We have proven that at least two of the selected points are within √2/2 units of each other in the given square.
We have,
To prove that at least two of the five selected points are within √2/2 units of each other, we can use the Pigeonhole Principle.
Let's divide the square into four smaller squares by drawing two perpendicular lines that intersect at the center of the square.
Each smaller square will have a side length of √2/2 units.
Now, we have four smaller squares and five selected points.
By the Pigeonhole Principle, if we distribute the five points into the four squares, at least two points must be in the same smaller square.
Since the side length of each smaller square is √2/2 unit, the maximum distance between any two points within the same smaller square is √2/2 units.
Therefore, at least two of the five selected points are within √2/2 units of each other.
Thus,
We have proven that at least two of the selected points are within √2/2 units of each other in the given square.
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solve the following recurrence relations:
a) an=an-1+3(n-1), a0=1
b) an=an-1+n(n-1), a0=3
c) an=an-1+3n^2, a0=10
a) To solve the recurrence relation an = an-1 + 3(n-1), a0 = 1, we can expand the recurrence relation iteratively.
a1 = a0 + 3(1-1) = 1 + 0 = 1
a2 = a1 + 3(2-1) = 1 + 3 = 4
a3 = a2 + 3(3-1) = 4 + 6 = 10
a4 = a3 + 3(4-1) = 10 + 9 = 19
...
We can observe a pattern from the values:
a0 = 1
a1 = 1
a2 = 4
a3 = 10
a4 = 19
We can notice that an = [tex]n^2 + 1.[/tex] We can prove this by induction, but in this case, we can also recognize that the given recurrence relation generates the triangular numbers (1, 3, 6, 10, 15, ...), which are defined by the formula Tn = n(n+1)/2.
Thus, the solution to the recurrence relation is an = [tex]n^2 + 1.[/tex]
b) For the recurrence relation an = an-1 + n(n-1), a0 = 3, we can again expand it iteratively:
a1 = a0 + 1(1-1) = 3 + 0 = 3
a2 = a1 + 2(2-1) = 3 + 2 = 5
a3 = a2 + 3(3-1) = 5 + 6 = 11
a4 = a3 + 4(4-1) = 11 + 12 = 23
...
Observing the values, we can notice that an =[tex]n(n^2 + 1).[/tex]
Thus, the solution to the recurrence relation is an = [tex]n(n^2 + 1).[/tex]
c) For the recurrence relation an = an-1 +[tex]3n^2[/tex], a0 = 10, we expand it iteratively:
a1 = a0 + [tex]3(1^2)[/tex] = 10 + 3 = 13
a2 = a1 + [tex]3(2^2) =[/tex] 13 + 12 = 25
a3 = a2 + [tex]3(3^2)[/tex]= 25 + 27 = 52
a4 = a3 + [tex]3(4^2)[/tex]= 52 + 48 = 100
...
We can notice that an = [tex]n^3 + 10.[/tex]
Thus, the solution to the recurrence relation is an = [tex]n^3 + 10.[/tex]
These are the solutions to the given recurrence relations.
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Maurice ran out of gas. The gas tank in his car can hold 13.6 gallons of gas, but the gas cans only hold 3.4 gallons. How many gas cans would he need to fill his tank
Answer: 4
Step-by-step explanation: 13.6 divided by 3.4 = 136 then divide by 34 you get 34 from 3.4 but divide by 34. Once you divide 136 and 34 you get your answer 4.
I’m not an expert at simplifying, so I need help
Answer: it's 1 1/2
please give me brainlest
3. Use telescoping or iteration to find a closed form for the recurrence relation c₂ = 2cn-1 - 1 with co = 2.
Using telescoping or iteration, the closed form for the recurrence relation c₂ = 2cn-1 - 1 with co = 2 is `cₙ = 2ⁿ+1 - 1` for `n ≥ 0`.
As the recurrence relation is `c₂ = 2cn-1 - 1` with `c₀ = 2`. In the closed form, first, we write out the first few terms of the sequence: `c₀ = 2, c₁ = 3, c₂ = 5, c₃ = 9, c₄ = 17, c₅ = 33, c₆ = 65, ...`
Let's try a pattern in the sequence. Observe that `c₁ = 2c₀ + 1 = 2(2) + 1 = 5`. Similarly, `c₂ = 2c₁ + 1 = 2(5) + 1 = 11`. Continuing this process, we can see that `cₙ = 2cₙ₋₁ + 1` for `n ≥ 1`.
