Class CBSE Class 12 Mathematics Applications of Derivatives Q #703

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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
Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area.
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Step-by-Step Solution

**1. Define the variables and constraints:** Let $x$ be the length of the side perpendicular to the partition, and $y$ be the length of the side parallel to the partition. The total length of the boundary material is 300 meters. This gives us the constraint: $3x + 2y = 300$
**2. Express $y$ in terms of $x$:** From the constraint, we can express $y$ in terms of $x$: $2y = 300 - 3x$ $y = 150 - \frac{3}{2}x$
**3. Define the area function:** The area $A$ of the rectangular installation is given by: $A = xy$ Substitute $y$ in terms of $x$: $A(x) = x(150 - \frac{3}{2}x) = 150x - \frac{3}{2}x^2$
**4. Find the critical points:** To find the critical points, we need to find the first derivative of $A(x)$ and set it to zero: $A'(x) = 150 - 3x$ Set $A'(x) = 0$: $150 - 3x = 0$ $3x = 150$ $x = 50$
**5. Find the second derivative:** To determine if the critical point is a maximum, we find the second derivative of $A(x)$: $A''(x) = -3$
**6. Apply the second derivative test:** Since $A''(x) = -3 < 0$, the critical point $x = 50$ corresponds to a maximum.
**7. Find the value of $y$ at the critical point:** $y = 150 - \frac{3}{2}(50) = 150 - 75 = 75$
**8. Find the maximum area:** The maximum area is given by: $A_{max} = xy = 50 \times 75 = 3750$

Correct Answer: Critical point at x=50, Maximum Area = 3750 square meters<\/strong>

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires students to apply their knowledge of calculus (derivatives) to solve a practical optimization problem. They must use the given constraints to formulate an area function and then use differentiation to find the maximum area.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a specific procedure (differentiation and optimization techniques) to find the maximum area. It involves knowing the steps to find derivatives, critical points, and using the second derivative test.
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 calculus concepts to a real-world scenario, which aligns with competency-based education principles. It goes beyond rote memorization and tests the student's problem-solving skills.

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