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Bogdan [553]
3 years ago
12

At the moment a hot iron rod is plunged into freezing water, the difference between the rod's and the water's temperatures is 10

0\degree100°100, degree Celsius. This causes the iron to cool and the temperature difference drops by 60\%60%60, percent every second. Write a function that gives the temperature difference in degrees Celsius, D(t)D(t)D, left parenthesis, t, right parenthesis, ttt seconds after the rod was plunged into the water.
Mathematics
2 answers:
Papessa [141]3 years ago
0 0

Answer:

D(t)=100(0.4)^t

Step-by-step explanation:

The temp is 100 at time t = 0

After 1 sec, the temp difference would be:

100-(\frac{100-60}{100})

After 2 sec, the temp difference would be:

100-(\frac{100-60}{100})^2

Similarly for 3 seconds, 4 seconds etc.

We notice that the parenthesis part is 40% of it, so we can also write:

100(40%)^t,

where

t is the time

40% can be written as 40/100  = 0.4

SO, the function is:

d(t)=100(0.4)^t

Artist 52 [7]3 years ago
0 0

Answer:

D(t)=100*0.4^t

Step-by-step explanation:

Initially, the difference between the rod's and the water's temperatures is 100°

i.e D(t)=100 When t=0

After 1 seconds, the temp drops by 60%.

Therefore, the new value of D will be the old value multiplied by (100-60)%.

D(1)=100 X (100%-60%) = 100*0.4

After 2 seconds, the temp difference would be:

D(2)=100*0.4*0.4= 100*0.4^2

We notice that for any t, the percentage at which the difference is reduced is raised to the power of t.

Therefore, temperature difference in degrees Celsius, D(t), t seconds after the rod was plunged into the water is given as:

D(t)=100*0.4^t

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Find a power series for the function, centered at c, and determine the interval of convergence. f(x) = 9 3x + 2 , c = 6
san4es73 [151]

Answer:

\frac{9}{3x + 2} = 1 - \frac{1}{3}(x - \frac{7}{3}) + \frac{1}{9}(x - \frac{7}{3})^2 - \frac{1}{27}(x - \frac{7}{3})^3 ........

The interval of convergence is:(-\frac{2}{3},\frac{16}{3})

Step-by-step explanation:

Given

f(x)= \frac{9}{3x+ 2}

c = 6

The geometric series centered at c is of the form:

\frac{a}{1 - (r - c)} = \sum\limits^{\infty}_{n=0}a(r - c)^n, |r - c| < 1.

Where:

a \to first term

r - c \to common ratio

We have to write

f(x)= \frac{9}{3x+ 2}

In the following form:

\frac{a}{1 - r}

So, we have:

f(x)= \frac{9}{3x+ 2}

Rewrite as:

f(x) = \frac{9}{3x - 18 + 18 +2}

f(x) = \frac{9}{3x - 18 + 20}

Factorize

f(x) = \frac{1}{\frac{1}{9}(3x + 2)}

Open bracket

f(x) = \frac{1}{\frac{1}{3}x + \frac{2}{9}}

Rewrite as:

f(x) = \frac{1}{1- 1 + \frac{1}{3}x + \frac{2}{9}}

Collect like terms

f(x) = \frac{1}{1 + \frac{1}{3}x + \frac{2}{9}- 1}

Take LCM

f(x) = \frac{1}{1 + \frac{1}{3}x + \frac{2-9}{9}}

f(x) = \frac{1}{1 + \frac{1}{3}x - \frac{7}{9}}

So, we have:

f(x) = \frac{1}{1 -(- \frac{1}{3}x + \frac{7}{9})}

By comparison with: \frac{a}{1 - r}

a = 1

r = -\frac{1}{3}x + \frac{7}{9}

r = -\frac{1}{3}(x - \frac{7}{3})

At c = 6, we have:

r = -\frac{1}{3}(x - \frac{7}{3}+6-6)

Take LCM

r = -\frac{1}{3}(x + \frac{-7+18}{3}+6-6)

r = -\frac{1}{3}(x + \frac{11}{3}+6-6)

So, the power series becomes:

\frac{9}{3x + 2} =  \sum\limits^{\infty}_{n=0}ar^n

Substitute 1 for a

\frac{9}{3x + 2} =  \sum\limits^{\infty}_{n=0}1*r^n

\frac{9}{3x + 2} =  \sum\limits^{\infty}_{n=0}r^n

Substitute the expression for r

\frac{9}{3x + 2} =  \sum\limits^{\infty}_{n=0}(-\frac{1}{3}(x - \frac{7}{3}))^n

Expand

\frac{9}{3x + 2} =  \sum\limits^{\infty}_{n=0}[(-\frac{1}{3})^n* (x - \frac{7}{3})^n]

Further expand:

\frac{9}{3x + 2} = 1 - \frac{1}{3}(x - \frac{7}{3}) + \frac{1}{9}(x - \frac{7}{3})^2 - \frac{1}{27}(x - \frac{7}{3})^3 ................

The power series converges when:

\frac{1}{3}|x - \frac{7}{3}| < 1

Multiply both sides by 3

|x - \frac{7}{3}|

Expand the absolute inequality

-3 < x - \frac{7}{3}

Solve for x

\frac{7}{3}  -3 < x

Take LCM

\frac{7-9}{3} < x

-\frac{2}{3} < x

The interval of convergence is:(-\frac{2}{3},\frac{16}{3})

6 0
2 years ago
Two angels re supplementary. The larger angle is 48 degrees more than the 10 times the smaller angle. Find the measure of each a
evablogger [386]
Supplementary angles are two angles that add up to 180°.

STEP 1:
find the value of x

x= smaller angle
10x + 48= larger angle

Add the two angles above to equal 180.

x + (10x + 48)= 180
combine like terms

11x + 48= 180
subtract 48 from both sides

11x= 132
divide both sides by 11

x= 12° smaller angle


STEP 2:
find the value of the second angle

=10x + 48
=10(12) + 48
=120 + 48
=168° larger angle


CHECK:
12° + 168°= 180°
180°= 180°


ANSWER: The smaller angle is 12° and the larger angle is 168°.

Hope this helps! :)
8 0
3 years ago
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