Answer is: option3
Increasing 6 miles per hourSolution:
\( x = 2 \) miles
\( y = 1.5 \)
\( z^2 = x^2 + y^2 \)
\( z = \sqrt{2^2 + (1.5)^2} = 2.5 \)
\(\frac{dx}{dt} = 60, \quad \frac{dy}{dt} = -70 \)
Differentiation:\( z^2 = x^2 + y^2 \)
\( 2z \frac{dz}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} \)
\( 2(2.5) \frac{dz}{dt} = 2(2)(60) + 2(1.5)(-70) \)
\( 5 \frac{dz}{dt} = 120 - 105 = 15 \)
\( \frac{dz}{dt} = \frac{15}{2.5} = 6 \)
Since \( \frac{dz}{dt} = 6 \) and is positive,
Final Answer: Increasing 6 miles per hour