Let's apply telescoping to this formula:```
c₁ = 2c₀ + 1c₂ = 2c₁ + 1= 2(2c₀ + 1) + 1
= 2²c₀ + 2 + 1c₃ = 2c₂ + 1
= 2(2²c₀ + 2 + 1) + 1
= 2³c₀ + 2² + 2 + 1c₄
= 2c₃ + 1= 2(2³c₀ + 2² + 2 + 1) + 1
= 2⁴c₀ + 2³ + 2² + 2 + 1
```Note that each term in this sum telescopes. In general, we can write `cₙ = 2ⁿc₀ + 2ⁿ-1 + ... + 2² + 2 + 1`. Simplifying this expression gives:```
cₙ = 2ⁿc₀ + (2ⁿ - 1)
= 2ⁿ(2) + (2ⁿ - 1)
= 2ⁿ+1 - 1 ```
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Find the slope And y-intercept
Answer:
Answer is (a)
Step-by-step explanation:
Well I didn’t really do any work but you can tell right away theres a ton of slope options so you can’t eliminate from there, BUT when you look at the y-intercept, 3 is the only y-intercept option here and the only choice with the y-intercept of 3 is (a).
what is the greatest common factor of 20 and 50
Answer:
the answer is 10
Step-by-step explanation:
Answer: 10
Why? Because 10x5=50 and then 10x2=20 ten can be used for both of them! Hope this helps^^ please give me brainlist if you could but you don’t have too!^^
Which value of r indicates a stronger correlation:r = 0.818 or r= -0.926? Explain your reasoning. Choose the correct answer below. O A. r= -0.926 represents a stronger correlation because 0.818 > -0.926. O B. r=0.818 represents a stronger correlation because | -0.926) > 10.818). OC. r= -0.926 represents a stronger correlation because | -0.926) > 10.818|- OD. r=0.818 represents a stronger correlation because 0.818 > -0.926.
Hence, option D is the correct answer to the given question.
The correct answer to the given question is the option D. i.e r=0.818 represents a stronger correlation because 0.818 > -0.926. Explanation: The strength of the correlation can be determined by the magnitude of the correlation coefficient. The correlation coefficient values vary between -1 to 1. If the value of correlation coefficient is close to -1 or 1, it indicates strong correlation. On the other hand, if the value of correlation coefficient is close to 0, it indicates a weak correlation.A correlation coefficient value of -0.926 indicates a strong negative correlation. A correlation coefficient value of 0.818 indicates a strong positive correlation. Hence, option D is the correct answer to the given question.
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Can someone help me with this question please
Answer:
61
Step-by-step explanation:
all angles that are in a triangle add up to 180
x+80+39=180
x+119=180
x=61
Hope that helps :)
A bathtub in the shape of A rectangular prism is 20 meters long 10 meters wide and 2 meters deep how much water can the bathtub hold
Answer:
Topic : CIRCLE
1. Capture/ cut outs pictures that represents a Circle then paste it in a long band paper.
2. Make a reflection in CERA form
C-Content (min. 5sentence)
E- experience(min. 5 sen.)
R-reflection (min.5 sent.)
A- Application (min.5
Please help. No files allowed or you will be reported
What is the slope of the line passing through the points (-9,-3) and (7,-7)
Answer:
The slope is [tex]\frac{1}{4}[/tex]
Step-by-step explanation:
[tex]\frac{Y_{2} -Y_{1} }{X_{2}-X_{1} }[/tex]
-7-(-3)= -7+3= 4
4/?
7-(-9)= 7+9 = 16
4/16=1/4
I hope this helps :-)
A binomial experiment consists of 12 trials. The probability of success on trial 5 is 0.16. What is the probability of success on trial 9?
A. 0.1
B. 0.18
C. 0.33
D. 0.25
E. 0.16
F. 0.68
A binomial experiment consists of 12 trials. The probability of success on trial 5 is 0.16. The probability of success on trial 9 is 0.16.
In a binomial experiment, there are two outcomes that are possible in each trial: a success and a failure.
A binomial experiment consists of a fixed number of trials.
The term "probability" refers to the measure of how likely an event is to happen.
The probability of success on trial 5 is 0.16.
We are required to calculate the probability of success on trial 9.
Since the experiment consists of a fixed number of trials, we can say that probability of success in each trial is the same.
Let us assume the probability of success is p.
Therefore, the probability of failure in each trial is (1-p).
The probability of success on trial 5 is 0.16.
Therefore, p = 0.16.
