Question

The length of a rectangle is 2 units and its width is square root of 2 unit. Is the area of the rectangle rational or irrational? Justify your answer.

Answer 1

i could use the help please ill award a medal

Answer 2

just logged on let me work it out

Answer 3

thank you

Answer 4

Irrational, and I assume you want the proof of why its irrational?

Answer 5

yes please

Answer 6

im guessing because the area is not a whole number

Answer 7

\[\sqrt{2} \] is irrational. Proof by way of contradiction. assume that \[\sqrt{2}\] is rational. Then is can be expressed as \[\frac{ a }{ b }\] where \[a,b \in \integer\] ,
\[b \neq 0\] when a and b have no common factors.

Answer 8

is the first part
let me finish

Answer 9

oh ok

Answer 10

\[\sqrt{2}=\frac{ a }{ b }\] \[2= [\frac{a}{b}]^2\] \[2b^2=a^2\] Because a is an integer it can be expressed as a=2k where k is some integer. by substituting 2k in the equation for a we have \[2b^2=(2k)^2\] \[2b^2=4k^2\] but this implies that a and b share a common factor of 2. which violates the premise that a and b share no common factors. therefore \[\sqrt{2}\] must be irrational. this leads us to conclude that \[2\sqrt{2}\] must also be irrational as any rational number times an irrational number is irrational Q.E.D.

Answer 11

thank you

Answer 12

do you think you can help with another question

Answer 13

I can try.

Answer 14

okay ill show you

Answer 15

A scientist is studying the growth of a particular species of plant. He writes the following equation to show the height of the plant f(n), in cm, after n days.

f(n) = 10(1.02)n

Part A: When the scientist concluded his study, the height of the plant was approximately 11.04 cm. What is a reasonable domain to plot the growth function? (4 points)

Part B: What does the y-intercept of the graph of the function f(n) represent? (2 points)

Part C: What is the average rate of change of the function f(n) from n = 1 to n = 5 and what does it represent? (4 points)

Answer 16

sorry i have big problems with functions i would really appreciate if you could help

Answer 17

Well i can try it, but what math class is this for, itll help me narrow down the approach your gonna want to use?

Answer 18

working with functions

Answer 19

oh im taking algebra

Answer 20

perfect, and one question, is the number (1.02) supposed to be a power by chance or are all of the given numbers 10, 1.02, n simply numbers?

Answer 21

umm its just the number that represents how much the plant grows and 10 represents the height of plant

Answer 22

f(n) = 10(1.02)n

Answer 23

alright

Answer 24

a reasonable range would be \[0 \le y \le 2\] because the equation \[y=10(1.02)n\] can be rewritten as \[y= 10.2n\] given the final value of 11.04 we substitute that into the equation for y \[11.04=10.2n\] \[\frac{11.04}{10.2} =n\] 1.08235.. = n so the researcher quit after one day. the domain is the number of days, so on a graph the number of days would be 0 to 2

Answer 25

okay so thats for part a

Answer 26

the y intercept repressents the height of the plant at the start of the experiment.
which is part b

Answer 27

oh okay and for part c it would be

Answer 28

doing the math now, give me a second

Answer 29

oh okay

Answer 30

The rate of change is the slope of the line from the points (1,10.2) to (5,51) [i got the y coordinate by plugging in the n values 1 and 5 into the equation and solving for y.
the slope is the rise over run of the points so:
\[\frac{51-10.2}{5-1}\] \[\frac{40.8}{4}\]
\[10.2\]
this number represents the average amount of growth per day of the plant

Answer 31

thank you so much i have questions but i dont bother you since you already helped me alot
thanks for your time