QUESTION

# A plant doubled its height each year until it reached its maximum height over the course of 12 years. How many years did it take for it to reach half of its maximum height?

Hint: Firstly, assume the height of the plant in the first year to be ‘h’ and then write the height for each year by simply doubling the previous year’s height till 12th year. Then divide the height of 12th year by 2 and see which year’s height is matching with it, you will get the final answer.

To solve the given question we will assume the height of the plant as ‘h’, therefore,
Height of the plant in 1st year = h
Now as per the given condition height of the plant doubles every year,
Therefore the height of the plant in 2nd year = 2h
Therefore the height of the plant in 3rd year = $2\times \left( 2h \right)$ = ${{2}^{2}}h$
Therefore the height of the plant in 4th year = $2\times \left( {{2}^{2}}h \right)$ = ${{2}^{3}}h$
Therefore the height of the plant in 5th year = $2\times \left( {{2}^{3}}h \right)$ = ${{2}^{4}}h$
Therefore the height of the plant in 6th year = $2\times \left( {{2}^{4}}h \right)$ = ${{2}^{5}}h$
Therefore the height of the plant in 7th year = $2\times \left( {{2}^{5}}h \right)$ = ${{2}^{6}}h$
Therefore the height of the plant in 8th year = $2\times \left( {{2}^{6}}h \right)$ = ${{2}^{7}}h$
Therefore the height of the plant in 9th year = $2\times \left( {{2}^{7}}h \right)$ = ${{2}^{8}}h$
Therefore the height of the plant in 10th year = $2\times \left( {{2}^{8}}h \right)$ = ${{2}^{9}}h$
Therefore the height of the plant in 11th year = $2\times \left( {{2}^{9}}h \right)$ = ${{2}^{10}}h$ ………………………………………….. (1)
Therefore the height of the plant in 12th year = $2\times \left( {{2}^{10}}h \right)$ = ${{2}^{11}}h$
As we have given that the maximum height of the plant will become in 12th year therefore from above equation we can write,
Maximum height of the Plant = ${{2}^{11}}h$
If we divide the above equation by 2 we will get half of the maximum height therefore we will get,
Half of the maximum height = $\dfrac{{{2}^{11}}h}{2}$
By using the formula $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$ in the above equation we will get,
Half of the maximum height = ${{2}^{11-1}}h$
By simplifying the above equation we will get,
Half of the maximum height = ${{2}^{10}}h$
If we compare the above equation with equation (1) we will get,
Therefore, the height of the plant in 11th year = Half of the maximum height
Therefore 11 years will be required for the plant to reach half of its maximum height.

Note: There is no need to write each year’s height, you can write the value for two to three years and then you can directly write the height for 12th year by reference as writing height for each year will require very much time.