Find the best tutors and institutes for Class 12 Tuition
Q26:
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is $\sin^{-1}(\frac{1}{3})$.
Solution :
Step 1: Establishing the Geometric Model
Let the cone have radius $r$, slant height $l$, vertical height $h$, and semi-vertical angle $\alpha$.
- $h = l \cos \alpha$ [By Trigonometric definition in $\triangle AOB$]
- $r = l \sin \alpha$
- Surface Area $S = \pi r^2 + \pi rl$ (Total Surface Area)
Step 2: Expressing $l$ and $h$ in terms of $r$ and $\alpha$
Given $S = \pi r^2 + \pi r(\frac{r}{\sin \alpha}) = \pi r^2(1 + \frac{1}{\sin \alpha})$.
Solving for $r^2$: $r^2 = \frac{S}{\pi(1 + \csc \alpha)}$.
The volume $V$ is given by $V = \frac{1}{3}\pi r^2 h$. Substituting $h = r \cot \alpha$:
$V = \frac{1}{3}\pi r^2 (r \cot \alpha) = \frac{1}{3}\pi r^3 \cot \alpha$.
Step 3: Optimization via Derivative
To maximize $V$, it is convenient to maximize $V^2$ (as $V>0$):
$V^2 = \frac{\pi^2}{9} r^6 \cot^2 \alpha = \frac{\pi^2}{9} \left( \frac{S}{\pi(1 + \csc \alpha)} \right)^3 \cot^2 \alpha$.
Let $f(\alpha) = \frac{\cot^2 \alpha}{(1 + \csc \alpha)^3} = \frac{\frac{\cos^2 \alpha}{\sin^2 \alpha}}{(\frac{\sin \alpha + 1}{\sin \alpha})^3} = \frac{\cos^2 \alpha \sin \alpha}{(1 + \sin \alpha)^3} = \frac{(1 - \sin^2 \alpha) \sin \alpha}{(1 + \sin \alpha)^3} = \frac{(1 - \sin \alpha)(1 + \sin \alpha) \sin \alpha}{(1 + \sin \alpha)^3}$.
$f(\alpha) = \frac{\sin \alpha - \sin^2 \alpha}{(1 + \sin \alpha)^2}$.
Step 4: Solving $f'(\alpha) = 0$
Let $x = \sin \alpha$. Then $f(x) = \frac{x - x^2}{(1 + x)^2}$.
Using the Quotient Rule $\frac{d}{dx}(\frac{u}{v}) = \frac{u'v - uv'}{v^2}$:
$f'(x) = \frac{(1 - 2x)(1 + x)^2 - (x - x^2) \cdot 2(1 + x)}{(1 + x)^4} = 0$.
$(1 - 2x)(1 + x) - 2(x - x^2) = 0$.
$1 + x - 2x - 2x^2 - 2x + 2x^2 = 0$.
$1 - 3x = 0 \implies x = \frac{1}{3}$.
Step 5: Conclusion
Since $x = \sin \alpha = \frac{1}{3}$, we have $\alpha = \sin^{-1}(\frac{1}{3})$.
By the Second Derivative Test, since the volume is bounded and we found a unique critical point, this corresponds to the local maximum [Per First Derivative Test/Global Extremum Theorem].
Final Solution: The semi-vertical angle is $\alpha = \sin^{-1}(\frac{1}{3})$.
