Advertisements
Advertisements
Question
The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?
Advertisements
Solution
Let ABC be the triangle such that the side BC = a = 4 cm. Also, the perimeter of the triangle is 10 cm.
i.e. a + b + c = 10
∴ 2s = 10
∴ s = 5
Also, 4 + b + c = 10
∴ b + c = 6
∴ b = 6 – c
Let Δ be the area of the trangle.
Then Δ = `sqrt(s(s - a)(s - b)(s - c)`
= `sqrt(5(5 - 4)(5 - 6 + c)(5 - c)`
= `sqrt(5(c - 1)(5 - c)`
∴ Δ2 = 5(c – 1)(5 – c)
= 5(5c – c2 – 5 + c)
∴ Δ2 = 5(– c2 + 6c – 5)
Differentiable both sides ww.r.t. c, we get
`2Δ(dΔ)/(dc) = 5d/"dc"(-c^2 ++ 6c - 5)`
= 5 (– 2c + 6 x 1 – 0)
= 5 (– 2c + 6)
∴ `(dΔ)/"dc" = (5(-c + 3))/Δ`
and
`(d^2Δ)/(dc^2) = 5d/"dc"((-c + 3)/Δ)`
= `5.(Δd/"dc"(– c + 3) – ( – c + 3)(dΔ)/"dc")/Δ^2`
= `5.(Δ(– 1 + 0) – ( – c + 3)(dΔ)/"dc")/Δ^2`
= `5/Δ^2(-Δ - (c + 3)(dΔ)/"dc")`
= `(-5)/Δ^2[Δ + (c + 3)(dΔ)/"dc"]`
For maximum Δ, `(dΔ)/"dc"` = 0
∴ `(5( - c + 3))/Δ` = 0
∴ – c + 3 = 0 ...[∵ Δ ≠ 0]
∴ c = 3
If c = 3,
Δ = `sqrt(5(3 - 1)(5 - 3)`
= `2sqrt(5)`
∴ `((d^2Δ)/(dc^2))_("at" c = 3)`
= `(-5)/(4 xx 5)[2sqrt(5) + (3 + 3)(0)]`
= `sqrt(5)/(2) < 0`
∴ by the second derivative test, Δ is maximum when c= 3.
When c = 3, b = 6 – c = 6 – 3 = 3
Hence, the area of the triangle is maximum when the other two sides are 3cm and 3cm.
APPEARS IN
RELATED QUESTIONS
Find the maximum and minimum value, if any, of the following function given by f(x) = (2x − 1)2 + 3.
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 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:
`g(x) = x/2 + 2/x, x > 0`
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) = (x −1)2 + 3, x ∈[−3, 1]
It is given that at x = 1, the function x4− 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.
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 \[\pi : (\pi + 2)\].
The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
Find the maximum and minimum of the following functions : f(x) = x log x
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 the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
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.
Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.
If f(x) = x.log.x then its maximum value is ______.
The function f(x) = x log x is minimum at x = ______.
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
A metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.
Solution: Let the dimensions of the rectangle be x cm and y cm.
∴ 2x + 2y = 36
Let f(x) be the area of rectangle in terms of x, then
f(x) = `square`
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme value, f'(x) = 0, we get
x = `square`
∴ f''`(square)` = – 2 < 0
∴ Area is maximum when x = `square`, y = `square`
∴ Dimensions of rectangle are `square`
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
If z = ax + by; a, b > 0 subject to x ≤ 2, y ≤ 2, x + y ≥ 3, x ≥ 0, y ≥ 0 has minimum value at (2, 1) only, then ______.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
The function y = 1 + sin x is maximum, when x = ______
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
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 maximum value of sin x . cos x is ______.
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`
If y `= "ax - b"/(("x" - 1)("x" - 4))` has a turning point P(2, -1), then find the value of a and b respectively.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
Range of projectile will be maximum when angle of projectile is
Divide 20 into two ports, so that their product is maximum.
Let f: R → R be a function defined by f(x) = (x – 3)n1(x – 5)n2, n1, n2 ∈ N. Then, which of the following is NOT true?
The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.
The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° 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)
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.
Divide the number 100 into two parts so that the sum of their squares is minimum.
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
