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

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

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Answer

Let ( r ) be the radius of the mint in mm and ( t ) be the time in minutes.

According to the problem, the rate of decrease of the radius is given by the relationship:

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

where ( k ) is a positive constant. By rearranging this equation, we can separate variables:

$r^2 dr = -k dt$

Now we will integrate both sides:

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

This results in:

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

To determine the constant of integration ( c ), we apply the initial conditions when ( t = 0 ) and ( r = 5 ):

$\frac{5^3}{3} = c \implies c = \frac{125}{3}$

Now, substituting ( c ) back into the equation, we get:

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

Rearranging gives us the final equation:

$r^3 = -3kt + 125$

Step 2

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

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Answer

To find the total time taken for the mint to completely dissolve, we need to establish when ( r = 0 ).

Using the derived equation:

$0 = -3kt + 125$

From the problem, after 4 minutes (( t = 4 )), the radius is 3 mm, so substituting ( r = 3 ) and ( t = 4 ):

$3^3 = -3k(4) + 125 \implies 27 = -12k + 125$

Solving for ( k ), we have:

$-12k = 27 - 125 \implies -12k = -98 \implies k = \frac{98}{12} = \frac{49}{6} \approx 8.1667$

Now we shall substitute this value of ( k ) back into the equation to find total time:

$0 = -3(\frac{49}{6})t + 125 \implies 3t \approx \frac{750}{49} \implies t \approx 51.43 \text{ minutes}$

Rounding to the nearest second:

Total time is approximately 51 minutes and 26 seconds.

Step 3

Suggest a limitation of the model.

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Answer

A suitable limitation of the model could be:

The model does not consider whether the mint is bitten.
It assumes a constant rate of dissolution, which might not be the case in reality.
The model does not account for variations in temperature or saliva production.
Potential changes in the shape of the mint are disregarded.
It does not incorporate the effect of time beyond certain limits, such as 5 minutes, where further behaviors may differ.

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

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

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

Suggest a limitation of the model.

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