The function (5n² - 9) / (n + 3) is a natural number for n = 3 and n = 15.
A function is given which is (5n² - 9) / (2n + 6).
We have to find the value of natural number n for which the given function is also natural number.
What do you mean by natural numbers ?
Natural numbers are known as the counting numbers which starts from 1 and end at infinity and also includes the whole numbers excluding 0.
As per the question ;
(5n² - 9) / (2n + 6) is a natural number for the value of n which is also a natural number.
Let's assume ;
(5n² - 9) / (2n + 6) = k
or
(5n² - 9) / (n + 3) = 2k
So , we can say that (5n² - 9) / (n + 3) is an even number.
We know that ;
Dividend ÷ Divisor = Quotient
If we divide (5n² - 9) by (n + 3) we will get ;
Quotient = 5n - 15
and
Remainder = 36
We can write it as ;
(5n² - 9) ÷ (n + 3) = (5n - 15) + {36 / (n+ 3)}
Here ;
36 / (n+ 3) is a natural number and n is a natural number.
Assume ;
n = 1 , 3 , 6 , 9 , 15 , 33
Let's check ;
for n = 1 ;
(5n - 15) + {36 / (n+ 3)} will be ;
= (5 - 15) + {36 / (1+ 3)}
= -1
for n = 3 ;
(5n - 15) + {36 / (n+ 3)} will be ;
= (15 - 15) + {36 / (3+ 3)}
= 6
for n = 6 ;
(5n - 15) + {36 / (n+ 3)} will be ;
= (30 - 15) + {36 / (6+ 3)}
= 19
for n = 9 ;
(5n - 15) + {36 / (n+ 3)} will be ;
= (45 - 15) + {36 / (9+ 3)}
= 33
for n = 15 ;
(5n - 15) + {36 / (n+ 3)} will be ;
= (75 - 15) + {36 / (15+ 3)}
= 62
for n = 33 ;
(5n - 15) + {36 / (n+ 3)} will be ;
= (165 - 15) + {36 / (33+ 3)}
= 151
So , we found that only for values of n = 3 and n = 15 , the value of (5n - 15) + {36 / (n+ 3)} is also an even number.
Thus , the function (5n² - 9) / (n + 3) is a natural number for n = 3 and n = 15.
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- 12(x+12) + 1177 = 10 + 15
Answer:
x = 84
Step-by-step explanation:
-12(x+12) + 1177 = 10 + 15
-12x - 144 + 1177 = 25
1033 = 25 + 12x
1008 = 12x
x = 84
There is a real number whose product with every real numbers equal zero
a. Some ___ has the property that is __.
b. There is a real number a such that the product of á __.
c. There is a real number á with the property that for every real number b __.
There is a real number whose product with every real number equals zero
a. Some real number has the property that its product with every real number equals 0.
b. There is a real number a such that the product of á and any real equals zero.
c. There is a real number á with the property that for every real number b, the product of a and b equals zero.
In August 2006, a report was issued about an Internet poll conducted on behalf of the Oxygen Television Network. The survey included 1400 women and 700 men, aged 15 to 49. The following is part of a press release about the findings.
"Apparently if it's a gadget, women gotta have it, according to a survey done by Oxygen Media, an American cable network owned and operated by women. The Women's Watch: Girls Gone Wired survey released in August indicates that 77 per cent of women fancy a new plasma TV over a diamond solitaire necklace, 78 per cent would rather have a sleek new cell phone than shoes, and 86 per cent would prefer a new digital video camera over designer footwear."
What can be said about these poll results?
Answer:
Results of the Assembly elections in five States are indeed a shot in the arm for the Bharatiya Janata Party (BJP), which won in all four States where it was in power. The Hindu editorial captures the broad trends that emerge out of these results. Prime Minister Narendra Modi has said these results have sealed the outcome of the 2024 Lok Sabha elections too in the BJP’s favour. He said people have rejected caste and dynastic politics.
o qualify for security officers’ training recruits are tested for stress tolerance. The scores are normally distributed, with a mean of 62 and a standard deviation of 8.
