Hypothetical Plants To compare logarithmic and exponential growth, we consider two

Chapter 1, Problem 5

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Hypothetical Plants To compare logarithmic and exponential growth, we consider two hypothetical plants that are of the same genus, but that exhibit rather different growth rates. Both plants produce a single leaf whose length continues to increase as long as the plant is alive. One plant is called ; the other one is called . The length L (measured in feet) of the leaf of at age t (measured in years) is given by L(t) = ln(t + 1), t 0 The length E (measured in feet) of the leaf of at age t (measured in years), is given by E(t) = et 1, t 0 (a) Find the length of each leaf after 1, 10, 100, and 1000 years. (b) How long would it take for the leaf of to reach a length of 233,810 mi, the average distance from the earth to the moon? (Note that 1 mi = 5280 ft.) How long would the leaf of then be? (c) How many years would it take the leaf of to reach a length of 233,810 mi? Compare this with the length of time since life appeared on earth, about 3500 million years. If had appeared 3,500 million years ago, and if there was a plant of this species that had actually survived throughout the entire period, how long would its leaf be today? (d) Plants started to conquer land only in the late Ordovician period, around 450 million years ago.2 If both and had appeared then, and there was a plant of each species that had actually survived throughout the entire period, how long would their respective leaves be today?