A spherical mint of radius 5 mm is placed in the mouth and sucked. Four minutes later, the radius of the mint is 3 mm. In a simple model, the rate of decrease of the radius of the mint is inversely proportional to the square of the radius. Using this model and all the information given, a) find an equation linking the radius of the mint and the time. (You should define the variables that you use.) b) Hence find the total time taken for the mint to completely dissolve. Give your answer in minutes and seconds to the nearest second. c) Suggest a limitation of the model.

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A spherical mint of radius 5 mm is placed in the mouth and sucked - Edexcel - A-Level Maths Pure - Question 12 - 2018 - Paper 2

Question 12

A spherical mint of radius 5 mm is placed in the mouth and sucked. Four minutes later, the radius of the mint is 3 mm. In a simple model, the rate of decrease of th... show full transcript

Worked Solution & Example Answer:A spherical mint of radius 5 mm is placed in the mouth and sucked - Edexcel - A-Level Maths Pure - Question 12 - 2018 - Paper 2

Step 1

a) find an equation linking the radius of the mint and the time.

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Answer

Let the radius of the mint at time $t$ be denoted as $r$ , where $r$ is in mm and $t$ is in minutes.

The problem states that the rate of decrease of the radius is inversely proportional to the square of the radius, which can be expressed mathematically as:

$\frac{dr}{dt} = -\frac{k}{r^2}$

Here, $k$ is the proportionality constant. To solve this differential equation, we can separate the variables and integrate:

$r^2 dr = -k dt$

Integrating both sides gives:

$\int r^2 dr = -k \int dt$

This leads to:

$\frac{r^3}{3} = -kt + C$

To determine the constant of integration $C$ , we use the initial conditions. At $t = 0$ (when the mint is first placed in the mouth), $r = 5$ mm. Thus,

$\frac{5^3}{3} = C\Rightarrow C = \frac{125}{3}$

Consequently, our equation becomes:

$\frac{r^3}{3} + kt = \frac{125}{3}$

Step 2

b) Hence find the total time taken for the mint to completely dissolve.

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Answer

We know that when the mint is completely dissolved, $r = 0$ .

Using the equation we derived previously:

$\frac{0^3}{3} + kt = \frac{125}{3}$

This simplifies to:

$0 + kt = \frac{125}{3} \Rightarrow kt = \frac{125}{3}$

We also need to find $k$ . From the information given, at $t = 4$ minutes, $r = 3$ mm:

Substituting these values into our equation:

$\frac{3^3}{3} + k(4) = \frac{125}{3} \Rightarrow \frac{27}{3} + 4k = \frac{125}{3}$

So:

$9 + 4k = \frac{125}{3} \Rightarrow 4k = \frac{125}{3} - 9 = \frac{125 - 27}{3} = \frac{98}{3}$

Thus:

$k = \frac{98}{12} = \frac{49}{6}$

Now substituting back to find $t$ :

$\frac{49}{6}t = \frac{125}{3} \Rightarrow t = \frac{125}{3} \times \frac{6}{49} = \frac{750}{49} \approx 15.31\text{ minutes}$

In minutes and seconds, this corresponds to 15 minutes and 19 seconds.

Step 3

c) Suggest a limitation of the model.

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Answer

One limitation of the model is that it assumes a constant rate of dissolution throughout the entire time period. In reality, the rate at which a mint dissolves may vary based on several factors, such as:

Changes in saliva concentration.
The physical state of the mint (e.g., if it is bitten).
The effect of temperature in the mouth on the dissolution rate.

Additionally, the model does not take into account the potential for the mint to be swallowed before completely dissolving.

A spherical mint of radius 5 mm is placed in the mouth and sucked - Edexcel - A-Level Maths Pure - Question 12 - 2018 - Paper 2

Worked Solution & Example Answer:A spherical mint of radius 5 mm is placed in the mouth and sucked - Edexcel - A-Level Maths Pure - Question 12 - 2018 - Paper 2

a) find an equation linking the radius of the mint and the time.

b) Hence find the total time taken for the mint to completely dissolve.

c) Suggest a limitation of the model.

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