Math, asked by nothaboncube1309, 8 months ago

how do I solve question 5. The topic is differentiation.Subtopic stationary points

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Answers

Answered by saounksh
1

Step-by-step explanation:

Given

Length of rectangle = y cm

Breadth of rectangle = x cm

Radius of the quarter circle = x cm

Perimeter of the figure = 60 cm

Calculation

1. y in term of x

Perimeter contributed by rectangle

= 2y + x

Perimeter contributed by quarter circle

= 2πx/4 + x

= πx/2 + x

Total perimeter(which is given as 60 cm)

= 2y + x + πx/2 + x = 60

⇒ 2y + 2x + πx/2 = 60

⇒ y + x + πx/4 = 30

y = 30 - x(1+π/4)

2. Area of plate

Area contributed by rectangle

= xy

Area contributed by quarter circle

= πx²/4

Total area A = xy + πx²/4

Using expression of y , we get

A = 30x - x²(1+ π/4) + πx²/4

= 30x - x² - πx²/4 + πx²/4

⇒ A = 30x - x²

Hence proved

3. Stationary point

Stationary point of a fuction is defined as that value of independent variable(s) where the derivative of the function becomes zero.

Let us take derivative of A wrt x

dA/dx = 30 - 2x

Equating dA/dx = 0 we get

30 - 2x = 0

or x = 15

4. Maxima or Minima

To check whether x = 15 is a point of minima or makina, we have to check the sign of d²A/dx².

d²A/dx² = -2 < 0

Since it is -ve, x = 15 is a point of local maxima.

The maximum value of A is given by

A(15) = 30*15-15²

= 450 - 225

= 225

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