Assignment:
Q1. (a) If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in terms of dr/dt
(b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1m/s how fast is the area of the spill increasing when the radius is 30 m?
Q2. If a snowball melts so that its surface area decreases at a rate of 1cm^2/min, find the rate at which the diameter decreases when the diameter is 10cm.
Q3. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1m higher than the bow of the boat. If the rope is pulled in at a rate of 1m/s, how fast is the boat approaching the dock when it is 8m from the dock?
Q4. A particle is moving along the curve y=√x. As the particle passes through the point (4, 2) its x-coordinate increases at a rate of 3 cm/s. How fast is the distance from the particle to the origin changing at this instant?
Q5. Two sides of a triangle are 12 m and 15 m. The angle between them is increasing at a rate of 2 degrees/min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60 degrees?
Provide complete and step by step solution for the question and show calculations and use formulas.