More Questions from Class 12 Mathematics Application of Derivatives EXERCISE 6.3
- Q1(i): Find the maximum and minimum values, if any, of the following functions given by (i) $f(x) = (2x – 1)^2 + 3$
- Q1(ii): Find the maximum and minimum values, if any, of the following functions given by (ii) $f(x) = 9x^2 + 12x + 2$
- Q1(iii): Find the maximum and minimum values, if any, of the following functions given by (iii) $f(x) = – (x – 1)^2 + 10$
- Q1(iv): Find the maximum and minimum values, if any, of the following functions given by (iv) $g(x) = x^3 + 1$
- Q2(i): Find the maximum and minimum values, if any, of the following functions given by (i) $f(x) = | x + 2 | – 1$
- Q2(ii): Find the maximum and minimum values, if any, of the following functions given by (ii) $g(x) = – | x + 1| + 3$
- Q2(iii): Find the maximum and minimum values, if any, of the following functions given by (iii) $h(x) = \sin(2x) + 5$
- Q2(iv): Find the maximum and minimum values, if any, of the following functions given by (iv) $f(x) = | \sin 4x + 3|$
- Q2(v): Find the maximum and minimum values, if any, of the following functions given by (v) $h(x) = x + 1, x \in (– 1, 1)$
- Q3(i): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (i) $f(x) = x^2$
- Q3(ii): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (ii) $g(x) = x^3 – 3x$
- Q3(iii): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (iii) $h(x) = \sin x + \cos x, 0 < x < \frac{\pi}{2}$
- Q3(iv): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (iv) $f(x) = \sin x – \cos x, 0 < x < 2\pi$
- Q3(v): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (v) $f(x) = x^3 – 6x^2 + 9x + 15$
- Q3(vi): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (vi) $g(x) = \frac{x}{2} + \frac{2}{x}, x > 0$
- Q3(vii): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (vii) $g(x) = \frac{1}{x^2 + 2}$
- Q3(viii): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (viii) $f(x) = x \sqrt{1-x}, 0 < x < 1$
- Q4(i): Prove that the following functions do not have maxima or minima: (i) $f(x) = e^x$
- Q4(ii): Prove that the following functions do not have maxima or minima: (ii) $g(x) = \log x$
- Q4(iii): Prove that the following functions do not have maxima or minima: (iii) $h(x) = x^3 + x^2 + x +1$
- Q5(i): Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i) $f(x) = x^3, x \in [– 2, 2]$
- Q5(ii): Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (ii) $f(x) = \sin x + \cos x, x \in [0, \pi]$
- Q5(iii): Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (iii) $f(x) = 4x - \frac{1}{2}x^2, x \in [-2, \frac{9}{2}]$
- Q5(iv): Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (iv) $f(x) = (x-1)^2 + 3, x \in [-3, 1]$
- Q6: Find the maximum profit that a company can make, if the profit function is given by $p(x) = 41 – 72x – 18x^2$
- Q7: Find both the maximum value and the minimum value of $3x^4 – 8x^3 + 12x^2 – 48x + 25$ on the interval $[0, 3]$.
- Q8: At what points in the interval $[0, 2\pi]$, does the function $\sin 2x$ attain its maximum value?
- Q9: What is the maximum value of the function $\sin x + \cos x$?
- Q10: Find the maximum value of $2x^3 – 24x + 107$ in the interval $[1, 3]$. Find the maximum value of the same function in $[–3, –1]$.
- Q11: It is given that at $x = 1$, the function $x^4 – 62x^2 + ax + 9$ attains its maximum value, on the interval $[0, 2]$. Find the value of $a$.
- Q12: Find the maximum and minimum values of $x + \sin 2x$ on $[0, 2\pi]$.
- Q13: Find two numbers whose sum is $24$ and whose product is as large as possible.
- Q14: Find two positive numbers $x$ and $y$ such that $x + y = 60$ and $xy^3$ is maximum.
- Q15: Find two positive numbers $x$ and $y$ such that their sum is $35$ and the product $x^2 y^5$ is a maximum.
- Q16: Find two positive numbers whose sum is $16$ and the sum of whose cubes is minimum.
- Q17: A square piece of tin of side $18$ cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.
- Q18: A rectangular sheet of tin $45$ cm by $24$ cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum ?
- Q19: Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
- Q20: Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
- Q21: Of all the closed cylindrical cans (right circular), of a given volume of $100$ cubic centimetres, find the dimensions of the can which has the minimum surface area?
- Q22: A wire of length $28$ m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
- Q23: Prove that the volume of the largest cone that can be inscribed in a sphere of radius $R$ is $\frac{8}{27}$ of the volume of the sphere.
- Q24: Show that the right circular cone of least curved surface and given volume has an altitude equal to $\sqrt{2}$ time the radius of the base.
- Q25: Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is $\tan^{-1} \sqrt{2}$.
