Question

"A rectangle has a height of 5 cm and its base is increasing at a rate" of 3/2 cm/min. When its area is 60 cm2, at what rate is the diagonal of the rectangle increasing?

Answer 1

Answer:

The diagonal is increasing at the rate of 119/104cm/min of the given rectangle.

Step-by-step explanation:

Dimensions of the rectangle

Height = 5cm

Rate of base = 3/2 cm/min

Area = 60cm^2

We know the area of a rectangle of given by = base* Height

b*h = 60

b*5 = 60

b = 12cm

Applying Pythagoras theorem while drawing a diagonal to the rectangle

  $b^2 +h^2 = D^2$

 $5^2 +12^2 = 13^2$

so our diagonal will be 13cm  

Upon differentiating the area of the rectangle  we get

b*h = A=60cm^2

using  the chain rule of differentiation

h*db/dt + b*dh/dt  = 0

b*dh/dt = -h*db/dt

12*dh/dt = -5*3/2

dh/dt = -5/8 cm//min

so the height of the rectangle is decreasing at the rate of -5/8cm/min

now we have all the measurements we need

b = 12 , db/dt = 3/2cm/min

h = 5 , dh/dt = -5/8 cm/min

$b^2 +h^2 = D^2$

Upon differentiating we get

2b*db/dt + 2h*dh/dt = 2D*dD/dt

b*db/dt + h*dh/dt = D*dD/dt

12*3/2 + 5*(-5/8) = 13*dD/dt

18 -25/8 = 13*dD/dt

$\frac{144-25}{8}$ = 13*dD/dt

dD/dt = $\frac{119}{104} cm/min$

Therefore the diagonal is increasing at the rate of 119/104cm/min of the given rectangle.