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?
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 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.
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.
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 f(x) = −(x − 1)2 + 10
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 3|
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) = x3 − 6x2 + 9x + 15
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:
`f(x) = xsqrt(1-x), x > 0`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
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]
What is the maximum value of the function sin x + cos x?
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−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.
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
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?
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
Find the maximum area of an isosceles triangle inscribed in the ellipse `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
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) = `logx/x`
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
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.
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
Determine the maximum and minimum value of the following function.
f(x) = x log x
Divide the number 20 into two parts such that their product is maximum.
If f(x) = x.log.x then its maximum value is ______.
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
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?
Divide the number 20 into two parts such that their product 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 minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
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 ______
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
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.
If x is real, the minimum value of x2 – 8x + 17 is ______.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
If y `= "ax - b"/(("x" - 1)("x" - 4))` has a turning point P(2, -1), then find the value of a and b respectively.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The function `f(x) = x^3 - 6x^2 + 9x + 25` has
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
The maximum value of the function f(x) = `logx/x` is ______.
Divide 20 into two ports, so that their product is maximum.
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 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 function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b is ______.
The minimum value of 2sinx + 2cosx is ______.
The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° is ______.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
Find the maximum and the minimum values of the function f(x) = x2ex.
A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.
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.
