You can use the fact that is irrational to answer the questions below. You can also use other facts proven within this exercise. (a) Prove that is irrational. (b) Prove that is irrational. (c) Is it true that the sum of two positive irrational numbers is also irrational
Answers
Answer:
See Explanation for proofs
Step-by-step explanation:
Given
See attachment for complete question
Solving (a):
[tex]\frac{\sqrt{2}}{2}[/tex] is irrational
[tex]\sqrt{2} =1.41421356237.....[/tex]
So, we have:
[tex]\frac{\sqrt{2}}{2} = \frac{1.41421356237.....}{2}[/tex]
[tex]\frac{\sqrt{2}}{2} = 0.70710678118....[/tex]
[tex]0.70710678118....[/tex] can not be represented as a fraction of whole numbers.
Hence:
[tex]\frac{\sqrt{2}}{2}[/tex] is irrational
Solving (b):
[tex]\sqrt{2} -2[/tex] is irrational
[tex]\sqrt{2} =1.41421356237.....[/tex]
So, we have:
[tex]\sqrt{2} -2 = 1.41421356237..... - 2[/tex]
[tex]\sqrt{2} -2 = -0.58578643763.....[/tex]
[tex]-0.58578643763.....[/tex] can not be represented as a fraction of whole numbers.
Hence:
[tex]\sqrt{2} -2[/tex] is irrational
Solving (c):
The sum of two positive irrational number is always irrational
Proof Below
If a is irrational and b is also irrational,
then
[tex]a + b = irrational[/tex]
Take for instance:
[tex]a = \sqrt{2[/tex]
[tex]b = \sqrt{8[/tex]
[tex]a + b = \sqrt{2} + \sqrt{8[/tex]
[tex]a + b = \sqrt{2} + \sqrt{4*2[/tex]
[tex]a + b = \sqrt{2} + \sqrt{4}*\sqrt{2[/tex]
[tex]a + b = \sqrt{2} + 2\sqrt{2[/tex]
[tex]a + b = 3\sqrt{2[/tex]
[tex]3\sqrt{2[/tex] is irrational and this is true for the sum of every positive irrational numbers
Related Questions
Rick and Lisa each have their own cookie jar Rick's cookie jar is 7/8 full and Lisa is 5/8. Fraction represents how much more of Rick's jar is full than Lisa?
Answers
Answer:
2/8 is the right answer your welcome
2/8 of a jar more full.
Tip: ALWAYS circle the numbers or key information to get the best results.
Watch what happens!
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Some key information in this problem is 7/8, and 5/8.
Since they asked us how much more and what fraction represents it, we should subtract 5/8 from 7/8 to find how much more full Rick's jar is than Lisa's.
7/8 - 5/8 = 2/8
So in summary, Rick's jar is 2/8 more full than Lisa's jar.
5 cm
12 cm
What is the area
of this rectangle?
And why
Answers
Answer:
60cm^2
Step-by-step explanation:
5X12=60
Help me pls pls pls
Answers
Answer:
1.61
Step-by-step explanation:
Answer:
Step-by-step explanation:
1.61
Which is the slowest reading rate?
A. 3 pages in 9 minutes
B. 4 pages in 10 minutes
C. 5 pages in 13 minutes
D. 7 pages in 20 minutes
Answers
Answer:
A. 3 pages in 9 minutes
Step-by-step explanation:
3 & 9 are all the lowest out of those options.
60÷(63÷7)=
I don't understand plz help
Answers
Answer:
answer is simple
first we will solve the bracket
(63÷7)
it is 9
and then 60÷ 9
which gives 6.67
subtract 10x^2-10x+2 from -8x^2+7x-7
Answers
Answer:
-18x^2+17x-9
Step-by-step explanation:
Jennifer has a hair ribbon that is 12 centimeters long. What is the length of the ribbon in millimeters? will mark brainliest
Answers
Answer:
120 mm
Step-by-step explanation:
Help me out please I need you’re help with this question ✏️
Answers
the answer to your question might be (B).
Lamar went to a baseball game. The game started at
11;50AM, and it was 3 hours 39minutes long. When did the game end?
Answers
11:50 + 3 hours = 2:50
2:50+ 39 mins = 3:29
Answer:
The game ends at 2:39
Step-by-step explanation:
11:50 + 3 hours (3:00) = 2:00
2:00+ 0:39 = 2:39
PLEASEEEEEE HELPPPPPP IM SO CONFUSED IVE BEEN TRYING TO SOLVE FOR DAYS
I WILL ALSO GIVE BRAINLIEST!!!
A biologist is studying the growth of a particular species of algae. She writes the following equation to show the radius of the algae, f(d), in mm, after d days:
f(d) = 9(1.04)d
Part A: When the biologist concluded her study, the radius of the algae was approximately 12.81 mm. What is a reasonable domain to plot the growth function?
