Answer is: option3
\( \frac{2}{3} \) cm/secSolution:
We are given a right triangle where the diagonal \( z \) is increasing at a rate of:
\[ \frac{dz}{dt} = 2 \text{ cm/sec}. \]We also have the relationship:
\[ \frac{dy}{dt} = 3 \frac{dx}{dt} \]We need to determine \( \frac{dx}{dt} \) when \( x = 3 \) cm and \( y = 4 \) cm.
Step 1: Pythagorean Theorem
The diagonal \( z \) of the rectangle follows:
\[ z^2 = x^2 + y^2 \]Differentiating both sides with respect to \( t \):
\[ 2z \frac{dz}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} \]Dividing by 2:
\[ z \frac{dz}{dt} = x \frac{dx}{dt} + y \frac{dy}{dt} \]Step 2: Substituting Given Values
At \( x = 3 \) cm and \( y = 4 \) cm, we use the Pythagorean theorem:
\[ z = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]We substitute \( z = 5 \), \( \frac{dz}{dt} = 2 \), \( x = 3 \), and \( y = 4 \):
\[ 5(2) = 3 \frac{dx}{dt} + 4 \left( 3 \frac{dx}{dt} \right) \]Since we know \( \frac{dy}{dt} = 3 \frac{dx}{dt} \), substitute this:
\[ 10 = 3 \frac{dx}{dt} + 12 \frac{dx}{dt} \] \[ 10 = 15 \frac{dx}{dt} \] \[ \frac{dx}{dt} = \frac{10}{15} = \frac{2}{3} \text{ cm/sec} \]Final Answer: \( \frac{2}{3} \) cm/sec.