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 Parametric Equations: 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

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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 ... 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: $\frac{1}{x} dx = -\frac{5}{2} dt$
Integrate both sides: $\int \frac{1}{x} dx = \int -\frac{5}{2} dt$ These integrals give: $\ln |x| = -\frac{5}{2} t + C$
Exponentiate to solve for x: $x = e^{-\frac{5}{2} t + C} = e^C e^{-\frac{5}{2} t}$ Let $k = e^C$ , therefore: $x = k e^{-\frac{5}{2} t}$
Use the initial condition: Given $x = 60$ when $t = 0$ , substitute these values: $60 = k e^{0} \Rightarrow k = 60$

Thus, the solution is: $x(t) = 60 e^{-\frac{5}{2} t}$

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

To find the time taken for the mass to decay from 60 grams to 20 grams, we set:

$20 = 60 e^{-\frac{5}{2} t}$

Rearranging the equation: $\frac{20}{60} = e^{-\frac{5}{2} t} \Rightarrow \frac{1}{3} = e^{-\frac{5}{2} t}$
Take the natural logarithm of both sides: $\ln\left(\frac{1}{3}\right) = -\frac{5}{2} t$
Solve for t: $t = -\frac{2}{5} \ln\left(\frac{1}{3}\right)$ Calculating this yields: $t \approx -\frac{2}{5} \times (-1.0986) \approx 0.4394 \text{ days}$
Convert days into minutes: Since 1 day = 1440 minutes: $0.4394 \text{ days} \times 1440 \text{ minutes/day} \approx 632.8 \text{ minutes}$

Rounding to the nearest minute gives us: $633 \text{ minutes}$ .

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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