Exponential Growth and Decay; Related Rates
Problem 1- The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample.
(a) Find the mass that remains after t years.
(b) How much of the sample remains after 100 years?
(c) After how long will only 1 mg remain?
Problem 2- A curve passes through the point (0, 5) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?
Problem 3- The radius of a sphere is increasing at the rate of 4 mm/s. How fast is the volume increasing when the diameter is 80 mm?
Problem 4- If x2 + y2 + z2 = 9, dx/dt = 5, and dy/dt = 4, find dz/dt when (x, y, z) = (2, 2, 1).
Problem 5- A particle moves along the curve y = 2 sin(πx/2). As the particle passes through the point (1/3, 1), its x-coordinate increases at a rate of √10 cm/s. How fast is the distance from the particle to the origin changing at this instant?
Problem 6- Water is leaking out of an inverted conical tank at a rate of 10,000 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank.