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
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
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]
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?
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Find the maximum and minimum of the following functions : f(x) = `logx/x`
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.
Solve the following:
A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.
Solve the following : Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/(3)`.
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.
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?
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`
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
The two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
Range of projectile will be maximum when angle of projectile is
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
Divide 20 into two ports, so that their product is maximum.
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.
If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
If the point (1, 3) serves as the point of inflection of the curve y = ax3 + bx2 then the value of 'a ' and 'b' are ______.
The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.
The minimum value of 2sinx + 2cosx is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
Divide the number 100 into two parts so that the sum of their squares is minimum.
If \[f\] has a maximum value at \[c\], what is \[c\] called?
Why must the left endpoint \[a\] and right endpoint \[b\] be checked for absolute extrema on \[[a,b]\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=0\]?
