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
valentinak56 [21]
3 years ago
8

The volume of sphere is increaseing

Mathematics
1 answer:
Artemon [7]3 years ago
4 0

Volume of a sphere:

<em>V</em> = 4/3 <em>π</em> <em>r</em> ³

Differentiate <em>V</em> with respect to time <em>t</em> :

d<em>V</em>/d<em>t</em> = 4<em>π</em> <em>r</em> ² d<em>r</em>/d<em>t</em>

Surface area of a sphere:

<em>A</em> = 4<em>π</em> <em>r</em> ²

Differentiate <em>A</em> with respect to <em>t</em> :

d<em>A</em>/d<em>t</em> = 8<em>π</em> <em>r</em> d<em>r</em>/d<em>t</em>

<em />

You're given that the volume is increasing at a constant rate, d<em>V</em>/d<em>t</em> = 10 cm³/s, so

10 cm³/s = 4<em>π</em> <em>r</em> ² d<em>r</em>/d<em>t</em>

Solve for d<em>r</em>/d<em>t</em> (the rate of change of the sphere's radius):

d<em>r</em>/d<em>t</em> = 5/(2<em>π</em> <em>r</em> ²) cm³/s

Substitute this into the equation for d<em>A</em>/d<em>t</em> :

d<em>A</em>/d<em>t</em> = 8<em>π</em> <em>r</em> (5/(2<em>π</em> <em>r</em> ²) cm³/s)

d<em>A</em>/d<em>t</em> = 20/<em>r</em> cm³/s

At the moment the radius is <em>r</em> = 5 cm, the surface area is increasing at a rate of

d<em>A</em>/d<em>t</em> = 20/(5 cm) cm³/s = 4 cm²/s

When the surface area is increasing at a rate of d<em>A</em>/d<em>t</em> = 2 cm²/s, the radius <em>r</em> is such that

2 cm²/s = 20/<em>r</em> cm³/s

which happens when the radius is

<em>r</em> = 20/2 (cm³/s) / (cm²/s) = 10 cm

You might be interested in
The radioactive isotope radium has a half-life of 1,600 years. The mass of a 100-gram sample of radium will be______ grams after
Olenka [21]
To solve this we are going to use the half life equation N(t)=N_{0} e^{( \frac{-0.693t}{t _{1/2} }) }
Where:
N_{0} is the initial sample
t is the time in years
t_{1/2} is the half life of the substance
N(t) is the remainder quantity after t years 

From the problem we know that:
N_{0} =100
t=200
t_{1/2} =1600

Lets replace those values in our equation to find N(t):
N(200) =100e^{( \frac{(-0.693)(200)}{1600}) }
N(200)=100e^{( \frac{-138.6}{1600} )}
N(200)=100e^{-0.086625}
N(200)=91.7

We can conclude that after 1600 years of radioactive decay, the mass of the 100-gram sample will be 91.7 grams.

8 0
3 years ago
What is the approximate circumference of the circle shown below<br><br>need help fast​
Lemur [1.5K]
C
All u have to do is add 23.5 to 23.5
7 0
3 years ago
A. 3s<br> B. s^2<br> C. s + 1<br> D. 2s + 1
Anna11 [10]
The answer is most likly to be B.
3 0
3 years ago
The sum of three numbers is 10. The first number minus the second plus the third is 6. The first minus the third is 2 more than
mina [271]
The answer to your question is 8
7 0
3 years ago
Please help me and show work so I can understand worth 20 points!!!
Alexus [3.1K]
The steps and final answer is shown in the attached image. I wrote it there so I could write out complex algebraic formulas. Let me know if you have any questions about any of the steps. Thank you.

8 0
2 years ago
Other questions:
  • Plssss answer! Your team is renting a photo booth for the 8th grade dance, but you have to buy the photo paper and other supplie
    15·1 answer
  • going into the final exam, which will count as two tests, amelia has test scores of 77 84 70 59 and 92. what score does amelia n
    7·1 answer
  • HELP PLZ<br> WILL NAME YOU
    8·1 answer
  • What is the slope and y-intercept of the graph of the equation?
    13·2 answers
  • D=5, a10=48. find a45
    9·1 answer
  • Find the volume of a cone that is 17.3ft and 4.7ft
    13·1 answer
  • In a c is ___<br> in b d is ___
    9·1 answer
  • Help! List all the pairs of triangles that are congruent by ASA (Angle-Side-Angle)
    6·1 answer
  • Please solve no links i will mark brailest
    14·1 answer
  • How to prove converse of. basic proportionality theorem​<br>please answer within one hour
    14·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!