✓ Board-verified
Mathematics · 2025 · 4 marks
CBSE 2025 · Region 1 · Set 1 · Q36
A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a…
A technical company is designing a rectangular solar panel installation on a roof using $\displaystyle 300$ 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 $\displaystyle x$ metres and with parallel to the partition be y metres. * $\displaystyle \begin{array}{r}\text { 口广 } \\ \text { sinc }\end{array}$ Based on this information, answer the following questions :
(i)
Write the equation for the total boundary material used in the boundary and parallel to the partition in terms of $\displaystyle x$ and $\displaystyle y$.
(ii)
Write the area of the solar panel as a function of $\displaystyle x$.
(iii)
(a)
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.
OR
(b)
Using first derivative test, calculate the maximum area the company can enclose with the $\displaystyle 300$ metres of boundary material, considering the parallel partition.
Marking-scheme solution
(i) \(\displaystyle 2 x+3 y=300\)
(ii) \(\displaystyle A=x y=\frac{x}{3}(300-2 x)\)
(iii) (a) \(\displaystyle A=\frac{x}{3}(300-2 x)=\frac{1}{3}\left(300 x-2 x^{2}\right)\)
\(\displaystyle \Rightarrow \frac{d A}{d x}=\frac{1}{3}(300-4 x)\)
For critical points, put \(\displaystyle \frac{d A}{d x}=0 \Rightarrow x=75\)
Also, \(\displaystyle \frac{\boldsymbol{d}^{\mathbf{2}} \boldsymbol{A}}{\boldsymbol{d} \boldsymbol{x}^{\mathbf{2}}}=-\frac{\mathbf{4}}{\mathbf{3}}<\mathbf{0}\). So, \(\displaystyle \boldsymbol{A}\) is maximum at \(\displaystyle \boldsymbol{x}=\mathbf{7 5}\)
Also, maximum area is \(\displaystyle \boldsymbol{A}=\frac{\mathbf{7 5}}{\mathbf{3}}(\mathbf{3 0 0}-\mathbf{1 5 0})=\mathbf{3 7 5 0 ~ m}^{\mathbf{2}}\)
\[\begin{aligned}
& \text { (iii)(b)A= } \frac{x}{3}(300-2 x)=\frac{1}{3}\left(300 x-2 x^{2}\right) \\
& \Rightarrow \frac{d A}{d x}=\frac{1}{3}(300-4 x)
\end{aligned}
\]
For critical points, put \(\displaystyle \frac{d A}{d x}=0 \Rightarrow x=75\)
As \(\displaystyle \frac{\boldsymbol{d A}}{\boldsymbol{d} \boldsymbol{x}}\) changes its sign from positive to negative as x passes through
\(\displaystyle \boldsymbol{x}=\mathbf{7 5}\) from left to right, whichmeans \(\displaystyle \boldsymbol{x}=\mathbf{7 5}\) is the point of maximum.
Also, maximum area is \(\displaystyle \boldsymbol{A}=\frac{\mathbf{7 5}}{\mathbf{3}}(\mathbf{3 0 0}-\mathbf{1 5 0})=\mathbf{3 7 5 0} \mathbf{m}^{\mathbf{2}}\)
\(\displaystyle 2 x+2 y=300\) or \(\displaystyle 2 x+4 y=300\) or \(\displaystyle 4 x+4 y=300\) or \(\displaystyle 4 x+3 y=300\)
Thesolutions of sub-parts will differ and marks maybe given accordingly.
All Application of Derivatives questions2025 question paperNCERT solutions: Application of DerivativesPractice Application of Derivatives →
More from Application of Derivatives
- Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach…2024
- Find the local maxima and local minima of the function f(x)=8/3 x^3-12 x^2+18 x+5.2025
- The least value of f(x)=x^3-12 x, x ∈[0,3] is2026
- Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function f(x) is strictly increasing in (a, b) if2024
- Show that the function f given by f(x)= sin x+ cos x, is strictly decreasing in the interval ((π)/4, (5 π)/4).2024
- A cylindrical tank is being filled with sand at a rate of 314 m^3 / h. If the radius of the tank is 10 m, then the height of sand in the tank…2026
- Find the sub-interval of (0, ∞) in which f(x)=x^2 e^-x is increasing.2026
- A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint…2025
CBSE Class 12 Mathematics past-paper question from the 2025 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.