The probability of failure on trial 5 is
1-p = 1-0.16
1-p = 0.84
The probability of success on trial 9 is given by:
P(success on trial 9) = p
P(success on trial 9) = 0.16.
Therefore, the probability of success on trial 9 is 0.16, which is option E.
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Please help with these questions... I've been trying to figure them out for like twenty minutes and at this point I just don't have the energy anymore... I will pick brainliest! Always do...
Answer:
1. y=2x^2-8X-1
2. y=-6x^2+36x-63
3. y=-3x^2+18x-34
4. y=9x^2+54x+76
5. y=-x^2+8x-22
6. y=6x^2-48x+90
7. y=8x^2+16x+5
8. y=2x^2-8x+1
Have a great day :)
A 2-g antibiotic vial stales "Reconstitute with 8.6 mL of sterile water for a final volume of 10 ml.* What is the concentration of the vial after reconstitution?
A. 2 g/8.6 mL
B. 232.6 mg/mL
C. 0.234 mg/mL
D. 200 mg/mL
The correct answer is option B. 232.6 mg/mL which is the concentration of the vial after reconstitution.
To determine that 232.6 mg/mL is the concentration of the antibiotic vial after reconstitution, we need to calculate the amount of antibiotic in grams divided by the final volume in milliliters.
The vial states that it needs to be reconstituted with 8.6 mL of sterile water for a final volume of 10 mL. This means that the antibiotic will be dissolved in 8.6 mL of sterile water.
Since the vial contains 2 grams of antibiotic, the concentration can be calculated as follows:
Concentration = Amount of antibiotic (g) / Final volume (mL)
Concentration = 2 g / 8.6 mL
Simplifying the expression, we get:
Concentration ≈ 0.2326 g/mL
Therefore, the correct option is B) 232.6 mg/mL, the concentration of the vial after reconstitution.
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Analyze and sketch a graph of the function. Find any intercepts, relative extrema, points of inflection, and asymptotes. (If an answer does not exist, enter DNE.)
x/ x^2 +25 intercept (x, y) =
relative minimum (x, y) =
relative maximum (x, y) =
points of inflection (x, y) = (smallest x-value)
(x, y) =
(x, y) = (largest x-value)
Find the equation of the asymptote.
Use a graphing utility to verify your results.
The given function is f(x) = x/(x^2 + 25).
Intercepts:
To find the intercepts, we set f(x) = 0 and solve for x. However, in this case, the function does not have any x-intercepts because the numerator x cannot be zero.
The y-intercept occurs when x = 0. Substituting x = 0 into the function, we have f(0) = 0/(0^2 + 25) = 0.
Relative Extrema:
To find the relative extrema, we take the derivative of the function and find the critical points where the derivative is zero or undefined. However, in this case, the function does not have any relative extrema because its derivative is always nonzero and defined for all x.
Points of Inflection:
To find the points of inflection, we need to analyze the second derivative of the function. However, in this case, the second derivative is always zero, indicating that there are no points of inflection.
Asymptotes:
The function has two asymptotes: a vertical asymptote and a horizontal asymptote.
The vertical asymptote occurs when the denominator of the function is equal to zero. Solving x^2 + 25 = 0, we find that there are no real solutions. Therefore, there is no vertical asymptote.
The horizontal asymptote can be found by examining the behavior of the function as x approaches positive or negative infinity. As x approaches positive or negative infinity, the function approaches zero. Hence, the horizontal asymptote is y = 0.
To verify these results, a graphing utility can be used to plot the function and visualize its behavior.
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The seesaw moves and the angle created by the left side of the seating board and the central support is now 80 degrees. Create and use a model, to find the distance from point Q to the ground when the angle created by the left side of the seating board and the central support is 80 degrees. Show your work or explain your answer. Round your final answer to the nearest tenth of a foot. Enter your model, your answer, and your work or explanations in the space provided.
Answer:5.7
Step-by-step explanation:
The tangent or tanθ in a right angle triangle is the ratio of its perpendicular to its base. The distance between the point Q and the ground is h-[L cos(80°)].
What is Tangent (Tanθ)?The tangent or tanθ in a right angle triangle is the ratio of its perpendicular to its base. it is given as,
[tex]\rm Tangent(\theta) = \dfrac{Perpendicular}{Base}[/tex]
where,
θ is the angle,
Perpendicular is the side of the triangle opposite to the angle θ,
The Base is the adjacent smaller side of the angle θ.
Let the height of the support be h and the height between the point q and the ground is x. Also, the distance between point Q and the support of the seesaw is L.
