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`
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
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
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:
`g(x) = 1/(x^2 + 2)`
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.
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?
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.
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 window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
Solve the following:
A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.
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`
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
The function y = 1 + sin x is maximum, when x = ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
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?
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 smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
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 ______.
If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
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 Mr. Rane order x chairs at the price p = (2x2 - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?
Solution: Let Mr. Rane order x chairs.
Then the total price of x chairs = p·x = (2x2 - 12x- 192)x
= 2x3 - 12x2 - 192x
Let f(x) = 2x3 - 12x2 - 192x
∴ f'(x) = `square` and f''(x) = `square`
f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0
∴ f is minimum when x = 8
Hence, Mr. Rane should order 8 chairs for minimum cost of deal.
