Answer:
dy/dt = - 0.0513 cm/s
Step-by-step explanation:
Given
dy/dt = ?
y = 3 cm (the height of the rectangle)
D = 5 cm (the diagonal of the rectangle)
dA/dt = 3/4 cm²/s
dD/dt = 1/3 cm/s
We can apply the formula
A = x*y ⇒ x = A/y
where x is the base and A is the area.
If we use Pythagoras' theorem
x² + y² = D² (i)
⇒ (A/y)² + y² = D²
we apply
((A/y)²)' + (y²)' = (D²)'
2*(A/y)*(((dA/dt)*y - A*(dy/dt))/y²) + 2*y*(dy/dt) = 2*D*(dD/dt)
⇒ (A/y)*(((dA/dt)*y - A*(dy/dt))/y²) + y*(dy/dt) = D*(dD/dt)
⇒ (dy/dt)*(y - (A²/y³)) = D*(dD/dt) - (A/y²)*(dA/dt)
⇒ dy/dt = (D*(dD/dt) - (A/y²)*(dA/dt)) / (y - (A²/y³)) (ii)
from eq. (i) we have
x² + (3 cm)² = (5 cm)² ⇒ x = 4 cm
we obtain A:
A = x*y ⇒ A = 4 cm* 3 cm
⇒ A = 12 cm²
Finally, we use eq. (ii)
dy/dt = (5 cm*(1/3 cm/s) - (12 cm²/(3 cm)²)*(3/4 cm²/s)) / (3 cm - ((12 cm²)²/(3 cm)³))
⇒ dy/dt = - 0.0513 cm/s
