Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
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.
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`. Also find maximum volume in terms of volume of the sphere
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:
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 −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.
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 right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.
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 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.
Determine the maximum and minimum value of the following function.
f(x) = x log x
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
If f(x) = x.log.x then its maximum value is ______.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
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) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.
The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The maximum value of `(1/x)^x` is ______.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
The maximum value of the function f(x) = `logx/x` is ______.
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?
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
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 ______.
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.
Sumit has bought a closed cylindrical dustbin. The radius of the dustbin is ‘r' cm and height is 'h’ cm. It has a volume of 20π cm3.
- Express ‘h’ in terms of ‘r’, using the given volume.
- Prove that the total surface area of the dustbin is `2πr^2 + (40π)/r`
- Sumit wants to paint the dustbin. The cost of painting the base and top of the dustbin is ₹ 2 per cm2 and the cost of painting the curved side is ₹ 25 per cm2. Find the total cost in terms of ‘r’, for painting the outer surface of the dustbin including the base and top.
- Calculate the minimum cost for painting the dustbin.
