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
George can read 32 pages in one hour. At this rate, how<br> many pages can he read in 90 minutes
LekaFEV [45]

Answer:

48 pages

Step-by-step explanation:

32 x 1.5 = 48

6 0
2 years ago
Read 2 more answers
I need help pleaseee!!!!!!!!!!
vovikov84 [41]
<h2>Answer:</h2>

Using the distributive property, we can simplify the expression.

4 × 3 = 12

4 × \frac{1}{4} = 1

4 × (-\frac{1}{2}) = -2

Here is the simplified expression:

12 + c - 2d

6 0
3 years ago
Cathy cut 2 apples into 6 slices each.She ate 9 slices.What expression matches the words?
olasank [31]
(6×2)-9=3
6×2=12
12-9=3
3 apple slices are left
5 0
3 years ago
Read 2 more answers
How many sides does a polygon have if each of its interior angle is 120
vesna_86 [32]
There are six sides
8 0
3 years ago
Compare 6.23 * 10^14 and 8.912 * 10^12
Masteriza [31]

Answer:

623,000,000,000,000 > 8,912,000,000,000

Step-by-step explanation:

8.912·10^12

The exponent is 12, making it 10 to the power of 12. As the exponent is positive, the solution is a number greater than the origin or base number. To find our answer, we move the decimal to the right 12 time(s):

8.912 -> 8,912,000,000,000

6.23·10^14

The exponent is 14, making it 10 to the power of 14. As the exponent is positive, the solution is a number greater than the origin or base number. To find our answer, we move the decimal to the right 14 time(s):

6.23 -> 623,000,000,000,000

6 0
3 years ago
Other questions:
  • What are the possible values of x and y for two distinct points, (5,-2) and (x, y), to represent a function?
    15·1 answer
  • The sum of x and y decreased by their product
    8·1 answer
  • 8x-5(4x+3)-(3-2x)=4-4(2x-3)-(9-3x)
    11·1 answer
  • What is the volume of a rectangular prism with the dimensions: base 2/1 2, height 1/1 2, and length 3/1 2?
    10·2 answers
  • Which graph represents the solution set of y = x2 − 4 and x + y + 2 = 0?
    9·1 answer
  • Whats pv = 100000(1.03)^{-6}
    6·1 answer
  • A jewelry box contains two gold hoop earrings and two silver hoop earrings. You randomly choose two earrings. What is the probab
    12·2 answers
  • Gwen bought a hardcover book for $16.50 not including taxes. Her mother told her the tax was 8% on all purchases. What was the a
    9·1 answer
  • Please help me with this question!!
    6·1 answer
  • Find the area of the rectangle.
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!