Exponential Functions Exercises

Subsection 2.1 Exponential Functions Exercises

Refer to $P(t)$ as given in Table 2.1.11.
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Table 2.1.11. Values of $P(t)$
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$t$ 0 3 6 9

$P(t)$ 5 135 3645 98415
1. Assume that $P$ is an exponential function, and write a function equation for $P(t)\text{.}$
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2. What is $P(4)\text{?}$
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3. Solve $P(t) \geq 2,000\text{.}$
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Refer to $g(x)$ as given in Table 2.1.12.
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Table 2.1.12. Values of $g(x)$
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$x$ 1 2 3 4

$g(x)$ 0.8 0.16 0.032 0.0064
1. Assume that $g(x)$ is an exponential function, and write a function equation for $g(x)\text{.}$
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2. What is $g(7)\text{?}$
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3. Solve $g(x) \geq 0.01\text{.}$
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Suppose you put $100 into a savings account paying 2.5% interest each year (compounded annually).
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1. What percent will you earn if you leave the money in the account for 3 years?
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2. Explain why the answer is NOT $2.5\% \times 3 = 7.5\%\text{.}$
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3. One student solved the problem this way: $100 \times 1.025 \times 1.025 \times 1.025 = 107.69\text{.}$ $107.69-100=7.69\text{.}$ So the answer is $7.69\%\text{.}$ Explain this student’s work. What does $7.69\%$ represent in this problem?
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The population of mosquitoes on a small island increases during the wet season. The population was measured once per week, as shown in Table 2.1.13.
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Table 2.1.13. Mosquito population, $S(t)$
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$t$ 0 1 2 3

$S(t)$ 100 1600 25600 409600
1. What is the average rate of growth of the mosquito population over the three weeks shown in the table?
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2. Assuming that the population continues to grow exponentially, write a function equation for $S(t)\text{.}$
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3. When will the population reach 1 million (1,000,000)?
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Maria is preparing envelopes for mailing. Maria has already prepared 50 envelopes this morning. At 1 pm, she returns from lunch. She can prepare 110 envelopes per hour.
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1. Write an equation for the total number of envelopes Maria has prepared $t$ hours after 1 pm.
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2. If Maria hopes to prepare 750 envelopes before going home, when can she expect to be done?
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Recall that the vertical position of falling objects can be modeled by the equation $h(t)=-16t^2+vt + c\text{,}$ where $t$ is the time in seconds, $h(t)$ is the height in feet, $c$ is the initial height of the object, and $v$ is the initial velocity of the object. A football is kicked from a height of 3 feet above the ground and has an initial velocity of 60 feet per second. [Note: Footballs are affected by air resistance (wind, etc), so this is not a very accurate model. There are more complicated models that take these factors into account.]
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1. What is the highest the football will go? At what time does this happen?
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2. How long is the football in the air?
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The graph below shows the height above ground, $h$ (in meters), of a Ferris wheel rider $t$ seconds after her ride starts (when she is at the 6 o’clock position on the wheel).

Figure 2.1.14. Graph of a periodic function.
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1. How far above the ground is the rider at $t=6$ seconds?
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2. What are the maximum and minimum heights reached by the rider? What do these heights tell you about the radius of the Ferris wheel?
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3. When is the rider 6 meters above ground?
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4. When will the rider be less than 8 meters above ground?
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5. Based on the graph of $h\text{,}$ how long does it take the rider to make one complete trip around the wheel?
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\(t\)	0	3	6	9
\(P(t)\)	5	135	3645	98415

\(x\)	1	2	3	4
\(g(x)\)	0.8	0.16	0.032	0.0064

\(t\)	0	1	2	3
\(S(t)\)	100	1600	25600	409600

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