- Q27: The point on the curve $x^2 = 2y$ which is nearest to the point $(0, 5)$ is
- Q28: For all real values of $x$, the minimum value of $\frac{1-x+x^2}{1+x+x^2}$ is
- Q29: The maximum value of $[x(x-1)+1]^{\frac{1}{3}}$, $0 \le x \le 1$ is
CBSE Solutions for Class 12 Mathematics Application of Derivatives
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Application of Derivatives
Top Tutor
Virugambakkam, Chennai
20 yrs of
Exp
46 students
1713 hrs of classes completed
600
per hour
Online Classes
Hi! I’m Vinotha Arulkumar, a passionate Maths Teacher from Chennai. My qualifications are M. Sc., M. Phil. and B. Ed in Mathematics. From my 10th standard onwards, I got many centum in maths. I’m the topper of my batch in my Under Graduation, M. Phil., and B. Ed. And also I’m the gold medalist in my M. Sc., Mathematics. I possess about 16 years of teaching experience in the field of education. One year in Arts and science college, seven years in Engineering colleges and taking tuition for 10th and 12th maths for more than 10 years. In Arts and Science College I handled B. Sc., Mathematics and M. Com. In Engineering colleges, I handled maths papers for BE, B. Tech., ME, M. Tech., MCA and MBA. While working in engineering colleges, I have received Proficiency Awards for 6 times as I produced 100% results in engineering mathematics. Due to the pandemic situation, presently I’m taking online classes only. I can teach in English and Tamil. My usual class duration is 1 Hr. The key element of my teaching method is to keep on interacting with students while teaching. Before getting in to a problem, usually I will make the students understand the concepts and fundamentals of the chapter first. Then, I will let the students to understand the problem and let them solve it with my guidance. So that they will never forget the concepts. I often conduct online MCQ assessments to evaluate the progress of my students. Bases on the assessment results, I will give individual attention to each student. I keep producing excellent results every year through my vast knowledge & experience in Mathematics. Lot of my students got centum marks in their board exams! Thank you and all the best for your bright future!
Good teaching skills. Explain the concepts clearly, student was able to understand easily. Good communication.
Super Tutor
Vasantha Vallabha Nagar, Bangalore
20 yrs of
Exp
38 students
2481 hrs of classes completed
400
per hour
Online Classes
I am a teacher with over 25 years of experience. Teaching is my passion. I teach across ICSE, ISC, IGCSE , AS&A Levels. I keep producing excellent results year after year through my vast knowledge&experience in the subjects I teach.
Nice teacher. Very punctual , always follow her schedule. Hold on her subject ie economics. Highly recommended.
Top Tutor
Adarsh Nagar, Sonipat
5 yrs of
Exp
6 students
284 hrs of classes completed
250
per hour
Online Classes
Student's Home
Tutor's home
I am an experienced, qualified teacher of Accountancy and Business studies. I am working as a teacher in a school and also providing tutions at home. I have experience of 7 years.I am certified in computer based teaching also. I will provide weekly assignments to the students.I am passionate to solve student's problems and to clear their doubts. I will provide assignments topic wise in order to make students understand the concepts easily.
Classes is going good. Tutor cleared all topics with good knowledge. Very friendly and supportive. Thank you.
Top Tutor
P847+PC6 Thrissur, Thrissur
17 yrs of
Exp
10 students
501 hrs of classes completed
600
per hour
Online Classes
I believe in conceptual learning and always relate the content to real life situations. I discourage the rote learning and promote the application of the concept in daily life situation.
I have enjoyed madam's classes a lot. Her sessions are very engaging. She has helped me to understand even the difficult concepts easily.
Top Tutor
Kharghar Sector 27, Mumbai
13 yrs of
Exp
6 students
1252 hrs of classes completed
650
per hour
Online Classes
Student's Home
Tutor's home
My approach to teaching Mathematics and Science is personalized and interactive. I focus on understanding each student's unique learning style and adapt my methods accordingly. My goal is to make Math and Science relatable and engaging, using real-life examples and practical applications to enhance understanding. I try to develop self-thinking on problems and encourage them to solve problems. **Key Skills:** - Strong grasp of Mathematics concepts across various levels - Ability to simplify complex topics for better understanding - Excellent communication and motivational skills - Patient and supportive, ensuring a comfortable learning environment - Regular progress assessments and constructive feedback **Services Offered:** - One-on-one personalized tutoring sessions - Homework and assignment assistance - Exam preparation and revision classes - Online and in-person tutoring options - Flexible scheduling to accommodate student needs **Achievements:** - Helped students improve their grades by an average of 25% - Successfully prepared students for competitive exams with success. Received positive feedback and high ratings from students and parents
He teaches from basics.Built up lot of confidence.Took lot of paper practice.Explains by giving practical life examples. Must go for it.
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Application of Derivatives EXERCISE 6.3 worksheets
Download Now
