✓ Board-verified
Mathematics · 2025 · 4 marks
CBSE 2025 · Region 5 · Set 1 · Q36
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…
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 required to paint the box on completion, he wants the surface area to be minimum. On the basis of the above information, answer the following questions :
(i)
Taking length = breadth = x m and height = y m, express the surface area ( S ) of the box in terms of x and its volume ( V ), which is constant.
(ii)
Find $\displaystyle \frac{\mathrm{dS}}{\mathrm{dx}}$.
(iii)
(a)
Find a relation between x and y such that the surface area (S) is minimum.
OR
(b)
If surface area (S) is constant, the volume (V) $\displaystyle =\frac{1}{4}\left(\mathrm{Sx}-2 \mathrm{x}^{3}\right)$, $\displaystyle \mathrm{x}$ being the edge of base. Show that volume (V) is maximum for $\displaystyle \mathrm{x}=\sqrt{\frac{\mathrm{S}}{6}}$. Case Study - $\displaystyle 2$
Marking-scheme solution
(i)
$\displaystyle \mathrm{V}=\mathrm{x}^{2} \mathrm{y} \Rightarrow \mathrm{y}=\frac{\mathrm{V}}{\mathrm{x}^{2}} \ldots \ldots \ldots \ldots$ (i)
Hence, $\displaystyle \mathrm{S}=2 \mathrm{x}^{2}+4 \mathrm{x} \mathrm{y}=2 \mathrm{x}^{2}+\frac{4 \mathrm{v}}{\mathrm{x}}$
(ii)
$\displaystyle \frac{\mathrm{dS}}{\mathrm{dx}}=4\left(\mathrm{x}-\frac{\mathrm{v}}{\mathrm{x}^{2}}\right)$
(iii)
$\displaystyle \frac{\mathrm{dS}}{\mathrm{dx}}=\mathrm{O} \Rightarrow \mathrm{V}=\mathrm{x}^{3} \Rightarrow \mathrm{x}^{2} \mathrm{y}=\mathrm{x}^{3} \Rightarrow \mathrm{y}=\mathrm{x} \frac{d^{2} \mathrm{S}}{d \mathrm{x}^{2}}=4\left(1+\frac{2 \mathrm{V}}{\mathrm{x}^{3}}\right)>0 \Rightarrow \mathrm{S}$ is minimum if $\displaystyle \mathrm{y}=\mathrm{x}$.
$\displaystyle \mathrm{V}=\frac{1}{4}\left(\mathrm{Sx}-2 \mathrm{x}^{3}\right) \Rightarrow \frac{\mathrm{dV}}{\mathrm{dx}}=\frac{1}{4}\left(\mathrm{~S}-6 \mathrm{x}^{2}\right)$
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
- Find the interval in which the function f(x)=x^4-4 x^3+10 is strictly decreasing.2024
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.