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