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07:3640–43. Population growth
A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N′(t) = 100e^−0.25t. Assume t is measured in days.
a. What is the population after 20 days? After 40 days?
Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.
b. Find the displacement of the object on the interval 0≤t≤6.
40–43. Population growth
Starting with an initial value of P(0)=55, the population of a prairie dog community grows at a rate of P′(t)=20−t/5 (prairie dogs/month), for 0≤t≤200, where t is measured in months.
b. Find the population P(t), for 0≤t≤200.
Find the area of the region (see figure) in two ways.
a. Using integration with respect to x.
Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.
a. Verify that the amount of blood pumped over a one-second interval is 70 mL.
Compressing and stretching a spring Suppose a force of 15 N is required to stretch and hold a spring 0.25 m from its equilibrium position.
b. How much work is required to compress the spring 0.2 m from its equilibrium position?
