The radius of a cone is increasing at a constant rate of 8 feet per second. The volume remains a constant 56 cubic feet. At the instant when the radius of the cone is 3 feet, what is the rate of change of the height? The volume of a cone can be found with the equation V = farh. Round your answer to three decimal places.

Answers

Answer 1

Answer:

[tex]\Rightarrow \frac{dh}{dt}\approx -31.690 [/tex]

Explanation:

The volume of a cone, V = 1/3πr^2h

Where r is the radius, and h is the height of the cone.

Given:

Volume is constant at 56 cubic feet.

Radius of cone increases at a constant rate of 8 feet per second.

The radius of the cone is 3 feet.

When radius is 3 feet

 [tex]\Rightarrow h= \frac{3V}{\pi r^2}[/tex]

 [tex]\Rightarrow h= \frac{3\times 56}{\pi \times 3^2} [/tex]

 [tex]\Rightarrow h= \frac{3\times 56}{\pi \times 9} [/tex]

 [tex]\Rightarrow h= \frac{ 56}{3\pi} [/tex]

Where h is in feets.

 [tex]\frac{dV}{dt}=\frac{d (\frac{1}{3}\pi r^2h) }{dt} [/tex]

 [tex]\Rightarrow \frac{dV}{dt}=(\frac{1}{3}\pi \frac{dr^2}{dt}h)+(\frac{1}{3}\pi r^2 \frac{dh}{dt}) [/tex]

 [tex]\Rightarrow \frac{dV}{dt}=(\frac{1}{3}\pi \frac{dr^2}{dr}.\frac{dr}{dt}h)+(\frac{1}{3}\pi r^2 \frac{dh}{dt}) [/tex]

 [tex]\Rightarrow \frac{dV}{dt}=(\frac{1}{3} \pi 2r\frac{dr}{dt}h)+(\frac{1}{3}\pi r^2 \frac{dh}{dt}) [/tex]

Since the volume is constant so  [tex] \frac{dV}{dt}=0 [/tex]

The rate of change of radius is  [tex]\frac{dr}{dt}=8 [/tex]

 [tex]\Rightarrow 0=(\frac{1}{3} \pi (2\times 3)(8)(\frac{56}{3\pi}))+(\frac{1}{3}\pi (3^2) \frac{dh}{dt}) [/tex]

 [tex]\Rightarrow 0=(\frac{56\times 16}{3})+(\frac{1}{3}\pi (9) \frac{dh}{dt}) [/tex]

 [tex]\Rightarrow 0=(\frac{56\times 16}{3})+(3\pi \frac{dh}{dt}) [/tex]

 [tex]\Rightarrow 3\pi \frac{dh}{dt}=-\frac{56\times 16}{3} [/tex]

 [tex]\Rightarrow \frac{dh}{dt}=-\frac{56\times 16}{3\times 3 \times \pi} [/tex]

 [tex]\Rightarrow \frac{dh}{dt}=-\frac{56\times 16}{9\pi} [/tex]

 [tex]\Rightarrow \frac{dh}{dt}=-31.689518 [/tex]

[tex]\Rightarrow \frac{dh}{dt}\approx -31.690 [/tex]

The minus indicates that the rate of change of height is decreasing.


Related Questions

The weights of babies born at a certain large hospital are normally distributed
with an average of 7.5 pounds and a standard deviation of 1.25 pounds.
What range of weights would put a baby born at this hospital in the middle 50%
in terms of weight?

Answers

Answer:

Option E

Explanation:

Formula for the confidence interval here will be;

CI = x¯ ± zσ

Where;

x¯ is mean = 7.5 pounds

σ is standard deviation = 1.25

z is critical value at confidence interval of 50% = 0.674

Thus;

CI = 7.5 ± (0.674 × 1.25)

CI = (7.5 - (0.674 × 1.25)), (7.5 + (0.674 × 1.25))

CI ≈ (6.7, 8.3)

Thus,the range of weights is between 6.7 pounds and 8.3 pounds.

A demographic transition graph is shown below. Mark on the graph where you would place Kenya, Japan, the United States, and India?


It is the last question on the pdf. Question 32

Answers

poste en français s’il vous plaît

Countries in stage three begin to industrialize, which results in increased urbanization and a decline in the overall fertility rate. Mexico, India, Colombia, and South Africa are nations that are currently in stage 3.

What demographic transition graph represent country?

The Demographic Transition Model (DTM) assumes that as a country's economy develops, its overall population growth rate cycles through several stages.

Kenya can most likely be categorized as being in Stage 3 of the Demographic Transition Model due to low death rates and declining birth rates.

Japan's population is rapidly aging and shrinking. The oldest population in the world is found in Japan, where the median age is 48.4 years. According to projections made by the Japanese government, there would be almost one elderly person for every person of working age by 2060.

Therefore,  It is based on historical population patterns of two demographic characteristics—the birth rate and death rate.

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The emergence of the BRICS (Brazil, Russia, India, China, South Africa) countries has the potential to most likely

ООООО

O eliminate financing challenges faced by developing nations.

end long-standing hostilities among these countries.

reduce the economic hegemony of the global north.

weaken connections between nations of the global north.

abolish the IMF and World Bank loan systems.

Answers

The correct answer is C) reduce the economic hegemony of the global north.

The emergence of the BRICS (Brazil, Russia, India, China, South Africa) countries has the potential to most likely reduce the economic hegemony of the global north.

For many years the northern hemisphere nations such as Western Europe, the United States, and Canada have dominated trade and the economic world in general.

However, experts have considered that the advent of the BRICS (Brazil, Russia, India, China, South Africa) can represent an important block to seriously compete with those northern hemisphere powers due to the size of their markets and cheap labor.

Although those BRICS countries have interesting numbers in Macroeconomy, they still have very poor people that in Microeconomics that contrasts the Macroeconomy scenario.

Which of the following is intended to prepare students for success at
colleges worldwide?
O A. dual enrollment
B. IB
C. ACT
O D. AP

Answers

The answer is B. IB :))

Answer:

the answer would be B.IB

Explanation:

saan nag mumula telang pinya jusi at sinamay

Answers

What? Is this a serious question?

Incumbents have all of the following advantages over their challengers EXCEPT

Answers

Answer: challengers have a clean political record, but the incumbents do not.

Explanation:

Incumbents simply refers to the person that's holding a particular position or office while the challenger is the person challenging the incumbents that want to take over their position.

While the incumbents have the experience in the role, we should note that the advantage that the Challenger has over the incumbent is that he or she will have a political record that's still clean. In a case whereby the incumbent have done some nasty things that the people didn't like, this may be of immense advantage to the challenger as he or she still has a clean record.

All nonrenewable resources have associated uses and costs. Which of the following statements best conveys
this?
A) Nonrenewable resources can be converted to nonmetallic minerals.
B) Nonrenewable resources can be converted to renewable resources for widespread use.
C) Nonrenewable resources can be mined, transported, exhausted and depleted
D) Nonrenewable resources can be recycled and reused

Answers

Answer:

C.) Nonrenewable resources can be mined, transported, exhausted and depleted

Explanation:

Got it right on Edge

I need help ASAP 100 points and brainliest, please do not put wrong answers I will fail the 9th grade if you do (note I just need help please do not do it all for me

Part 1

For the following examples, pick the best piece of Evidence from the choices available to use to support the given argument. (3 pts. Each)



ARGUMENT

The police and state government are making the argument that Jim killed his friend Robert.



EVIDENCE

One of Robert’s friends witnessed Jim having an argument with Robert the day Robert was killed.



It’s a statistical fact that most people who are murdered are killed by someone they know.



At the time of Robert’s death, Jim owed Robert $5547 from a business deal they had done together.



Evidence Choice:



ARGUMENT

A politician named Kevin Smith wishes you to elect him senator from your state.



EVIDENCE

Kevin Smith has held some form of elected office for the last 20 years.



Under his leadership as mayor of Austin, Texas the unemployment rate for the city went from 12% to 9%.



The current governor, a member of his political party who is retiring, gives him his unconditional support as the next governor of the state.



Evidence Choice:



ARGUMENT

“Buy the new Nike Nighttime Nitro Trainer shoe. It’ll take you higher than ever before.” ---- NIKE Corporation



EVIDENCE

Professional basketball player Leronn Jameson says, “When I play, it’s only in my Nitro Trainers.”



The Nighttime Nitro Trainer is made from industrial grade rubber infused with micro cushioning air pockets and double stitched to a ½ inch canvas top.



Over 42% of amateur, recreational basketball players around the country have already chosen the Nitro Trainer to play in.



Part II

For the following arguments, create what you think is the best piece of Evidence to support each of them. (Note: Please CREATE each piece of Evidence yourself, do not do research and find pieces of Evidence.) (7 pts. Each)



ARGUMENT

Students in public school should be required to wear school uniforms.



Evidence: _________________________________________________________________________________________________________________________________________



_________________________________________________________________________________________________________________________________________



________________________________________________________________________________________________________________________________________



ARGUMENT

Food and beverages containing high levels of sugar should be prohibited from being sold on public school campuses.



Evidence:

_______________________________________________________________________________________________________________________________________



______________________________________________________________________________________________________________________________________



______________________________________________________________________________________________________________________________________



ARGUMENT

The legal driving age across the United States should be lowered from its current age of 16 to 14.



Evidence:

_______________________________________________________________________________________________________________________________________



_______________________________________________________________________________________________________________________________________



______________________________________________________________________________________________________________________________________

Answers

Answer:

AHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

Explanation:

Soil compaction results from
A) the reduction or elimination of pore space in soil
B) the weight of heavy machinery on soil
C) livestock walking over soil
D industrialization
E all of the above

Answers

Explanation: ALL OF THE ABOVE:E

Compaction of soil can be defined as the compression of the soil particles into smaller volume which will reduce the pore space and it will provide more space for water and less space for air.

It can be done by many ways. Some of the ways are suggested here. The reduction of pore space in the soil.

The weight of heavy machinery on the soil will reduce the pore space of the soil.

The livestock walking also reduces the pore space of the soil and helps in soil compaction

Soil compaction results from all of the above (reduction or elimination, weight of heavy machinery on soil, livestock walking over soil,  industrialization). Thus, option (e) is correct.

When soil particles are compressed, pore space is reduced and soil compaction takes place. Putting weight on the soil, which occurs frequently in agricultural areas due to foot traffic, livestock tramples, or the enormous weight of agricultural machinery, is the main factor in soil compaction.

Other factors that contribute to soil compaction include heavy foot traffic, cultivating and walking on damp ground in gardens, compression caused by heavy machinery, vehicles parked under trees, and any activity that puts pressure on the soil and weakens its structure.

Therefore, option (e) is correct.

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