Answers
Answer:
9 days
Step-by-step explanation:
1= 9*(1.04)^d
12.81/9= (1.04)^d
1.423= (1.04)^d
log(1.423)= d *log(1.04)
0.1532=d * 0.017
d= 0.1532/0.017
d=9
The reasonable domain to plot the growth function is in the interval [0,9].
Answer:
9 days
Step-by-step explanation:
Solve the inequality 8c + 6 < -6c - 1
Answers
Answer:
c < - 0,5
Step-by-step explanation:
8c + 6 < - 6c - 1
// move the terms
8c + 6c < - 1 - 6
// collect the terms
14c < - 7
// divide both sides by 7
2c < -1
// divide both sides by 2
c < - 0,5
8c + 6c < -1 - 6
14c < -7
c < -7/14
c < -1/2
Hope it helps!
Help me out please mathematic ✏️
Answers
To solve this you need to take the two degree values you already have, 90 degrees and 25 degrees. From there you can subtract 90 from 25.
Find cosX and cosZ. Write each answer as a fraction and as a decimal. Round to four decimals places, if necessary. Fractions to reduce and use /.
Answers
Answer/Step-by-step explanation:
Recall the acronym, SOHCAHTOA.
✔️Cos X = adj/hyp (CAH)
Adj = 10
Hyp = 26
Cos X = 10/26
Simplify
Cos X = 5/13 (as a fraction)
Cos X = 0.3846 (as a decimal to 4 d.p)
✔️Cos Z = adj/hyp (CAH)
Adj = 24
Hyp = 26
Cos Z = 24/26
Cos Z = 12/13 (as a fraction)
Cos Z = 0.9231 (as a decimal to 4 d.p)
Find a. Give your answer in
the simplest form.
[?]
a
zva
d
а
45
30
b
с
Answers
45 45 90 triangles are x x and x sqrt(2) so a is 7
Answer:
a=7
Step-by-step explanation:
7(sq rt 2)=(sq rt 98)
sin 45=a/(sq rt 98)
0.7071=a/(sq rt 98)
a=0.7071(sq rt 98)
a=7
Can you look at the photo and help please.?
Answers
Answer:
sorry it aint werking for meeee
Step-by-step explanation:
the length of a rectangle garden is 5 m the area is 176 m2 what is the width of the garden
Answers
Answer:
Step-by-step explanation:
L = 5 m
area = L×W = 176 m²
5W = 176
W = 176/5 = 35.2 m
Solve this system by substitution. y = 3x - 12 6x + y = 6 *Remember to write your answer as a coordinate voint (x, y).
Answers
6x+3x-12=6
9x-12=6
+12= 18/9 x=2
Y=6-12
Y=-6 (2,-6
The Statistical Abstract of the United States reported that the average cost per day of owning an automobile in the United states is $7.62. This includes the cost of the car, general maintenance, gasoline, and insurance. A researcher claims that college students' average daily ownership expenses are less than the national average. A random sample of 54 college students who own cars found the average cost per day to be $6.78 with standard deviation $1.77. Use a 5% level of significance to test the claim that a college student's average daily ownership expenses are less than the national average. State the Null and Alternate Hypotheses, calculate the test statistic, compute the P value for this test, state the alpha value, then state your conclusion. Is there enough evidence to support the claim?
Answers
Answer:
Step-by-step explanation:
The null and the alternative can be computed as:
[tex]H_o : \mu = 7.62[/tex]
[tex]H_1 : \mu < 7.62[/tex]
[tex]Claim: \mu < 7.62[/tex]
[tex]\alpha= 0.05[/tex]
The critical value for a left-tailed test at [tex]\alpha= 0.05[/tex] = -1.645
The test statistics can be computed as:
[tex]Z = \dfrac{\overline x - \mu}{\dfrac{\sigma}{\sqrt{n}} }[/tex]
[tex]Z = \dfrac{6.78 - 7.62}{\dfrac{1.77}{\sqrt{54}} }[/tex]
[tex]Z = -3.487[/tex]
The P-value = (Z< -3.487)
The P-value = 0.00024
Decision Rule: TO reject [tex]\mathbf{H_o}[/tex] at [tex]\alpha= 0.05[/tex], if test statistics is less than the critical value (left tailed)
Conclusion: We reject [tex]\mathbf{H_o}[/tex] at ∝ = 5%, thus there is enough evidence to support the claim that the college students average daily ownership expenses are less than the national average.
Pls help me pls pls pls I’ll make brainliest
Answers
Answer:
Alr what do you need help with
please help giving lot of points
Answers
Answer:
They would need to sell 32 calendars to EXCEED their goal.
Explanation:
8 x 32= 256, which is more than the $250 needed!
