Advertisements
Advertisements
Question
Solve the following:
A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.
Advertisements
Solution
Let r be the radius of the circle and x be the length of the side of the square.
Then,
Total length of the wire = circumference of the circle + perimeter of the square = l
∴ 2πr + 4x = l
∴ r = `(l - 4x)/(2pi)`
A = area of the circle + area of the square
= πr2 + x2
= `pi((l - 4x)/(2pi))^2 + x^2`
= `x^2+ (1)/(4pi) (l - 4x)^2`
= f(x) ...(Say)
Then f'(x) = `2x + (1)/(4pi) xx 2(l - 4x)(- 4)`
= `2x - (2)/pi(l - 4x)`
and
f"(x) = `2 - (2)/pi( - 4)`
= `2 + (8)/pi`
Now, f'(x) = 0 when `2x - (2)/pi (l - 4x)` = 0
i.e. when 2πx – 2l + 8x = 0
i.e when 2(π + 4)x = 2l
i.e. when x = `l/(pi + 4)`
and
f"`(l/(pi + 4)) = 2 + (8)/pi > 0`
∴ By the second derivative test, f has a minimum,
When x = `l/(pi + 4)`.
For this value of x,
r = `(l - 4(l/(pi + 4)))/(2pi)`
= `(pil + 4l - 4l)/(2pi(pi + 4)`
= `l/(2(pi + 4)`
= `x/(2)`
This shows that the sum of the areas of circle and square is least, when radius of the circle = `(1/2)` side of the square.
APPEARS IN
RELATED QUESTIONS
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
Find the maximum and minimum value, if any, of the following function given by h(x) = x + 1, x ∈ (−1, 1)
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
`h(x) = sinx + cosx, 0 < x < pi/2`
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = sinx − cos x, 0 < x < 2π
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
`g(x) = 1/(x^2 + 2)`
Prove that the following function do not have maxima or minima:
f(x) = ex
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = sin x + cos x , x ∈ [0, π]
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
What is the maximum value of the function sin x + cos x?
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.
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?
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Divide the number 20 into two parts such that sum of their squares is minimum.
Show that among rectangles of given area, the square has least perimeter.
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
Determine the maximum and minimum value of the following function.
f(x) = x log x
The function f(x) = x log x is minimum at x = ______.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.
The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.
Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is ______.
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
The maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
The minimum value of the function f(x) = xlogx is ______.
A straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.
Read the following passage:
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
|
Based on the above information, answer the following questions:
- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
OR
(b) For maximum volume, h > r. State true or false and justify. (2)
Check whether the function f : R `rightarrow` R defined by f(x) = x3 + x, has any critical point/s or not ? If yes, then find the point/s.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
If Mr. Rane order x chairs at the price p = (2x2 - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?
Solution: Let Mr. Rane order x chairs.
Then the total price of x chairs = p·x = (2x2 - 12x- 192)x
= 2x3 - 12x2 - 192x
Let f(x) = 2x3 - 12x2 - 192x
∴ f'(x) = `square` and f''(x) = `square`
f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0
∴ f is minimum when x = 8
Hence, Mr. Rane should order 8 chairs for minimum cost of deal.
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
The shortest distance between the line y - x = 1and the curve x = y2 is
