CBSE 2025, Set 3

Question

The relation between the height of the plant ($y	ext{ cm}$) 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 ?

Edvaya · Accepted Answer

The correct answer is Subjective. **Step 1:** Find the rate of growth. The height of the plant is given by $y = 4x - \frac{1}{2}x^2$. The rate of growth with respect to sunlight ($x$ days) is the first derivative $\frac{dy}{dx}$. $$\frac{dy}{dx} = \frac{d}{dx}\left(4x - \frac{1}{2}x^2ight)$$ $$\frac{dy}{dx} = 4 - x$$ So, the rate of growth is $4 - x 	ext{ cm/day}$. **Step 2:** Find the critical point for maximum height. To find the maximum height, we set the first derivative to zero: $$\frac{dy}{dx} = 0 \implies 4 - x = 0 \implies x = 4$$ **Step 3:** Verify the maximum using the second derivative. Differentiate the first derivative again with respect to $x$: $$\frac{d^2y}{dx^2} = \frac{d}{dx}(4 - x) = -1$$ Since $\frac{d^2y}{dx^2} = -1 < 0$ for all $x$, the function attains a **local maximum** at $x = 4$. Thus, the plant will attain its maximum height in **$4$ days**. **Step 4:** Calculate the maximum height. Substitute $x = 4$ back into the original height equation: $$y_{max} = 4(4) - \frac{1}{2}(4)^2$$ $$y_{max} = 16 - \frac{1}{2}(16)$$ $$y_{max} = 16 - 8 = 8$$ Therefore, the maximum height attained by the plant is **$8	ext{ cm}$**.

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Mathematics > Calculus > Application Of Derivatives

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