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
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
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.
Prove that the following function do not have maxima or minima:
f(x) = ex
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]
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Find two numbers whose sum is 24 and whose product is as large as possible.
A square piece of tin of side 18 cm is to made into a box without a top by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
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.
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
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
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?
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.
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit 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 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 rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
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.
AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
If y = x3 + x2 + x + 1, then y ____________.
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
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 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 ______.
The sum of all the local minimum values of the twice differentiable function f : R `rightarrow` R defined by
f(x) = `x^3 - 3x^2 - (3f^('')(2))/2 x + f^('')(1)`
The maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, is ______.
Read the following passage:
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
|
Based on the above information, answer the following questions:
- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
OR
(b) For maximum volume, h > r. State true or false and justify. (2)
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
If x + y = 8, then the maximum value of x2y is ______.
If \[\mathrm{A}+\mathrm{B}=\frac{\pi}{2}\] then the maximum value of cosA.cosB is
The absolute maximum value of the function f(x) = 2x3 − 3x2 − 36x + 9 defined on [−3, 3] is ______.
