The relation between the height of the plant (\(y\) cm) with respect to exposure to sunlight is governed by the equation
\[
y = 4x - \frac{1}{2} x^2,
\]
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?
Show Hint
The maximum or minimum of a function occurs when the first derivative is zero.
(i) Rate of Growth of the Plant:
The rate of growth of the plant with respect to sunlight is the derivative of the height function \(y\) with respect to time \(x\).
The height function is given by:
\[
y = 4x - \frac{1}{2} x^2.
\]
Differentiating with respect to \(x\), we get:
\[
\frac{dy}{dx} = 4 - x.
\]
Thus, the rate of growth of the plant with respect to sunlight is:
\[
\frac{dy}{dx} = 4 - x.
\]
(ii) Maximum Height:
To find when the plant reaches its maximum height, we set the rate of growth \(\frac{dy}{dx}\) to zero:
\[
4 - x = 0 \quad \Rightarrow \quad x = 4.
\]
Thus, the plant reaches its maximum height in 4 days.
Now, substitute \(x = 4\) into the height equation to find the maximum height:
\[
y = 4(4) - \frac{1}{2} (4)^2 = 16 - 8 = 8 \, \text{cm}.
\]
Hence, the maximum height of the plant is 8 cm.