Now if we look at the ΔQAR, the measure of the ∠QRA is 80°, therefore, the sine of ∠QRA can be written as,
[tex]\rm Cos(\theta)=\dfrac{Base}{Perpendicular}\\\\Cos(\angle QRA) = \dfrac{AR}{QR}\\\\Cos(\angle QRA) = \dfrac{h-x}{L}\\\\L \times Cos(80^o)=h-x\\\\x=h-[L\ Cos(80^o)][/tex]
Hence, the distance between the point Q and the ground is h-[L cos(80°)].
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Please help me to finish the table! D:
Answer: 32, 27, 20
Step-by-step explanation: This is a quadratic table.
The Mavericks football team was losing when they got
control of the ball at midfield in the last minute of a game.
They lost 17 yards on the first play they ran. Then they
gained 3 yards on the second play. On the third play, the
Mavericks lost 21 yards, and on the fourth play, they lost
8 yards. What was the position of the ball relative to its
starting point after those four plays?
A 33 yd
B -50 yd
C 9 yd
D- 43 yd
Pls help I’m doing a renaissance test
Answer:
-43 yards is the answer...
Not yet anvered Which of the following What if analysis scenarios can be solved using a two way data and out of 20 ago Select one: O a. to determine how the monthly payments will depend on the amount borrowed O b. to display how the profit of a lemonade stand will change when the price per cup and the count per cup chama Pc None of the mentioned d. to display how the demand and the profit of a lemonade stand change when the price per cup changes tion 13 Not yet answered Marked out
The scenario that can be solved using a two-way data table out of the options provided is to display how the profit of a lemonade stand will change when the price per cup and the count per cup change.
A two-way data table is a useful tool for conducting sensitivity analysis and exploring how changing two input variables affects the output or result of a formula or calculation. In this case, the profit of a lemonade stand is the output variable, while the price per cup and the count per cup are the two input variables. By creating a two-way data table with different values for the price per cup and the count per cup, we can systematically analyze how these two factors impact the profit of the lemonade stand. Each combination of the price per cup and the count per cup will be evaluated, and the corresponding profit will be calculated. This analysis allows us to observe the relationship between the input variables (price per cup and count per cup) and the output variable (profit). We can identify the optimal price per cup and count per cup that maximize the profit or explore different scenarios to understand how changes in these variables affect the overall profitability of the lemonade stand. Therefore, the option "b. to display how the profit of a lemonade stand will change when the price per cup and the count per cup change" can be solved using a two-way data table.
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If 5.90 × 1.8 = 10.62, what does 106.2 ÷ 59 equal?
Answer:
1.8
Step-by-step explanation:
- hope this helps!!
The desired accumulated is $150,000 after 6 years invested in an
account with 2% interest compounded quarterly.
The amount to be invested now, or the present value needed, is
$__. Round to the nearest
The present value needed to accumulate $150,000 after 6 years with a 2% interest rate compounded quarterly is $131,488
To calculate the present value needed to accumulate $150,000 after 6 years with a 2% interest rate compounded quarterly, we can use the formula for compound interest:
Present Value = Future Value / (1 + r/n)^(n*t)
Where:
Future Value = $150,000
Interest Rate (r) = 2% = 0.02 (converted to decimal)
Number of Compounding Periods per Year (n) = 4 (quarterly compounding)
Number of Years (t) = 6
Substituting these values into the formula, we have:
Present Value = $150,000 / (1 + 0.02/4)^(4*6)
Calculating inside the parentheses first:
Present Value = $150,000 / (1 + 0.005)^(24)
Next, calculate the exponent:
Present Value = $150,000 / (1.005)^(24)
Using a calculator or spreadsheet, we can evaluate the expression inside the parentheses and then divide $150,000 by that result to find the present value. Rounding to the nearest dollar, the present value needed is $131,488.
Therefore, the amount to be invested now or the present value needed is $131,488.
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Solve for X, assume that all segmented that appear to be tangent are tangent.
Answer:
Step-by-step explanation:
x = 9
Which expression is equivalent to 7(a+2) when a= 6?
O 7(6)
916)
0 7(6)+2
7(6)+ 14
Answer:
when a = 6, substitute 6 in the equation for a:
7(a + 2) = 7(6)+ 14
Step-by-step explanation:
Answer:
56
Step-by-step explanation:
When a = 6
7(a+2)
7(6+2)
7(8)
56
3.
Mrs Rita buys 6 cartons of soya bean milk. Each carton contains 1.251
of soya bean milk. Does Mrs Rita have enough soya bean milk to fill up
20 mugs with a capacity of 320 ml each? Explain.
