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Chapter 6: Problem 7

Give two examples of processes that are modeled by exponential growth.

Short Answer

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Question: Provide two examples of processes that are modeled by exponential growth. Answer: Two examples of processes modeled by exponential growth are population growth of a species and compound interest in finance. In population growth, the growth rate is constant, and the population increases by a certain factor or percentage with each time period. In compound interest, the total amount of money grows exponentially as both the initial investment and accrued interest earn additional interest with each compounding period.

Step by step solution

01

Example 1: Population Growth

In our first example, we can consider population growth of a species. If there are sufficient resources and an environment with no limiting factors, the population of a species will grow exponentially. In this case, the growth rate, which is affected by birth and death rates, is constant, and the population increases by a certain factor or percentage with each time period.
02

Example 2: Compound Interest

Another example that can be modeled by exponential growth is compound interest in finance. When an initial amount of money (called the principal) is invested at a fixed interest rate, the total amount of money will grow exponentially over a certain period, as long as the interest is compounded (i.e., added to the principal). The exponential growth is due to the fact that not only the initial investment, but also the accrued interest, earns additional interest with each compounding period.

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Most popular questions from this chapter

Consider the functions \(f(x)=a \sin 2 x\) and \(g(x)=(\sin x) / a,\) where \(a>0\) is a real number. a. Graph the two functions on the interval \([0, \pi / 2],\) for \(a=\frac{1}{2}, 1\) and 2. b. Show that the curves have an intersection point \(x^{*}\) (other than \(x=0)\) on \([0, \pi / 2]\) that satisfies \(\cos x^{*}=1 /\left(2 a^{2}\right),\) provided \(a>1 / \sqrt{2}\) c. Find the area of the region between the two curves on \(\left[0, x^{*}\right]\) when \(a=1\) d. Show that as \(a \rightarrow 1 / \sqrt{2}^{+}\). the area of the region between the two curves on \(\left[0, x^{*}\right]\) approaches zero.

Kelly started at noon \((t=0)\) riding a bike from Niwot to Berthoud, a distance of \(20 \mathrm{km},\) with velocity \(v(t)=15 /(t+1)^{2}\) (decreasing because of fatigue). Sandy started at noon \((t=0)\) riding a bike in the opposite direction from Berthoud to Niwot with velocity \(u(t)=20 /(t+1)^{2}\) (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours. a. Make a graph of Kelly's distance from Niwot as a function of time. b. Make a graph of Sandy's distance from Berthoud as a function of time. c. When do they meet? How far has each person traveled when they meet? d. More generally, if the riders' speeds are \(v(t)=A /(t+1)^{2}\) and \(u(t)=B /(t+1)^{2}\) and the distance between the towns is \(\vec{D},\) what conditions on \(A, B,\) and \(D\) must be met to ensure that the riders will pass each other? e. Looking ahead: With the velocity functions given in part (d), make a conjecture about the maximum distance each person can ride (given unlimited time).

Suppose the cells of a tumor are idealized as spheres each with a radius of \(5 \mu \mathrm{m}\) (micrometers). The number of cells has a doubling time of 35 days. Approximately how long will it take a single cell to grow into a multi- celled spherical tumor with a volume of \(0.5 \mathrm{cm}^{3}(1 \mathrm{cm}=10,000 \mu \mathrm{m}) ?\) Assume that the tumor spheres are tightly packed.

A strong west wind blows across a circular running track. Abe and Bess start running at the south end of the track, and at the same time, Abe starts running clockwise and Bess starts running counterclockwise. Abe runs with a speed (in units of \(\mathrm{mi} / \mathrm{hr}\) ) given by \(u(\varphi)=3-2 \cos \varphi\) and \(\mathrm{Bess}\) runs with a speed given by \(v(\theta)=3+2 \cos \theta,\) where \(\varphi\) and \(\theta\) are the central angles of the runners. a. Graph the speed functions \(u\) and \(v,\) and explain why they describe the runners' speeds (in light of the wind). b. Compute the average value of \(u\) and \(v\) with respect to the central angle. c. Challenge: If the track has a radius of \(\frac{1}{10} \mathrm{mi}\), how long does it take each runner to complete one lap and who wins the race?

A simple model (with different parameters for different people) for the flow of air in and out of the lungs is $$V^{\prime}(t)=-\frac{\pi}{2} \sin \frac{\pi t}{2}$$ where \(V(t)\) (measured in liters) is the volume of air in the lungs at time \(t \geq 0, t\) is measured in seconds, and \(t=0\) corresponds to a time at which the lungs are full and exhalation begins. Only a fraction of the air in the lungs in exchanged with each breath. The amount that is exchanged is called the tidal volume. a. Find and graph the volume function \(V\) assuming that $$ V(0)=6 \mathrm{L} $$ b. What is the breathing rate in breaths/min? c. What is the tidal volume and what is the total capacity of the lungs?
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