Question

Length of a rectangle varies with time as l=sin t and breadth varies with time as b=cos t, if time has domain[0,90°]. Find a time t at which area of rectangle is maximum.

Answer 1

length of the rectangle , l = sint

breadth of the rectangle, b = cost

we know, area of a rectangle = length × breadth

so, area of the rectangle , A = l × b

= sint × cost

= (2sint. cost)/2

= (sin2t)/2

here it is clear that area will be maximum when sin2t will be maximum.

we know, maximum value of sine function is 1.

so, maximum value of sin2t = 1

or, sin2t = 1 = sin(π/2)

or, 2t = nπ

or, t = nπ/2

but domain of time is [0, 90°]

so, t = π/2 = 90°

hence, at t = 90° area of rectangle will be maximum.