If only the top 15% of recruits are selected, find the cutoff score.
The cutoff score is mathematically given as
x_n=70.32 (cutoff)
This is further explained below.
What is the cutoff score.?Generally, the equation for is mathematically given as
[tex]&P\left(X > x_0\right)=0.15 \Rightarrow P\left(X < x_0\right)=0.85 \\[/tex]
[tex]&\Rightarrow P\left(\frac{X-\mu}{\sigma} < \frac{x_0-\mu}{\sigma}\right)=0.85 \\[/tex]
[tex]&\Rightarrow P\left(Z < \frac{x_0-\mu}{\sigma}\right)=0.85 \\\\&\Rightarrow \frac{x_0-\mu}{\sigma}=\text { invNorm }(0.85) \\\\&\Rightarrow x_0=\mu+\sigma \times \text { invNorm }(0.85)[/tex]
Use z-table to get inv Norm() = 1.04
x_0=62+8 *(1.04)=70.32
In conclusion, Top 15% are above x_n=70.32 (cutoff)
Clearly this is 85th percentile
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What is the slope of a line perpendicular to the line whose equation is x+y=-6x. Fully reduce your answer.
The slope of the equation is -7. Then the slope of the perpendicular line will be 1/7.
What is the equation of a perpendicular line?Let the equation of the line be y = mx + c. Then the equation of the perpendicular line that is perpendicular to the line y = mx + c is given as y = -(1/m) + d. If the slope of the line is m, then the slope of the perpendicular line will be negative 1/m.
The equation of the line is given as,
y = - 7x
The slope of the equation is -7. Then the slope of the perpendicular line will be
⇒ -(1/(-7))
⇒ 1/7
The slope of the equation is -7. Then the slope of the perpendicular line will be 1/7.
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100 POINTS PLEASE ANSWER QUICK
Use the box plot below to answer the questions.
Part A: Estimate the IQR for the male data.
Part B: Estimate the difference between the median values of each data set.
Part C: Describe the distribution of the data and if the mean or median would be a better measure of center for each.
The median is a better measure of center for the male's data, while the mean is a better for the female's data.
What are quartiles?When we get data which can be compared relatively with each other, for finding quartiles, we arrange them in ascending or descending order.
Quartiles are then selected as 3 points such that they create four groups in the data, each groups approximately possessing 25% of the data.
(a) The IQR of the male's data
From the given male's boxplot, we have the following data elements;
Upper quartile (Q3) = 14
Lower quartile (Q1) = 2
Then the IQR is calculated as:
IQR = Q3 - Q1
So, we have:
IQR = 14 - 2
IQR = 12
Hence, the IQR for the given males' data is 12
(b) The difference between the median values can be calculated as;
Median of male's data = 14
Median of female's data = 9
The difference (d) between the median values is;
d = 14 - 9
d = 5
Hence, the difference between the median values of each data set is 5
(c) The distribution of data
It is found that the median is a better measure of center for the male's data, while the mean is a better measure of center for the female's data.
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The median is a better measure of center for the male's data, while the mean is a better for the female's data.
What are quartiles?
When we get data which can be compared relatively with each other, for finding quartiles, we arrange them in ascending or descending order.
Quartiles are then selected as 3 points such that they create four groups in the data, each groups approximately possessing 25% of the data.
(a) The IQR of the male's data
From the given male's boxplot, we have the following data elements;
Upper quartile (Q3) = 14
Lower quartile (Q1) = 2
Then the IQR is calculated as:
IQR = Q3 - Q1
So, we have:
IQR = 14 - 2
IQR = 12
Hence, the IQR for the given males' data is 12
(b) The difference between the median values can be calculated as;
Median of male's data = 14
Median of female's data = 9
The difference (d) between the median values is;
d = 14 - 9
d = 5
Hence, the difference between the median values of each data set is 5
(c) The distribution of data
It is found that the median is a better measure of center for the male's data, while the mean is a better measure of center for the female's data.
