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valentina_108 [34]
3 years ago
10

A sample of a radioactive isotope had an initial mass of 640 mg in the year 2010 and decays exponentially over time. A measureme

nt in the year 2014 found that the sample's mass had decayed to 440 mg. What would be the expected mass of the sample in the year 2024, to the nearest whole number?
Mathematics
1 answer:
Leni [432]3 years ago
4 0

Answer: 172

Step-by-step explanation:

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\huge\underline{\underline{\boxed{\mathbb {ANSWER:}}}}

◉ \large\bm{ -4}

\huge\underline{\underline{\boxed{\mathbb {SOLUTION:}}}}

Before performing any calculation it's good to recall a few properties of integrals:

\small\longrightarrow \sf{\int_{a}^b(nf(x) + m)dx = n \int^b _{a}f(x)dx +  \int_{a}^bmdx}

\small\sf{\longrightarrow If \: a \angle c \angle b \Longrightarrow \int^{b} _a  f(x)dx= \int^c _a f(x)dx+  \int^{b} _c  f(x)dx }

So we apply the first property in the first expression given by the question:

\small \sf{\longrightarrow\int ^3_{-2} [2f(x) +2]dx= 2 \int ^3 _{-2} f(x) dx+ \int f^3 _{2} 2dx=18}

And we solve the second integral:

\small\sf{\longrightarrow2 \int ^3_{-2} f(x)dx + 2 \int ^3_{-2} f(x)dx = 2 \int ^3_{-2} f(x)dx + 2 \cdot(3 - ( - 2)) }

\small \sf{\longrightarrow 2 \int ^3_{-2} f(x)dx + 2 \int ^3_{-2} 2dx  = 2 \int ^3_{-2} f(x)dx +   2 \cdot5 = 2 \int^3_{-2} f(x)dx10 = }

Then we take the last equation and we subtract 10 from both sides:

\sf{{\longrightarrow 2 \int ^3_{-2} f(x)dx} + 10 - 10 = 18 - 10}

\small \sf{\longrightarrow 2 \int ^3_{-2} f(x)dx  = 8}

And we divide both sides by 2:

\small\longrightarrow \sf{\dfrac{2  {  \int}^{3} _{2}  }{2}  =  \dfrac{8}{2} }

\small \sf{\longrightarrow 2 \int ^3_{-2} f(x)dx=4}

Then we apply the second property to this integral:

\small \sf{\longrightarrow 2 \int ^3_{-2} f(x)dx + 2 \int ^3_{-2} f(x)dx + 2 \int ^3_{-2} f(x)dx = 4}

Then we use the other equality in the question and we get:

\small\sf{\longrightarrow 2 \int ^3_{-2} f(x)dx  =  2 \int ^3_{-2} f(x)dx  = 8 +  2 \int ^3_{-2} f(x)dx  = 4}

\small\longrightarrow \sf{2 \int ^3_{-2} f(x)dx =4}

We substract 8 from both sides:

\small\longrightarrow \sf{2 \int ^3_{-2} f(x)dx -8=4}

• \small\longrightarrow \sf{2 \int ^3_{-2} f(x)dx =-4}

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How do I write a whole number in standard form
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Step-by-step explanation:

hope this helps if yes then name brainliest laav :))

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Step-by-step explanation:

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