Question

ALGEBRA 1 HELP ASAP PLEASE I GIVE BRAINLIST AND EXTRA POINTS AND THANKS

Answer 1

Concept Used :-

HOW TO FIND MAXIMUM AND MINIMUM VALUE OF A FUNCTION

Let f(x) be a given function.

Differentiate the given function w. r. t. x.

Put f'(x) = 0, to find critical point or points say 'a'.

Then, find the second derivative f''(x).

Apply those critical points in the second derivative.

The function f (x) is maximum when f''(a) < 0.

The function f (x) is minimum when f''(a) > 0.

$\large\underline{\bold{Solution-}}$

Let Length of rectangle be 2x

and

Let Breadth of rectangle be y.

So,

Radius of semi - circular portion be 'x'.

Since,

it is given that Perimeter of window = 12 feet.

Therefore,

$\tt \: 2x + y + y + \dfrac{1}{2} (2\pi \: x) = 12$

$\tt \: 2x + 2y + \pi \:x = 12$

$\therefore \: \tt \: y \: = \: \dfrac{12 - \pi \:x - 2x}{2} - - (1)$

Now,

Window is in the form of rectangle surmounted by a semi circle.

So,

Area of window, A

$\tt \: A = Area_{(rectangle)} + Area_{(semi - circle)}$

$\tt \: A = 2xy + \dfrac{1}{2} \pi \: {x}^{2}$

Now, Substitute the value of y, evaluated in equation (1),

$\tt \: A = 2x\bigg( \dfrac{12 - \pi \:x - 2x}{2} \bigg) + \dfrac{\pi \:}{2} {x}^{2}$

$\tt \: A = 12x - 2 {x}^{2} - \pi \: {x}^{2} + \dfrac{\pi \: {x}^{2} }{2}$

$\tt \: A = 12x - 2{x}^{2} - \dfrac{\pi \: {x}^{2} }{2}$

On differentiating w. r. t. x, we get

$\tt \: \dfrac{dA}{dx} = 12 - 4x - \pi \:x - - (2)$

For maxima or minima,

$\tt \: \dfrac{dA}{dx} =0$

$\therefore \: \tt \: 12 - 4x - \pi \:x = 0$

$\therefore \: \tt \: 12 = 4x + \pi \:x$

$\therefore \: \tt \: x \: = \: \dfrac{12}{\pi \: + 4} - - (3)$

On differentiating both sides w. r. t. x, equation (2), we get

$\tt \: \dfrac{d^{2} A}{dx^{2} } = \: - 4 - \pi \: < 0$

$\therefore \bf \: A \: is \: maximum$

Now, Substitute the value of 'x' evaluated in equation (3) in equation (1), we get

$\tt \: y \: = \: \dfrac{12 - (\pi \: + 2)x}{2}$

$\tt \: y = \dfrac{12 - (\pi \: + 2) \times \dfrac{12}{\pi \: + 4} }{2}$

$\tt \: y = \dfrac{12\pi \: + 48 - 12\pi \: - 24}{2(\pi \: + 4)}$

$\therefore \: \tt \: y = \dfrac{12}{\pi \: + 4}$

Hence,

Dimensions of window are

$\tt \: Length = 2x = \dfrac{24}{\pi \: + 4} \: ft.$

and

$\tt \: Breadth = y = \dfrac{12}{\pi \: + 4} \: ft.$