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Lesson 1.17
Quadrilateral ABCD is congruent to quadrilateral
A'B'C'D'. Describe a sequence of rigid motions that
takes A to A', B to B', C to C', and D.
please im begging you please help!!!
Answer:
You will translate A up until it is equal with A', then you will reflect the figure over the line of symmetry that will be half way between line segments CD and C'D'
Step-by-step explanation:
I am not sure if the polygon is drawn on a coordinate plane. I cannot see any horizontal or vertical lines to tell you exactly how many units to move up.
Find the vertex of the quadratic function f(x)=(x+3)2−4
The vertex form of the quadratic equation is y + 4 = (x + 3)² and the vertex of the parabola is (h, k) = (- 3, - 4).
How to find the vertex of the parabola from the expression of a quadratic equation
Herein we find the expression of a quadratic equation in vertex form, whose definition is shown below:
4 · p · (y - k) = (x - h)² (1)
Where:
h, k - Coordinates of the vertex.
p - Distance from the vertex to the focus.
Then, the vertex form of the quadratic equation is y + 4 = (x + 3)² and the vertex of the parabola is (h, k) = (- 3, - 4).
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PLEASE HELP in the figure below, what is the value of x? Show all work
Answer:
x=19
Step-by-step explanation:
All of the angles add up to 180 degrees since it is a straight angle. So, the equation is:
3x+7+90+x+7=180
You combine like terms:
4x+104=180
Then you subtract on both sides:
4x=76
Then you divide by 4 on both sides:
x=19
Quantitative Reasoning :')
Answer:
Step-by-step explanation:
more taking a shower you use almost 3 gallons per a shower
Which of the following statements are true?
a) 25:35 = 45:55
b) 105: 30 = 49: 14
c) 45:48 = 60:64
HELP 50 POINTS (see image attached)
5
1
2
8
-2
The value of the function f(3) is 1
How to determine the value of f(3)?The graph represents the given parameter
The value of f(3) implies that we calculate the value of the function when x = 3
From the graph, we have
f(x) = 1, when x = 3
This means that
f(3) = 1
The above is calculated by substituting 3 for x in the equation f(x) = 1
Hence, the value of f(3) is 1
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Mackenzie has
22
ft of decorative fencing to line her rectangular flower garden. If her garden must have a width of
4
ft, what will the length of her garden have to be to use all
22
ft of fencing?
Answer:
the length of her garden will have to be 5.5ft
22÷4=5.5
Mathematics in the modern world
Answer:
Mathematics shaping the modern world.
2.
If a train travels between two cities in 3 hours at an average speed of 65 miles per hour, how long
would it take at an average speed of 80 miles per hour?
Answer:
At 65mph the car traveled 195 miles in 3hrs.
So divide 195 miles by 80mph to get 2.4375 hrs
Find the slope P=(-4,-1) Q=(-3,-3)
The slope of P=(-4,-1) Q=(-3,-3) is -2.
Given,
P=(-4,-1)
Q=(-3,-3)
The slope between two points P=(x₁, y₁)
Q=(x₂ ,y₂)
slope = (y₂-y₁)/(x₂-x₁)
So slope between P=(-4,-1) Q=(-3,-3) is
[(-3)-(-1) ]/ [(-3)-(-4)]
=-2/1
=-2
A line's steepness and direction are measured by the line's slope. Without actually using a compass, determining the slope of lines in a coordinate plane can assist in forecasting whether the lines are parallel, perpendicular, or none at all.
Any two different points on a line may be used to compute the slope of any line. The ratio of "vertical change" to "horizontal change" between two separate locations on a line is calculated using the slope of a line formula.
