plot the graph of \(\displaystyle{y}={2}^{{x}}\) and \(\displaystyle{y}={4}{x}\) and shows that the only other solution is between 0 and 1.

Jeffery Autrey

Answered 2022-01-06
Author has **3319** answers

If your intention is to 'convince' and not to prove, I'd draw a graph with the functions \(\displaystyle{y}={2}^{{x}}\ \text{ and }\ {y}={4}{x}\).The growth rate of each function should make clear that they intersect only at two points, being the first between 0 and 1 (and hence, not being an integer).

Vasquez

Answered 2022-01-11
Author has **8850** answers

For positive n, we have two growing sequences

1,2,4,8,16,32,64,128,256⋯

0,4,8,12,16,20,24,28,32⋯

This shows them that the ''curves'' cross each other at 16, and it seems that the first grows faster.

Indeed, taking the ratios of successive terms

\(\frac{2^{n+1}}{2^n}=2>\frac{4(n+1)}{4n}=1+\frac{1}{n}\)

asked 2021-08-15

The following data was collected about students in Mr. Rexinger's high school statistics class.
Wearing jeansNot wearing jeans Has long hair(below chin)77 Does not have long hair513

a. Mr. Rexinger is playing a game with his students. He randomly chooses a mystery student from his class roster. If a player guesses the hair length of the mystery student correctly, the player gets an early-lunch pass. Madeline is the next player. To have the greatest chance of winning an early-lunch pass, should she guess that the student has long hair? Explain. b. Mr. Rexinger tells Madeline that the mystery student is wearing jeans. Would you advise Madeline to change her guess? Explain. c. In a previous course, you may have studied the association of two numerical variables by analyzing scatterplots and least squares regression lines. Associations between categorical events, like having long hair or wearing jeans, are determined by independence-if two events are independent, then they are not associated. Are the events {not having long hair} and {wearing jeans} associated for the students in Mr. Rexinger's class today? Explain the independence relationship using P(A given B) = P(A). d. Are the events {not having long hair} and {wearing jeans} mutually exclusive? Explain.

a. Mr. Rexinger is playing a game with his students. He randomly chooses a mystery student from his class roster. If a player guesses the hair length of the mystery student correctly, the player gets an early-lunch pass. Madeline is the next player. To have the greatest chance of winning an early-lunch pass, should she guess that the student has long hair? Explain. b. Mr. Rexinger tells Madeline that the mystery student is wearing jeans. Would you advise Madeline to change her guess? Explain. c. In a previous course, you may have studied the association of two numerical variables by analyzing scatterplots and least squares regression lines. Associations between categorical events, like having long hair or wearing jeans, are determined by independence-if two events are independent, then they are not associated. Are the events {not having long hair} and {wearing jeans} associated for the students in Mr. Rexinger's class today? Explain the independence relationship using P(A given B) = P(A). d. Are the events {not having long hair} and {wearing jeans} mutually exclusive? Explain.

asked 2021-08-09

asked 2021-08-14

The two-way table summarizes information about eye color and gender in a random sample of 200 high school students.

asked 2021-05-28

The two-way table summarizes data on whether students at a certain high school eat regularly in the school cafeteria by grade level. \text{Grade}\ \text{Eat in cafeteria} \begin{array}{l|r|r|r|r|r} & 9 \mathrm{th} & 10 \mathrm{th} & 11 \mathrm{th} & 12 \mathrm{th} & \text { Total } \ \hline \text { Yes } & 130 & 175 & 122 & 68 & 495 \ \hline \text { No } & 18 & 34 & 88 & 170 & 310 \ \hline \text { Total } & 148 & 209 & 210 & 238 & 805 \end{array} If you choose a student at random, what is the probability that the student eats regularly in the cafeteria and is not a 10th-grader?

asked 2021-06-23

Some high school physics students dropped a ball and measured the distance fallen (in centimeters) at various times (in seconds) after its release. If you have studied physics, you probably know that the theoretical relationship between the variables is distance \(=490(time)^2.\) Which of the following scatter-plots would not approximately follow a straight line?

(a) A plot of distance versus \(time^2\)

(b) A plot of radic distance versus time

(c) A plot of distance versus radic time

(d) A plot of \(\ln\)(distance) versus \(\ln(time)\)

(e) A plot of \(\log\)(distance) versus \(\log(time)\)

asked 2021-06-29

The following data was collected about students in Mr. Rexinger's high school statistics class. Wearing jeansNot wearing jeans Has long hair(below chin)77 Does not have long hair 513

a. Mr. Rexinger is playing a game with his students. He randomly chooses a mystery student from his class roster. If a player guesses the hair length of the mystery student correctly, the player gets an early-lunch pass. Madeline is the next player. To have the greatest chance of winning an early-lunch pass, should she guess that the student has long hair? Explain.

b. Mr. Rexinger tells Madeline that the mystery student is wearing jeans. Would you advise Madeline to change her guess? Explain.

c. In a previous course, you may have studied the association of two numerical variables by analyzing scatterplots and least squares regression lines. Associations between categorical events, like having long hair or wearing jeans, are determined by independence-if two events are independent, then they are not associated. Are the events \(\{\text{not having long hair}\}\) and \(\{\text{wearing jeans}\}\) associated for the students in Mr. Rexinger's class today? Explain the independence relationship using \(P(A\ \text{given}\ B) = P(A)\).

d. Are the events \(\{\text{not having long hair}\}\) and \(\{\text{wearing jeans}\}\) mutually exclusive? Explain.