Sam goes on a diet. He assumes that his mass, $m$ kg after $t$ days, decreases at a rate that is inversely proportional to the cube root of his mass. (a) Construct a differential equation involving $m$, $t$ and a positive constant $k$ to model this situation. (b) Explain why Sam's assumption may not be appropriate.

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Sam goes on a diet - AQA - A-Level Maths Mechanics - Question 6 - 2017 - Paper 1

Question 6

Sam goes on a diet. He assumes that his mass, $m$ kg after $t$ days, decreases at a rate that is inversely proportional to the cube root of his mass. (a) Construct ... show full transcript

Worked Solution & Example Answer:Sam goes on a diet - AQA - A-Level Maths Mechanics - Question 6 - 2017 - Paper 1

Step 1

Construct a differential equation involving $m$, $t$ and a positive constant $k$

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Answer

To model Sam's weight loss, we start with the statement that the rate of change of his mass is inversely proportional to the cube root of his mass. This translates mathematically to:

$\frac{dm}{dt} = -k \cdot \frac{1}{\sqrt[3]{m}}$

This equation captures the essence of the problem: as Sam's mass decreases, the rate of decrease is influenced by how much mass he has at that moment.

Step 2

Explain why Sam's assumption may not be appropriate

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Answer

Sam's assumption may not be appropriate because it implies that his mass will continue to decrease indefinitely. This is unrealistic as, upon eating, his mass will increase rather than decrease, thereby violating the assumption of continuous weight loss outlined in the model.

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