Calculate an estimate for the mean temperature

Mathematics

1 answer:

disa [49]4 years ago

3 0

The temperature was in the range of 10-13 on 7 of the days.
Give it the weight of the middle of the range = 11.5 for 7 times = 80.5

The temperature was in the range of 14-17 on 9 of the days.
Give it the weight of the middle of the range = 15.5 for 9 times = 139.5

The temperature was in the range of 18-21 on 16 of the days.
Give it the weight of the middle of the range = 19.5 for 16 times = 312

The temperature was in the range of 22-25 on 22 of the days.
Give it the weight of the middle of the range = 23.5 for 22 times = 517

The temperature was in the range of 26-29 on 6 of the days.
Give it the weight of the middle of the range = 27.5 for 6 times = 165

Total weight for all ranges = (80.5 + 139.5 + 312 + 517 + 165) = 1,214

Total number of days = 60

Estimated mean temperature = (1,214) / (60) = <u>20.23°C </u>

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Find the sum of the convergent series 7 0.7 0.007

mafiozo [28]

$7+0.7+0.007+\cdots=7\left(1+\dfrac1{10}+\dfrac1{100}+\cdots\right)$

Let $S_n$ denote the $n$ th partial sum of the series, i.e.

$S_n=1+\dfrac1{10}+\dfrac1{100}+\cdots+\dfrac1{100^n}$

Then

$\dfrac1{10}S_n=\dfrac1{10}+\dfrac1{100}+\dfrac1{1000}+\dfrac1{10^{n+1}}$

and subtracting from $S_n$ we get

$S_n-\dfrac1{10}S_n=\dfrac9{10}S_n=1-\dfrac1{10^{n+1}}$
$\implies S_n=\dfrac{10}9\left(1-\dfrac1{10^{n+1}}\right)$

As $n\to\infty$ , the exponential term vanishes, leaving us with

$\displaystyle\lim_{n\to\infty}S_n=\dfrac{10}9$

and so

$7+0.7+0.007+\cdots=7.777\ldots=\dfrac{70}9$