Advertisements
Advertisements
प्रश्न
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`
Advertisements
उत्तर
Let ΔABC be the right angled triangle in which ∠B = 90°
Let AC = x, BC = y
∴ AB = `sqrt(x^2 - y^2)`
∠ACB = θ
Let Z = x + y ....(Given)
Now area of ΔABC, A = `1/2 xx "AB" xx "BC"`
⇒ A = `1/2 y * sqrt(x^2 - y^2)`
⇒ A = `1/2 y * sqrt(("Z" - y)^2 - y^2)`
Squaring both sides, we get
⇒ A2 = `1/4 y^2 [("Z" - y)^2 - y^2]`
⇒ A2 = `1/4 y^2 ["Z"^2 + y^2 - 2"Z" y - y^2]`
⇒ P = `1/4 y^2 ["Z"^2 - 2"Z"y]`
⇒ P = `1/4 [y^2"Z"^2 - 2"Z"y^3]` ....[A2 = P]
Differentiating both sides w.r.t. y we get
`"dP"/"dy" = 1/4 [2y"Z"^2 - 6"Z"y^2]` .....(i)
For local maxima and local minima,
`"dP"/"dy"` = 0
∴ `1/4 (2y"Z"^2 - 6"Z"y^2)` = 0
⇒ `(2y"Z")/4 ("Z" - 3y)` = 0
⇒ yZ(Z – 3y) = 0
⇒ yZ ≠ 0 .....(∵ y ≠ 0 and Z ≠ 0)
⇒ Z – 3y = 0
⇒ y = `"Z"/3`
⇒ y = `(x + y)/3` .....(∵ Z = x + y)
⇒ 3y = x + y
⇒ 3y – y = x
⇒ 2y = x
⇒ `y/x = 1/2`
⇒ cos θ = `1/2`
∴ θ = `pi/3`
Differentiating eq. (i) w.r.t. y,
We have `("d"^2"P")/("dy"^2) = 1/4 [2"Z"^2 - 12"Z"y]`
`("d"^2"P")/("dy"^2)` at y = `"Z"/3 = 1/4 [2"Z"^2 - 12"Z" * "Z"/3]`
= `1/4 [2"Z"^2 - 4"Z"^2]`
= `(-"Z"^2)/2 < 0`
Hence, the area of the given triangle is maximum when the angle between its hypotenuse and a side is `pi/3`.
APPEARS IN
संबंधित प्रश्न
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
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.
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.
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) = (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 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) = x + 1, x ∈ (−1, 1)
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
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.
Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening
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 height of a cylinder, which is open at the top, having a given surface area and greatest volume, 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
Divide the number 30 into two parts such that their product is maximum.
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.
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 : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
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 ______.
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 distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
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)
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, 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 minimum value of the function f(x) = xlogx is ______.
Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.
If \[\mathrm{A}+\mathrm{B}=\frac{\pi}{2}\] then the maximum value of cosA.cosB is
