# The number of teams y remaining in a single elimination tournament can be found using the exponential function y = 128 (frac{1}{2})^x , where x is the number of rounds played in the tournament. a. Determine whether the function represents exponential growth or decay. Explain. b. What does 128 represent in the function? c. What percent of the teams are eliminated after each round? Explain how you know. d. Graph the function. What is a reasonable domain and range for the function? Explain.

Question
Exponential growth and decay
The number of teams y remaining in a single elimination tournament can be found using the exponential function $$\displaystyle{y}={128}{\left({\frac{{{1}}}{{{2}}}}\right)}^{{x}}$$ , where x is the number of rounds played in the tournament. a. Determine whether the function represents exponential growth or decay. Explain. b. What does 128 represent in the function? c. What percent of the teams are eliminated after each round? Explain how you know. d. Graph the function. What is a reasonable domain and range for the function? Explain.

2021-02-17
(a) For an exponential function in the form $$\displaystyle{y}={a}\cdot{b}^{{x}}$$, If $$\displaystyle{b}{>}{1}$$, the function is increasing and is an exponential growth function. If $$\displaystyle{0}{<}{b}{<}{1}$$</span>, the function is decreasing and is an exponential decay function. Because $$\displaystyle{b}={\frac{{{1}}}{{{2}}}}{<}{1}$$</span>, then the function represents an exponential decay. (b) In an exponential function $$\displaystyle{y}={a}\cdot{b}^{{x}}$$, a represents the initial value so 128 means that there were initially 128 teams in the tournament. (c) The exponential decay function is given by: $$\displaystyle{A}{\left({t}\right)}={a}{\left({1}-{r}\right)}^{{t}}$$ where a is the initial amount, $$\displaystyle{\left({1}-{r}\right)}$$ is the decay factor, and r is the rate of decay. Using the value of $$\displaystyle{b},{b}={1}-{r}\rightarrow{\frac{{{1}}}{{{2}}}}={1}-{\frac{{{1}}}{{{2}}}}$$ So, the rate of decay is: $$\displaystyle{r}={\frac{{{1}}}{{{2}}}}={0.5}{\quad\text{or}\quad}{50}\%$$ This means that 50% of the teams are eliminated after each round. (d) Both the round numbers and the number of teams must be positive integer values and 1 winner is determined after 7 rounds. Hence, a reasonable domain is the integer values from 0 to 7. A reasonable range is from 1 to 128.
2021-06-11

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The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
$$\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}$$
where A is the cross-sectional area of the vehicle and $$\displaystyle{C}_{{d}}$$ is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity $$\displaystyle\vec{{{v}}}$$. Find the power dissipated by form drag.
Express your answer in terms of $$\displaystyle{C}_{{d}},{A},$$ and speed v.
Part B:
A certain car has an engine that provides a maximum power $$\displaystyle{P}_{{0}}$$. Suppose that the maximum speed of thee car, $$\displaystyle{v}_{{0}}$$, is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power $$\displaystyle{P}_{{1}}$$ is 10 percent greater than the original power ($$\displaystyle{P}_{{1}}={110}\%{P}_{{0}}$$).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, $$\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}$$, is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
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