The rate of decay of the mass of a particular substance is modelled by the differential equation $$\frac{dx}{dt} = -\frac{5}{2}x, \quad t > 0$$ where $x$ is the mass of the substance measured in grams and $t$ is the time measured in days. Given that $x = 60$ when $t = 0$, (a) solve the differential equation, giving $x$ in terms of $t$. You should show all steps in your working and give your answer in its simplest form. (b) Find the time taken for the mass of the substance to decay from 60 grams to 20 grams. Give your answer to the nearest minute.

A-Level Edexcel Maths Pure Applications of Differentiation: The rate of decay of the mass of

The rate of decay of the mass of a particular substance is modelled by the differential equation $$\frac{dx}{dt} = -\frac{5}{2}x, \quad t > 0$$ where $x$ is the mass of the substance measured in grams and $t$ is the time measured in days - Edexcel - A-Level Maths Pure - Question 5 - 2016 - Paper 4

Question 5

$The-rate-of-decay-of-the-mass-of-a-particular-substance-is-modelled-by-the-differential-equation--$$\frac{dx}{dt}-=--\frac{5}{2}x,-\quad-t->-0$$--where-$x$-is-the-mass-of-the-substance-measured-in-grams-and-$t$-is-the-time-measured-in-days-Edexcel-A-Level Maths Pure-Question 5-2016-Paper 4.png$

The rate of decay of the mass of a particular substance is modelled by the differential equation $$\frac{dx}{dt} = -\frac{5}{2}x, \quad t > 0$$ where $x$ is the ma... show full transcript

Worked Solution & Example Answer:The rate of decay of the mass of a particular substance is modelled by the differential equation $$\frac{dx}{dt} = -\frac{5}{2}x, \quad t > 0$$ where $x$ is the mass of the substance measured in grams and $t$ is the time measured in days - Edexcel - A-Level Maths Pure - Question 5 - 2016 - Paper 4

Step 1

solve the differential equation, giving x in terms of t

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Answer

To solve the differential equation, we start with:

$\frac{dx}{dt} = -\frac{5}{2}x$

Separate the variables:

$\frac{dx}{x} = -\frac{5}{2} dt$
Integrate both sides:

$\int \frac{dx}{x} = -\frac{5}{2} \int dt$

This gives us:

$\ln |x| = -\frac{5}{2}t + C$
Solve for $x$ :

Exponentiating both sides, we have:

$x = e^{C} e^{-\frac{5}{2}t}$

Let $A = e^{C}$ , so:

$x = A e^{-\frac{5}{2}t}$
Use the initial condition $x(0) = 60$ :

Substituting into the equation gives:

$60 = A e^{0} \Rightarrow A = 60$
Final answer:

Now we can write:

$x = 60 e^{-\frac{5}{2}t}$

Step 2

Find the time taken for the mass of the substance to decay from 60 grams to 20 grams

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Answer

We need to find the time $t$ when $x$ decays from 60 grams to 20 grams:

Set the equation:

$20 = 60 e^{-\frac{5}{2}t}$
Divide both sides by 60:

$\frac{20}{60} = e^{-\frac{5}{2}t} \Rightarrow \frac{1}{3} = e^{-\frac{5}{2}t}$
Take the natural logarithm of both sides:

$\ln\left(\frac{1}{3}\right) = -\frac{5}{2}t$
Solve for $t$ :

$t = -\frac{2}{5} \ln\left(\frac{1}{3}\right)$
Calculate $t$ :

Using a calculator:

$\ln\left(\frac{1}{3}\right) \approx -1.0986$

Thus,

$t \approx -\frac{2}{5} \times (-1.0986) \approx 0.4394$
Convert to days:

Since the unit is in days, multiply by 24 hours/day:

$0.4394 \times 24 \approx 10.626 \text{ hours}$
Convert to minutes:

$10.626 \times 60 \approx 637.56 ext{ minutes}$
Round to the nearest minute:

The final answer is approximately 638 minutes.

The rate of decay of the mass of a particular substance is modelled by the differential equation $$\frac{dx}{dt} = -\frac{5}{2}x, \quad t > 0$$ where $x$ is the mass of the substance measured in grams and $t$ is the time measured in days - Edexcel - A-Level Maths Pure - Question 5 - 2016 - Paper 4

solve the differential equation, giving x in terms of t

Find the time taken for the mass of the substance to decay from 60 grams to 20 grams

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