Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.
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 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) = x3 − 6x2 + 9x + 15
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) =x^3, x in [-2,2]`
Find two numbers whose sum is 24 and whose product is as large as possible.
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
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?
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.
Show that the minimum length of the hypotenuse is `(a^(2/3) + b^(2/3))^(3/2).`
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.`
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 box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?
The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
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 ______.
Determine the maximum and minimum value of the following function.
f(x) = x log 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
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] 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`
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
The function `"f"("x") = "x" + 4/"x"` has ____________.
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 function `f(x) = x^3 - 6x^2 + 9x + 25` has
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 ______.
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then `(4/π + 1)`k is equal to ______.
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 ______.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
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 ______.
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 ______.
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 lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, 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 ______.
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
Find the maximum and the minimum values of the function f(x) = x2ex.
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
The absolute maximum value of the function f(x) = 2x3 − 3x2 − 36x + 9 defined on [−3, 3] is ______.
