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.

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
asked 2021-02-16
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.

Answers (2)

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.
0
2021-06-11

Answer of this question:

0

Relevant Questions

asked 2021-05-29
Determine whether each function represents exponential growth or exponential decay. Identify the percent rate of change.
\(\displaystyle{g{{\left({t}\right)}}}={2}{\left({\frac{{{5}}}{{{4}}}}\right)}^{{t}}\)
asked 2021-05-08
Write an exponential growth or decay function to model each situation. Then find the value of the function after the given amount of time. A new car is worth $25,000, and its value decreases by 15% each year; 6 years.
asked 2021-03-06

Determine whether each equation represents exponential growth or exponential decay. Find the rate of increase or decrease for each model. Graph each equation. \(y=5^x\)

asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
asked 2021-05-05
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with \(\displaystyle\mu={1.5}\) and \(\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}\).
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than \(\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}\).
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of \(\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}\)? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
asked 2020-12-12

Tell whether the function represents exponential growth or exponential decay. Then graph the function. \(f(x)=(1.5)^{x}\)

asked 2021-02-25
We will now add support for register-memory ALU operations to the classic five-stage RISC pipeline. To offset this increase in complexity, all memory addressing will be restricted to register indirect (i.e., all addresses are simply a value held in a register; no offset or displacement may be added to the register value). For example, the register-memory instruction add x4, x5, (x1) means add the contents of register x5 to the contents of the memory location with address equal to the value in register x1 and put the sum in register x4. Register-register ALU operations are unchanged. The following items apply to the integer RISC pipeline:
a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.
b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.
c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.
d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.
Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.
asked 2020-10-23
The close connection between logarithm and exponential functions is used often by statisticians as they analyze patterns in data where the numbers range from very small to very large values. For example, the following table shows values that might occur as a bacteria population grows according to the exponential function P(t)=50(2t):
Time t (in hours)012345678 Population P(t)501002004008001,6003,2006,40012,800
a. Complete another row of the table with values log (population) and identify the familiar function pattern illustrated by values in that row.
b. Use your calculator to find log 2 and see how that value relates to the pattern you found in the log P(t) row of the data table.
c. Suppose that you had a different set of experimental data that you suspected was an example of exponential growth or decay, and you produced a similar “third row” with values equal to the logarithms of the population data.
How could you use the pattern in that “third row” to figure out the actual rule for the exponential growth or decay model?
asked 2021-02-05
The value of a home y (in thousands of dollars) can be approximated by the model, \(y= 192 (0.96)^t\) where t is the number of years since 2010.
1. The model for the value of a home represents exponential _____. (Enter growth or decay in the blank.)
2. The annual percent increase or decrease in the value of the home is ______ %. (Enter the correct number in the blank.)
3. The value of the home will be approximately $161,000 in the year
asked 2021-05-09
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.
...