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
scoray [572]
3 years ago
8

Suppose you are given the task of constructing a soup can that holds 400mL of water (1 mL = 1 cm3 ). The material for the base a

nd lid of the can costs $.01/100cm2 , and the material for the side of the can costs $.02/100cm2 . (a) (6 pts) Find the dimensions of a can that would minimize the cost to produce it.
Mathematics
1 answer:
Vilka [71]3 years ago
5 0

Answer:

Dimensions:

r = ~^3\sqrt{\frac{400}{\pi}}

h = ~^3\sqrt{\frac{400}{\pi}}

Step-by-step explanation:

Volume is given by  

\text{Volume of cylinder} = \pi r^2h = 400~cm^3\\\\h = \displaystyle\frac{400}{\pi r^2}

where r is the radius of can and h is the height of the can.

Area of lid and base = \pi r^2 + \pi r^2 = 2\pi r^2

Area of curved surface = 2\pi rh

Cost function :

C(r) = \displaystyle\frac{0.01}{100}2\pi r^2 + \frac{0.02}{100}2\pi rh\\\\= \frac{0.02}{100}\pi r^2 + \frac{0.04}{100}\pi r \frac{400}{\pi r^2}\\\\= \frac{0.02}{100}\pi r^2 + \frac{0.16}{r}

First, we differentiate C(r) with respect to r, to get,

\displaystyle\frac{d(C(r))}{dr} = \displaystyle\frac{0.04}{100}\pi r - \frac{0.16}{r^2}

Equating the first derivative to zero, we get,

\displaystyle\frac{d(C(r))}{dr} = 0\\\\\frac{0.04}{100}\pi r - \frac{0.16}{r^2} = 0

Solving, we get,

\displaystyle\frac{0.04}{100}\pi r - \frac{0.16}{r^2} = 0\\\\\frac{0.04}{100}\pi r = \frac{0.16}{r^2}\\\\r^3 = \frac{0.16\times 100}{0.04\pi}\\\\r =~^3\sqrt{\frac{400}{\pi}} = \bigg(\frac{400}{\pi}\bigg)^{\frac{1}{3}}

Again differentiation C(r), with respect to r, we get,

\displaystyle\frac{d^2(C(r))}{dr^2} =\displaystyle\frac{0.04}{100}\pi+\frac{0.32}{r^3}

At r = \bigg(\displaystyle\frac{400}{\pi}\bigg)^{\frac{1}{3}},

\displaystyle\frac{d^2(C(r))}{dr^2} > 0

Thus, by double derivative test, the minima occurs for C(r) at  r = \bigg(\displaystyle\frac{400}{\pi}\bigg)^{\frac{1}{3}}.

Dimensions:

r = \bigg(\displaystyle\frac{400}{\pi}\bigg)^{\frac{1}{3}}

h = \displaystyle\frac{400}{\pi r^2}\\\\h = \frac{400}{\pi (\frac{400}{\pi})^{\frac{2}{3}}} = ~^3\sqrt{\frac{400}{\pi}}

You might be interested in
Convert to standard form: F(x) = 2x + 4
mote1985 [20]
The answer will be F=2x/x + 4/x
7 0
3 years ago
CAN SOMEONE PLEASE HELP ME???<br> ↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓∵
nata0808 [166]

Answer:

it is number 3 (-1/5)

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
Acceleration of 10kg Maas's pushed by a 5n force
malfutka [58]

Answer:

<h3>The answer is 0.5 m/s²</h3>

Step-by-step explanation:

To find the acceleration of an object given it's mass and the force acting on it can be found by using the formula

acceleration =  \frac{force}{mass}  \\

From the question

force = 5 N

mass = 10 kg

So we have

acceleration =  \frac{5}{10 }  =  \frac{1}{2}  \\

We have the final answer as

<h3>0.5 m/s²</h3>

Hope this helps you

7 0
3 years ago
Dawn is selling her MP3 player for 3/4 of the original price. The original price for the MP3 player was $40. How much is she sel
IRISSAK [1]

Answer:

$30

Step-by-step explanation:

Have a Nice Day! :D

3 0
3 years ago
Read 2 more answers
Drag each number to show whether it is equal to 90%, 9100, or neither.
Misha Larkins [42]

Answer:

90% = 0.90, 9/10

9/100 = 0.09, 9%

Other = 90/10, 0.9%

7 0
2 years ago
Read 2 more answers
Other questions:
  • Francisco has a savings account balance of 2,033.88. The interest rate on the account is 2.9% compounded monthly. If he opened t
    13·1 answer
  • A sign is in the shape of a rhombus with a 60° angle and sides of 12 cm long. Find its area to the nearest tenth. Select one: a.
    9·1 answer
  • HELPPPPPP PLEASEEEEEEEEEE
    5·1 answer
  • Help i need the answers by tomorrow ​
    15·1 answer
  • Which graph represents the equation y−3.6=1.6(x−1)?<br> show work, please!
    13·2 answers
  • Enter an equation in the form of y = MX + b that represents the function described by the graph ​
    7·1 answer
  • Can someone help me with this question
    6·1 answer
  • Can someone please help me???????????????????
    6·2 answers
  • Sasha is twice as old as Kapila. If Kapila is x years old, find their total age.
    5·1 answer
  • Find the term that must be added to the equation x2 4x=1 to make it into a perfect square.
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!