tangent line and word problem

Question

Answer 1

3)

y = 2 √ (5x + 1)

Take derivative with respect to x .

dy/dx = d/dx [ 2 √ (5x + 1)]

         = 2 d/dx [ √ (5x + 1)]

         = 2 [ 1/2√ (5x + 1)] d/dx (5x +1)

         =5√ (5x + 1)

Find the slope of the tangent line at (3, 8).

m = (dy/dx) at (3, 8)

    = 5 √ [5(3) + 1]

    = 5 √ 16

     = 20

Answer 2

4)

Rate of change in radius , dr/dt = 5 feet/ minute

Radius of the circle  , r = 14 feet

Area of the circle, A = π r²

Take derivative both sides with respect to t,

dA/dt = d/dt (π r²)

         = 2π r dr/dt

         = 2π (14)( 5 )

         = 140π square feet /minute

Therefore, the increasing rate of change in area is 140π square feet /minute.