1. Use implicit differentiation to find an equation of the tangent line to the curve
2 = -x2 + 2√3xy + y2
at the point (0, -√2.
2.
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. During an exhibition, artist Damien Hirst pours red paint onto the ground and it spreads' in a circular fashion. If the radius of this paint is increasing at a constant rate of 8 in/s, how fast is the area increasing when the radius is 29 in?
3. A rectangle of fixed perimeter is has length l increasing at 5 cm/s and the width w is decreasing at 5 cm/s. How fast is the area increasing when the length is 20 cm centimeters and the width is 30 cm? When the length is 40 cm and the width is 10 cm, is the area increasing or decreasing?
4. A perfectly spherical balloon is being inflated so that the radius is increasing at a rate of 3 mm/s. How fast is the volume increasing when the radius is 75mm?
5. At 12:00em a FedEx truck is 100 miles east of a UPS truck. The FedEx truck is driving west at a 55 mi/h and the UPS truck is driving north at 75 mi/h. How fast is the distance between these trucks changing at 2:00PM?
6. Water is being pumped into an inverted conical tank at a constant rate of 10ft3/min. At the exact same time, water is leaking out of the tank at a constant rate.
The tank is 7 ft tall and has a radius of 9 ft. If the water level is decreasing at a rate of 1 ft/min when the height of the water is 3 ft, what is the rate at which the water is leaking out of the tank?
7. A ball is dropped from a height of 20 meters and 12 meters away from the top of a 20-meter lamp post. The ball's shadow, caused by the light at the top of a lamppost, is moving along level ground.
a. Draw a diagram representing this scenario.
b. How fast is the shadow moving 1 second after the ball is released?