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
katrin [286]
2 years ago
10

A punch glass is in the shape of a hemisphere with a radius of 5 cm. If the punch is being poured into the glass so that the cha

nge in height of the punch is 1,5 cm/sec, at what rate is the exposed area of the punch changing when the height of the punch is 2 cm.

Mathematics
1 answer:
Galina-37 [17]2 years ago
5 0

Answer:

28.27 cm/s

Step-by-step explanation:

Though Process:

  • The punch glass (call it bowl to have a shape in mind) is in the shape of a hemisphere
  • the radius r=5cm
  • Punch is being poured into the bowl
  • The height at which the punch is increasing in the bowl is \frac{dh}{dt} = 1.5
  • the exposed area is a circle, (since the bowl is a hemisphere)
  • the radius of this circle can be written as 'a'
  • what is being asked is the rate of change of the exposed area when the height h = 2 cm
  • the rate of change of exposed area can be written as \frac{dA}{dt}.
  • since the exposed area is changing with respect to the height of punch. We can use the chain rule: \frac{dA}{dt} = \frac{dA}{dh} . \frac{dh}{dt}
  • and since A = \pi a^2 the chain rule above can simplified to \frac{da}{dt} = \frac{da}{dh} . \frac{dh}{dt} -- we can call this Eq(1)

Solution:

the area of the exposed circle is

A =\pi a^2

the rate of change of this area can be, (using chain rule)

\frac{dA}{dt} = 2 \pi a \frac{da}{dt} we can call this Eq(2)

what we are really concerned about is how a changes as the punch is being poured into the bowl i.e \frac{da}{dh}

So we need another formula: Using the property of hemispheres and pythagoras theorem, we can use:

r = \frac{a^2 + h^2}{2h}

and rearrage the formula so that a is the subject:

a^2 = 2rh - h^2

now we can derivate a with respect to h to get \frac{da}{dh}

2a \frac{da}{dh} = 2r - 2h

simplify

\frac{da}{dh} = \frac{r-h}{a}

we can put this in Eq(1) in place of \frac{da}{dh}

\frac{da}{dt} = \frac{r-h}{a} . \frac{dh}{dt}

and since we know \frac{dh}{dt} = 1.5

\frac{da}{dt} = \frac{(r-h)(1.5)}{a}

and now we use substitute this \frac{da}{dt}. in Eq(2)

\frac{dA}{dt} = 2 \pi a \frac{(r-h)(1.5)}{a}

simplify,

\frac{dA}{dt} = 3 \pi (r-h)

This is the rate of change of area, this is being asked in the quesiton!

Finally, we can put our known values:

r = 5cm

h = 2cm from the question

\frac{dA}{dt} = 3 \pi (5-2)

\frac{dA}{dt} = 9 \pi cm/s// or//\frac{dA}{dt} = 28.27 cm/s

You might be interested in
Globalization leads to more trade between countries
Firlakuza [10]

Answer:

I BELIEVE ITS C

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
Which sequence shows a pattern where each term is 1.5 times the previous term
Zanzabum
1.5, 2.25, 3.375, 5.0625, 7.59375... and so on
6 0
3 years ago
Read 2 more answers
What is the value of t?
frutty [35]

Answer:

-39 2/3

Step-by-step explanation:

t - 122= 2t - 3

subtract 't' from each side to get:

-122 = 3t - 3

add 3 to each side to get:

-119 = 3t

divide each side by 3 to get:

t = -39 2/3

5 0
2 years ago
Complete the proof by providing the missing reasons. given:cb bisects bd,de bisects ec, cb=de. prove dbc=ced. . . CB=DE,CB BISEC
Alexus [3.1K]
Base on your question were the are proofs that is given are this are: CB bisects BD, DE bisects EC, CB=DE. So answer your question you must understand first your given statements and realize that CBD and DEC are both right triangles because they are a perpendicular segment so i conclude that the answer should be CBD=DEC because they are both right angles
5 0
2 years ago
Read 2 more answers
Bonita makes $11.75/hour and Bruce
lbvjy [14]
C is the answer you get while doing the math
4 0
3 years ago
Other questions:
  • Find the volume of the pyramid
    15·1 answer
  • Solve for x.<br><br><br><br> 2x/3 + 1 = 3<br><br> x = 1 <br> x = 6<br> x = 3<br> x = 2
    10·1 answer
  • Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar. The simplest form o
    14·1 answer
  • Simplify 7/8 divided by 8
    6·2 answers
  • A rectangle on a coordinate plane is translated 5 units up and 3 units to the left. Which rule describes the translation?
    9·2 answers
  • The height of a cylinder is twice the radius of its base.
    15·1 answer
  • For which of the following compound inequalities is there no solution? ​
    9·1 answer
  • Find the lenghts of the following segement ​
    13·1 answer
  • ALGEBRA By bundling cellular, land line, high-speed Internet
    8·1 answer
  • There is a picture below. Help me please!
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!