A-Level Edexcel Maths Pure Sequences & Series: The mass, m grams, of a radioact

The mass, m grams, of a radioactive substance, t years after first being observed, is modelled by the equation m = 25e^{0.05t} According to the model, a) find the mass of the radioactive substance six months after it was first observed, b) show that $ \frac{dm}{dt} = km $, where k is a constant to be found. - Edexcel - A-Level Maths Pure - Question 6 - 2017 - Paper 2

Question 6

$The-mass,-m-grams,-of-a-radioactive-substance,-t-years-after-first-being-observed,-is-modelled-by-the-equation--m-=-25e^{0.05t}--According-to-the-model,--a)-find-the-mass-of-the-radioactive-substance-six-months-after-it-was-first-observed,--b)-show-that-$-\frac{dm}{dt}-=-km-$,-where-k-is-a-constant-to-be-found.-Edexcel-A-Level Maths Pure-Question 6-2017-Paper 2.png$

The mass, m grams, of a radioactive substance, t years after first being observed, is modelled by the equation m = 25e^{0.05t} According to the model, a) find the... show full transcript

Worked Solution & Example Answer:The mass, m grams, of a radioactive substance, t years after first being observed, is modelled by the equation m = 25e^{0.05t} According to the model, a) find the mass of the radioactive substance six months after it was first observed, b) show that $ \frac{dm}{dt} = km $, where k is a constant to be found. - Edexcel - A-Level Maths Pure - Question 6 - 2017 - Paper 2

Step 1

find the mass of the radioactive substance six months after it was first observed

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Answer

To find the mass of the radioactive substance six months after it was first observed, we first convert six months into years. Six months is equivalent to 0.5 years.

Now substitute ( t = 0.5 ) into the equation:

$m = 25e^{0.05 \times 0.5}$

Calculating this, we have:

$m = 25e^{0.025}$

Using a calculator, we find:

$m \approx 24.4 \text{ grams}$

Thus, the mass of the radioactive substance six months after it was first observed is approximately 24.4 grams.

Step 2

show that $ \frac{dm}{dt} = km $, where k is a constant to be found

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Answer

To show that ( \frac{dm}{dt} = km ), we start by differentiating the equation ( m = 25e^{0.05t} ) with respect to ( t ).

Using the chain rule, we get:

$\frac{dm}{dt} = 25 \cdot 0.05e^{0.05t}$

This simplifies to:

$\frac{dm}{dt} = 1.25e^{0.05t}$

Next, we substitute ( m = 25e^{0.05t} ) into the equation:

$\frac{dm}{dt} = 0.05m$

Thus, we can write:

\text{where } k = 0.05 $$ This shows that \( \frac{dm}{dt} = km \) with \( k \) being a constant, which is 0.05.

find the mass of the radioactive substance six months after it was first observed

show that \( \frac{dm}{dt} = km \), where k is a constant to be found

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