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Mathematics · 2025 · 5 marks
CBSE 2025 · Region 4 · Set 1 · Q33
The relation between the height of the plant ( ycm ) with respect to exposure to sunlight is governed by the equation y =4 x -1/2 x^2, where x is the…
The relation between the height of the plant ( ycm ) with respect to exposure to sunlight is governed by the equation $\displaystyle \mathrm{y}=4 \mathrm{x}-\frac{1}{2} \mathrm{x}^{2}$, where x is the number of days exposed to sunlight.
(i)
Find the rate of growth of the plant with respect to sunlight.
(ii)
In how many days will the plant attain its maximum height ? What is the maximum height?
Marking-scheme solution
(i)
$\displaystyle \mathrm{y}=4 \mathrm{x}-\frac{1}{2} \mathrm{x}^{2} \Rightarrow \frac{d \mathrm{y}}{d \mathrm{x}}=(4-\mathrm{x}) \mathrm{cm} /$ day
(ii)
For maximum height, $\displaystyle \frac{d \mathrm{y}}{d \mathrm{x}}=0 \Rightarrow \mathrm{x}=4$ days as $\displaystyle \frac{d^{2} \mathrm{y}}{d \mathrm{x}^{2}}<0$, number of days $\displaystyle =4$
Now, Maximum height $\displaystyle =\mathrm{y}(4)=16-\frac{1}{2}(16)=8 \mathrm{~cm}$
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