is consistently filled with liquid, describe the cor
    6·1 answer
  • Helen rolls a dice and flips a coin calculate the probibility that she gets a 4 and tails
    10·2 answers
  • Create two linear functions that are parallel.
    7·1 answer
  • Find x <br> 3x-5 <br> 32<br> 5x-5<br> 56
    5·2 answers
  • Let u = ⟨–2, –3⟩ and v = ⟨3, –1⟩. Which graph shows u – v?
    15·2 answers
  • Intersecting Secant (Exterior) <br> Find X =
    7·1 answer
  • A right triangle is removed from a rectangle to create the shaded region shown below. Find the area of the shaded region. Be sur
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!