Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let AB be the diameter and C be any point on the circle with radius r.
∠ACB = 90° ......[angle in the semi-circle is 90°]
Let AC = x
∴ BC = `sqrt("AB"^2 - "AC"^2)`
⇒ BC = `sqrt((2"r")^2 - x^2)`
⇒ BC = `sqrt(4"r"^2 - x^2)` ....(i)
Now area of ∆ABC
A = `1/2 xx "AC" xx "BC"`
⇒ A = `1/2 x * sqrt(4"r"^2 - x^2)`
Squaring both sides, we get
A2 = `1/4 x^2 (4"r"^2 - x^2)`
Let A2 = Z
∴ Z = `1/4 x^2(4"r"^2 - x^2)`
⇒ Z = `1/4(4x^2"r"^2 - x^4)`
Differentiating both sides w.r.t. x, we get
`"dZ"/"dx" = 1/4 [8x"r"^2 - 4x^3]` ....(ii)
For local maxima and local minima `"dZ"/"dx"` = 0
∴ `1/4 [8x"r"^2 - 4x^3]` = 0
⇒ `x[2"r"^2 - x^2]` = 0
x ≠ 0
∴ 2r2 – x2 = 0
⇒ x2 = 2r2
⇒ x = `sqrt(2)"r"`
= AC
Now from equation (i) we have
BC = `sqrt(4"r"^2 - 2"r"^2)`
⇒ BC = `sqrt(2"r"^2)`
⇒ BC = `sqrt(2)"r"`
So AC = BC
Hence, ∆ABC is an isosceles triangle.
Differentiating equation (ii) w.r.t. x, we get
`("d"^2"Z")/("dx"^2) = 1/4 [8"r"^2 - 12x^2]`
Put x = `sqrt(2)"r"`
∴ `("d"^2"Z")/("dx"^2) = 1/4 [8"r"^2 - 12 xx 2"r"^2]`
= `1/4[8"r"^2 - 24"r"^2]`
= `1/4 xx (-16"r"^2)`
= `-4"r"^2 < 0` maxima
Hence, the area of ∆ABC is maximum when it is an isosceles triangle.
APPEARS IN
संबंधित प्रश्न
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
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 f(x) = |sin 4x + 3|
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:
f(x) = x2
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)`
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]
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
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 the maximum possible?
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?
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
Solve the following : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
If x + y = 3 show that the maximum value of x2y is 4.
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima
Solution: f(x) = x3 – 9x2 + 24x
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme values, f'(x) = 0, we get
x = `square` or `square`
∴ f''`(square)` = – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = `square`
∴ f''`(square)` = 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = `square`
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
The two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`
A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
Read the following passage and answer the questions given below.
|
|
- Is the function differentiable in the interval (0, 12)? Justify your answer.
- If 6 is the critical point of the function, then find the value of the constant m.
- Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
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 ______.
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)
A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.
Sumit has bought a closed cylindrical dustbin. The radius of the dustbin is ‘r' cm and height is 'h’ cm. It has a volume of 20π cm3.
- Express ‘h’ in terms of ‘r’, using the given volume.
- Prove that the total surface area of the dustbin is `2πr^2 + (40π)/r`
- Sumit wants to paint the dustbin. The cost of painting the base and top of the dustbin is ₹ 2 per cm2 and the cost of painting the curved side is ₹ 25 per cm2. Find the total cost in terms of ‘r’, for painting the outer surface of the dustbin including the base and top.
- Calculate the minimum cost for painting the dustbin.
For a function defined on an interval \[I\], which condition means that \[f\] has a maximum value at \[c\in I\]?
