Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Feb-Mar | (P1-9709/12) | Q#3

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Question

On a certain day, the height of a young bamboo plant was found to be 40 cm. After  exactly one day its height was found to be 41.2 cm. Two different models are used to  predict its height exactly 60 days after it was first measured.

·       Model A assumes that the daily amount of growth continues to be constant at the  amount found for the first day.

·       Model B assumes that the daily percentage rate of growth continues to be  constant at the percentage rate of growth found for the first day.


i.       
Using model A, find the predicted height in cm of the bamboo plant exactly 60  days after it was first measured.

ii.       Using model B, find the predicted height in cm of the bamboo plant exactly 60  days after it was first measured.

Solution


i.
 

From the given information, we can collect following information.

The height measured on 1st day;

The height measured on 2nd day;

Since model A assumes that the daily amount of growth continues to be constant at the amount found for the first day.

It is evident that this model represents Arithmetic Progression (A.P).

Therefore, daily amount of growth is difference between 2nd and 1st day  measurements.

The expression for difference  in Arithmetic Progression (A.P) is:

Hence;

We are required to find the predicted height in cm of the bamboo plant exactly 60  days after it was first measured. Hence, we are interested in 61st term of Arithmetic  Progression (A.P).

Expression for the general term  in the Arithmetic Progression (A.P) is:

Therefore;


ii.
 

Since model B assumes that the daily percentage rate of growth continues to be  constant at the percentage rate of growth found for the first day. 

We need to find the percentage rate of growth found for the first day

The height measured on 1st day;

The height measured on 2nd day;

Expression for Common Ratio () in a Geometric Progression (G.P) is;

Hence;

It is evident that this model represents Geometric Progression (G.P).

We are required to find the predicted height in cm of the bamboo plant exactly 60  days after it was first measured. Hence, we are interested in 61st term of Geometric  Progression (G.P).

Expression for the general term  in the Geometric Progression (G.P) is:

Hence;

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