Step-by-step explanation:
it cost 0.60 cents to run a fan for 4 hours how much does it cost to run the fan for 6 hours I need an answer asap
Answers
Answer:
$0.90
Step-by-step explanation:
So first you need to find the rate (showing how much money it costs to run for 1 hour) so you get 0.60 and divide it by 4 and its 0.15, then multiply 0.15 by 6 and get 0.90
work out yhe value of 6.105×10^8/3.7×10^3 give your answer in standard form
Answers
Hope this helps
Jonny had ? cakes. He eats 1/2 of
his total amount of cakes. How much
cake does he have left?
Answers
Answer:
1/2 * (?)
Step-by-step explanation:
Divide (?) Cakes in half
Answer:
How many cakess did he have? impossible to answer otherwise
Step-by-step explanation:
You are given the following equation: (x+3)^2 + (y+9)^2 = 144 What is the
(h,k)
Answers
Answer:
Step-by-step explanation:
Equation of circle: (x - h)² + (y -k)² = r²
(x+ 3)² + (y + 9)² = 144
(x+ 3)² + (y + 9)² = 12²
h = -3 ; k = -9
A group decides to go white-water rafting on a river. The map they have of the river has the scale 3 inches:7 miles. On the map, the distance between the start point and the end point of their trip is 15 inches. What is the actual distance?
Answers
what is 240 x 343 x 234 x 123 x 23 - 89
Answers
4x+3(2x-1)-4x i need help
Answers
Answer:
x=0.5
Step-by-step explanation:
Simplifying
4x + -3(2x + -1) + -4x = 0
Reorder the terms:
4x + -3(-1 + 2x) + -4x = 0
4x + (-1 * -3 + 2x * -3) + -4x = 0
4x + (3 + -6x) + -4x = 0
Reorder the terms:
3 + 4x + -6x + -4x = 0
Combine like terms: 4x + -6x = -2x
3 + -2x + -4x = 0
Combine like terms: -2x + -4x = -6x
3 + -6x = 0
Solving
3 + -6x = 0
Solving for variable 'x'.
Move all terms containing x to the left, all other terms to the right.
Add '-3' to each side of the equation.
3 + -3 + -6x = 0 + -3
Combine like terms: 3 + -3 = 0
0 + -6x = 0 + -3
-6x = 0 + -3
Combine like terms: 0 + -3 = -3
-6x = -3
Divide each side by '-6'.
x = 0.5
Simplifying
x = 0.5
I need the answer to this problem please
Answers
The value of a boat is depreciating at a rate of 7% per year. In 2006, the boat was worth $17,300. Find the value of the boat in 2011. Enter the exponential decay function for this situation.
The exponential decay function is y = .
In 2011 the boat will be worth $ .
Answers
Answer:
Step-by-step explanation:
[tex]V=Ar^t\\ \\ V=17300((100-7)/100)^t\\ V=17300(0.93)^{(y-2006)}\\ V(2011)=17300(0.93)^{(2012-2006)}=\$ 11192.29[/tex]
Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. x2 9y2
Answers
This question is incomplete, the complete question is;
Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.
x² + 9y² = 9
(a) About
y = 3
(b) About
x = 3
Answer:
a) the volume of solid revolution is 18π² = 177.652879 ≈ 177.65288
b) the volume of solid revolution is 18π² = 177.652879 ≈ 177.65288
Step-by-step explanation:
a) the volume of solid of revolution
about y=3
Using SHELL METHOD;
Volume V = 2π . [tex]\int\limits^1_{y=-1} \,[/tex] (3-y) (2.3√(1-y²)) dy
= 12π . [tex]\int\limits^1_{y=-1} \,[/tex] (3-y) √(1-y²) dy
= 12π . [tex]\int\limits^1_{y=-1} \,[/tex] (( 3√(1-y²) - y(√(1-y²)) dy
= 12π[ [tex]\frac{3}{2}[/tex]π - 0 ]
= 18π² = 177.652879 ≈ 177.65288
Therefore, the volume of solid revolution is 18π² = 177.652879 ≈ 177.65288
b)
the volume of solid of revolution
about x=3
Using SHELL METHOD;
Volume V = [tex]\frac{4}{3}[/tex]π . [tex]\int\limits^3_{y=-3} \,[/tex] (3-x) ([tex]2.\frac{1}{3}[/tex]√(9-x²)) dx
= [tex]\frac{4}{3}[/tex]π . [tex]\int\limits^3_{y=-3} \,[/tex] (3-x) √(9-x²) dx
= [tex]\frac{4}{3}[/tex]π . [tex]\int\limits^3_{y=-3} \,[/tex] (3√9-x² -x√(9-x²) dx
= [tex]\frac{4}{3}[/tex]π [ [tex]\frac{27}{2} \pi[/tex] - 0 ]
= 18π² = 177.652879 ≈ 177.65288
Therefore, the volume of solid revolution is 18π² = 177.652879 ≈ 177.65288