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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

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Answer:

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b

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c

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Step-by-step explanation:

From the question we are told we are told that  

   The  equation for the temperature of a cup of coffee place on a counter is

         H= f(t)

Here  t is the time in minutes the  coffee was put on the counter

Generally  f(t)'  is the derivative of  f(t) at it represents the change of the temperature of the cup of coffee with respect to time

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Generally the unit of the temperature is in degrees Celsius while that of time is  in minutes

  So the change in temperature with time f(35)' will be in degrees Celsius/minutes (i.e  degC/min)

  From the question we are told that

        |f(35)'| =  1.5

i.e  the rate of change of temperature after 35 minutes is  1.5

and

         |f(35)| =  68

i.e the temperature of the cup of coffee after 35 minutes is  67 degC

Now after another   t=  30  seconds =  \frac{30}{60}= 0.5 \ minutes the  rate  of change of the temperature of the cup of  coffee is  

          R =  |f(35)'| *  t

=>        R = 1.5  *  0.5

=>        R = 0.75    

So

At <u>35</u> minutes after the coffee was put on the counter, its temperature is <u>68</u> and will <u>decrease</u> by about <u>0.75</u> in the next 30 seconds.  

     

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Step-by-step explanation:

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The balance on a credit card, that charges a 15.5%APR interest rate, over a 1 month period is given inthe following table:Days 1
Karo-lina-s [1.5K]

First step is to get the average daily balance.

Take the sum of each day's balances.

If a payment has been made, the sign will be negative.

From Days 1 - 5 :

Day 1 = $200

Day 2 = $200

...

Day 5 = $200

That's $200 x 5 days = $1000

From Days 6 - 20 :

Day 6 = $350

Day 7 = $350

...

Days 20 = $350

That's $350 x 15 days = $5250

From Days 21 - 30 :

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Day 22 = $150

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Day 30 = $150

That's $150 x 10 days = $1500

A total of :

$1000 + $5250 + $1500 = $7750

Now divide this total to the number of days to get the ADB or Average Daily Balance.

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Next Step is to calculate for the finance charge using the formula :

\text{Charge}=\text{ADB}\times r\times d

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From the problem, APR is 15.5%, it means that we need to divide the APR by 365 days.

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The answer is $3.29

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