JEE Mains · Maths · STD 12 - 6. Application of derivatives
Coasider a cuboid of sides \(2 x , 4 x\) and \(5 x\) and a closed hemisphere of radius \(r\). If the sum of their surface areas is a constant \(k\), then the ratio \(x: r\), for which the sum of their volumes is maximum, is
- A \(2: 5\)
- B \(19:45\)
- C \(3: 8\)
- D \(19: 15\)
Answer & Solution
Correct Answer
(B) \(19:45\)
Step-by-step Solution
Detailed explanation
Surface area \(=76 x ^{2}+3 \pi r^{2}=\) constant \(( K )\) \(V =40 x ^{3}+\frac{2}{3} \pi r ^{3}\) \({\left[76 x ^{2}+3 \pi r ^{2}= K \right]}\) \(r ^{2}=\frac{ K -76 x ^{2}}{3 \pi}\) \(r =\left(\frac{ K -76 x ^{2}}{3 \pi}\right)^{\frac{1}{2}}\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(y(x)=\left|\begin{array}{ccc}\sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1\end{array}\right|, x \in \mathbb{R}\), then \(\frac{d^2 y}{d x^2}+y\) is equal toJEE Mains 2025 Easy
- If the matrix \(A=\left(\begin{array}{cc}0 & 2 \\ K & -1\end{array}\right)\) satisfies \(A\left(A^{3}+3 I\right)=2 I\) then the value of \(\mathrm{K}\) is :JEE Mains 2021 Hard
- The area (in square units) bounded by the curves \(y = \sqrt x \) and \(2y - x + 3 = 0\) and \(X-\) axis and lying in the first quadrant is :JEE Mains 2013 Medium
- Let \(a = lm\left( {\frac{{1 + {z^2}}}{{2iz}}} \right)\), where \(z\) is any non-zero complex number. The set \(A = \{ a:\left| z \right| = 1\,and\,z \ne \pm 1\} \) is equal toJEE Mains 2013 Hard
- If \(1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\ldots\) upto \(\infty=2\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)\), where \(a\) and \(b\) are integers with \(\operatorname{gcd}(a, b)=1\), then \(11 a+18 b\) is equal to ...............JEE Mains 2024 Hard
- Let \(\alpha, \beta\) be the roots of the equation \(x^2-\sqrt{2} x+2=0\). Then \(\alpha^{14}+\beta^{14}\) is equal toJEE Mains 2023 Hard
More PYQs from JEE Mains
- A software company sets up \(m\) number of computer systems to finish an assignment in \(17\) days. If \(4\) computer systems crashed on the start of the second day, \(4\) more computer systems crashed on the start of the third day and so on, then it took \(8\) more days to finish the assignment. The value of \(m\) is equal to :JEE Mains 2024 Medium
- Equation of two diameters of a circle are \(2 x-3 y=5\) and \(3 x-4 y=7\). The line joining the points \(\left(-\frac{22}{7},-4\right)\) and \(\left(-\frac{1}{7}, 3\right)\) intersects the circle at only one point \(P(\alpha, \beta)\). Then \(17 \beta-\alpha\) is equal toJEE Mains 2024 Hard
- In the expansion of \((1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0\), the sum of the coefficient of \(x^3\) and \(x^{-13}\) is equal toJEE Mains 2024 Hard
- If \(\int {\frac{{\sqrt {1 - {x^2}} }}{{{x^4}}}} dx\, = \,A\,(x)\,{(\sqrt {1 - {x^2}} )^m}\, + \,C,\) for a suitable chosen integer \(m\) and a function \(A(x),\) where \(C\) is a constant of integration, then \((A(x))^m\) equalsJEE Mains 2019 Hard
- Let \(X _{1}, X _{2}, \ldots, X _{18}\) be eighteen observations such that \(\sum_{ i =1}^{18}\left( X _{ i }-\alpha\right)=36 \quad\) and \(\sum_{i=1}^{18}\left(X_{i}-\beta\right)^{2}=90,\) where \(\alpha\) and \(\beta\) are distinct real numbers. If the standard deviation of these observations is \(1,\) then the value of \(|\alpha-\beta|\) is ...... .JEE Mains 2021 Hard
- Let \(f ( x )=\frac{ x }{\left(1+ x ^{ n }\right)^{\frac{1}{ n }}}, x \in R -\{-1\}, n \in N , n > 2\). If \(f ^{ n }( x )= (fofof \ldots \ldots\) upto \(n\) times) \(( x )\), then \(\operatorname{Lim}_{n \rightarrow \infty} \int \limits_0^1 x^{n-2}\left(f^n(x)\right) d x\) is equal to \(...............\).JEE Mains 2023 Hard