Yes, it is possible to fill all twenty mugs.
Can someone please tell me if I’m correct, quick
Answer: I believe your answer is correct .
Answer:
yes i belive u got it right
Step-by-step explanation:
Help Me Please Quick Will Mark Brainliest if correct : - )
Answer:
y=2x+3
Step-by-step explanation:
Lets verify
[tex]\\ \sf\longmapsto -1=2(-2)+3=-4+3=-1[/tex]
[tex]\\ \sf\longmapsto 1=2(-1)+3=-2+4=1[/tex]
Hence verified
Answer:
C): y = 2x + 3
Step-by-step explanation:
• when x is -2:
[tex]{ \tt{y = 2(- 2) + 3}} \\ { \tt{y = - 1}}[/tex]
• when x is -1:
[tex]{ \tt{y = 2( - 1) + 3}} \\ { \tt{y = 1}}[/tex]
• since y holds for -2 and -1, it'll hold for other positive integers.
Choose the correct definition. Group of answer choices Sensitivity refers to the probability that a test correctly classifies as positive individuals who have preclinical disease. Sensitivity refers to the probability that a test correctly classifies individuals without preclinical disease as negative Specificity refers to the probability that a test correctly classifies individuals with preclinical disease as positive Specificity refers to the probability that a test correctly classifies as positive individuals who have preclinical disease.
Answer:
the probability that a test correctly classifies as positive individuals who have preclinical disease
Step-by-step explanation:
Sensitivity represents the true positive percentage
Lets take an example if there is 90% sensitivity that means there are 90% people who have the target disease considered the test positive
So according to the given options, it is the probability in which if the test is done in a correct manner so this means that the positive individuals have the preclinical disease
Therefore the first option is correct
can someone help me complete this???
Answer:
Part A
750, 2250, 6750, 20250, 60750
Part B
1640250
An unknown radioactive element decays into non-radioactive substances. In 560 days the radioactivity of a sample decreases by 36 percent.
(a) What is the half-life of the element?
half-life:
(b) How long will it take for a sample of 100 mg to decay to 55 mg?
time needed:
After taking the given data into consideration we conclude that
a) the half-life of the unknown radioactive element is approximately 1921.7 days.
b) approximately 1011.4 days, or about 2.77 years, for a sample of 100 mg to decay to 55 mg.
(a) The half-life of a radioactive element is the time needed for half of the material to decay. To find the half-life of the unknown radioactive element, we can use the fact that in 560 days, the radioactivity of a sample decreases by 36 percent. Let T be the half-life of the element. Then, we have:
[tex]0.5 = (1 - 0.36)^{(560/T)}[/tex]
Simplifying this equation, we get:
[tex]0.5 = 0.64^{(560/T)}[/tex]
Taking the natural logarithm of both sides, we get:
[tex]ln(0.5) = ln(0.64)^{(560/T)}[/tex]
[tex]ln(0.5) = (560/T) * ln(0.64)[/tex]
Solving for T, we get:
[tex]T = -560 / (ln(0.64) * ln(0.5))[/tex]
[tex]T \approx 1921.7 days[/tex]
(b) To find how long it will take for a sample of 100 mg to decay to 55 mg, we can use the half-life formula:
[tex]N = N_0 * (1/2)^{(t/T)}[/tex]
where:
N is the final amount, which is 55 mg in this case
[tex]N_0[/tex] is the initial amount, which is 100 mg in this case
t is the time it takes for the sample to decay from [tex]N_0[/tex] to N
T is the half-life of the element, which we found to be approximately 1921.7 days
Substituting the values, we get:
[tex]55 = 100 * (1/2)^{(t/1921.7)}[/tex]
Simplifying this equation, we get:
[tex]0.55 = (1/2)^{(t/1921.7)}[/tex]
Taking the natural logarithm of both sides, we get:
[tex]ln(0.55) = (t/1921.7) * ln(1/2)[/tex]
Solving for t, we get:
[tex]t = -1921.7 * ln(0.55) / ln(1/2)[/tex]
[tex]t \approx 1011.4 days[/tex]
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[tex] \sqrt[3]{625} [/tex]
how would I simplify this without decimals & show work?
Answer:
[tex]5\sqrt[3]{5}[/tex]
Step-by-step explanation:
[tex]\sqrt[3]{625}[/tex] factor [tex]625^{1/3}[/tex]=[tex]5(5^{1/3} )[/tex]=5\sqrt[3]{5}
Hope that helps :)