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
AlladinOne [14]
3 years ago
6

A cone-shaped paper drinking cup is to be made to hold 27 cm of water. Find the height h and radius r of the cup that will use t

he smallest amount of paper. You will find that the domain of your function will not be a closed set this time. For this one, use the Second Derivative Test. Please write your answer accurate to three-decimal places. A recommendation: Write the area function A in terms of h, and then let T be the square of A. You should now have an expression that is free of radicals. Optimize this function T of h
Mathematics
1 answer:
andreyandreev [35.5K]3 years ago
8 0

Answer:

  • r = 2.632 cm
  • h = 3.722 cm

Step-by-step explanation:

The formula for the volume of a cone of radius r and height h is ...

  V = (1/3)πr²h

Then r² can be found in terms of h and V as ...

  r² = 3V/(πh)

The lateral surface area of the cone is ...

  A = (1/2)(2πr)√(r² +h²) = πr√(r² +h²)

The square of the area is ...

  T = A² = π²r²(r² +h²)

Substituting for r² using the expression above, we have ...

  T = π²(3V/(πh))((3V/(πh) +h²) = 9V²/h² +3πVh

We want to find the minimum, which we can do by setting the derivative to zero.

  dT/dh = -18V²/h³ +3πV

This will be zero when ...

  3πV = 18V²/h³

  h³ = 6V/π . . . . . multiply by h³/(3πV)

For V = 27 cm³, the value of h that minimizes paper area is ...

  h = 3∛(6/π) ≈ 3.7221029

The corresponding value of r is ...

  r = √(3V/(πh)) = 9/√(π·h) ≈ 2.6319242

The optimal radius is 2.632 cm; the optimal height is 3.722 cm.

_____

The second derivative test applied to T finds that T is always concave upward, so the value we found is a minimum.

__

Interestingly, the ratio of h to r is √2.

You might be interested in
Solve y - 3x + 15 = 10
dimulka [17.4K]

Answer:


Step-by-step explanation:


3 0
3 years ago
Ivan purchased n notebooks. They were 4 dollars each. Write an equation to represent the total cost c that Ivan paid.
xenn [34]

Answer:

4n=c

Step-by-step explanation:

your trying to figure out how how much he paid and how much money it cost total so you have to multiply the cost of one by the number of notebooks you have n and by doing that you would be able to get your total cost c

3 0
3 years ago
Read 2 more answers
Juan has $10 to spend. He needs $13 to purchase the
alexandr402 [8]

Answer:

JUST ADD 3$

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
One - eighth of 7.92 x 10^10
liraira [26]
The answer is 9900000000 (thats 8 zeros) What you do is you find 10^10 which is 10000000000 (ten zeros) and multiply that by 7.92 to get 79200000000 then you just multiply it by 1/8 to get the answer.
3 0
3 years ago
Is -12/6 a rational number?
stiv31 [10]
A rational number is any number that can be turned into a fraction....it can be negative or positive.

Therefore, -12/6 is a rational number
3 0
3 years ago
Other questions:
  •  The price of Stock A at 9 A.M. was ​$13.56. Since​ then, the price has been increasing at the rate of ​$0.12 each hour. At noon
    9·2 answers
  • An object is taken out of a 21 degree C room and placed outside where the temperature is 4 degree C. Twenty-five minutes later t
    9·2 answers
  • I cant find answer pls help me :)
    15·2 answers
  • Each lapel on a suit jacket is in the shape of a triangle two of the three angles of each triangle measures 47 and 68 what is th
    13·1 answer
  • Two thirds plus a number plus 4 is 7
    15·2 answers
  • Helppppp (8th) plsss
    7·2 answers
  • Help pls its iready ill give brainiest
    15·2 answers
  • I will give brainliest
    5·1 answer
  • How many miles can you drive in 1/2 hour
    6·1 answer
  • Which statement is the clearest translation of 4 j - 9 = 1?
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!