A-Level Edexcel Maths Pure Binomial Expansion: 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 7 - 2017 - Paper 2

Question 7

$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 7-2017-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) fin... 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 7 - 2017 - Paper 2

Step 1

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

96%

114 rated

Answer

To find the mass of the radioactive substance six months after it was first observed, we need to first convert six months into years. Since there are 12 months in a year, six months is equal to 0.5 years.

Substituting this value into the equation, we get:

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

Calculating the exponent:

$0.05 \times 0.5 = 0.025$

Thus, we have:

$m = 25e^{0.025}$

Now, calculating the value of ( e^{0.025} \approx 1.0253 ):

$m \approx 25 \times 1.0253 \approx 24.4$

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

Step 2

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

99%

104 rated

Answer

To prove that ( \frac{dm}{dt} = km ), we begin by differentiating the mass equation with respect to time t:

$m = 25e^{0.05t}$

Taking the derivative:

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

Using the chain rule:

$\frac{d}{dt}(e^{0.05t}) = 0.05e^{0.05t}$

Thus,

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

This simplifies to:

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

Now, since ( m = 25e^{0.05t} ), we can substitute m into our derivative:

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

Here, we can identify that k is the constant 0.05. Therefore, it follows that:

$\frac{dm}{dt} = km \text{ where } k = 0.05$ .

(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

Join the A-Level students using SimpleStudy...