Class CBSE Class 12 Mathematics Applications of Derivatives Q #702

Read the Passage

A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections.

Let the length of the side perpendicular to the partition be $x$ metres and with parallel to the partition be $y$ metres.,
COMPETENCY BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) SUBJECTIVE
Write the area of the solar panel as a function of $x$
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Step-by-Step Solution

**1. Define the variables:** - Let $x$ be the length of the side perpendicular to the partition. - Let $y$ be the length of the side parallel to the partition.
**2. Express the total length of the boundary material:** The total length of the boundary material is given as 300 meters. This includes the perimeter of the rectangle and the partition. Therefore, $$3x + 2y = 300$$
**3. Solve for $y$ in terms of $x$:** $$2y = 300 - 3x$$ $$y = \frac{300 - 3x}{2} = 150 - \frac{3}{2}x$$
**4. Express the area $A$ of the solar panel as a function of $x$:** The area of the rectangular solar panel is given by $A = x \cdot y$. Substituting the expression for $y$ in terms of $x$, we get: $$A(x) = x \left(150 - \frac{3}{2}x\right)$$ $$A(x) = 150x - \frac{3}{2}x^2$$

Correct Answer: $A(x) = 150x - \frac{3}{2}x^2$

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply their knowledge of area and perimeter to a practical scenario and formulate a function.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concepts of area, perimeter, and how to represent them algebraically. It involves relating these concepts to form a function.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question is designed to assess the student's ability to apply mathematical concepts to a real-world problem, which aligns with competency-based education.

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