The increase divided by the run, or the ratio of the rise to the run, is known as the line's slope. In the coordinate plane, it describes the slope of the line. Finding the slope between two separate locations and calculating the slope of a line are related tasks. In general, we require the values of any two separate coordinates of a line in order to determine its slope.
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FG is perpendicular to DM, if DF = 4R, FM = r + 21, and DF = FM, what is the value of R?
Answer:
(r+21 ) / 4
Step-by-step explanation:
as we can see DF = FM
and DF = 4R , FM =r+21
so,
4R = r+21
=> R= (r+21)/4
change the subject. make x the subject
Answer:
X= 1/y
Step-by-step explanation:
[tex]y = 1 - \frac{1}{ \sqrt{1} - \frac{1}{1 - x} } \\ y = 1 - \frac{1}{1 - \frac{1}{1 - x} } \\ y - 1 = - \frac{1}{1 - \frac{1}{1 - x} } \\ y - 1 = - \frac{1}{ \frac{1 - x - 1}{1 - x} } \\ y - 1 = - \frac{1 - x}{ - x} \\ y - 1 = \frac{1 - x}{x} \\ yx - x = 1 - x \\ yx = 1 \\ x = \frac{1}{y} [/tex]
using associative property
(y+12)+28
Find the area of the shaded region. Help please!
[tex]\textit{area of a sector of a circle}\\\\ A=\cfrac{\theta \pi r^2}{360} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ r=10\\ \theta =85 \end{cases}\implies \begin{array}{llll} A=\cfrac{(85)\pi (10)^2}{360} \\\\\\ A=\cfrac{425\pi }{18}\implies A\approx 74.18~cm^2 \end{array}[/tex]
Help asap please!! I don’t understand
Answer: 2+[tex]\frac{6x-2}{x(x-3)}[/tex] ==> x[tex]\neq[/tex]0, x[tex]\neq[/tex]3
Step-by-step explanation:
f(x)=2x^2-2
g(x)=x^2-3x
(f/g)(x)=
[tex]\frac{2x^2-2}{x^2-3x}[/tex]=
[tex]\frac{2x^2-6x+6x-2}{x^2-3x}[/tex]=
[tex]\frac{2x^2-6x+6x-2}{x^2-3x}[/tex]=
2+[tex]\frac{6x-2}{x^2-3x}[/tex]=2+[tex]\frac{6x-2}{x(x-3)}[/tex] ==> x[tex]\neq[/tex]0, x[tex]\neq[/tex]3
6 What types of decimal numbers are rational? What types are irrational?
7 Write down two more examples (decimal, fraction, etc.) of each type of number: rational and irrational. Explain your choices.
Solve these questions
1) x +2x + x = 68
2) x+y+3y= 45
3) x+y-4z= -32
4) x+y+z= ?
5) -x + 8x= 63
6) 4x+3x-x=42
Same questions as the pic
Answer:
here is the answers
Step-by-step explanation:
sample:
1. 4x = 68
x = 68/4
x = 17
Solution :
1) x + 2x + x = 68
>> 4x = 68
>> x = 68 / 4
>> x = 17
2) x + y + 3y = 45
>> x + 4y = 45
>> 17 + 4y = 45
>> 4y = 45 - 17
>> 4y = 28
>> y = 28 / 4
>> y = 7
3) x + y - 4z = -32
>> 17 + 7 - 4z = -32
>> 24 - 4z = -32
>> -4z = -32 - 24
>> -4z = -56
>> z = 56 / 4
>> z = 14
4) x + y + z = ?
>> 17 + 7 + 14
>> 24 + 14
>> 38
5) -x + 8x = 63
>> 7x = 63
>> x = 63 / 7
>> x = 9
6) 4x + 3x - x = 42
>> 7x - x = 42
>> 6x = 42
>> x = 42 / 6
>> x = 7
Given q = √(5p²-1)²-2 find p in terms of q.
Answer:
[tex]\sf p =\sqrt{\dfrac{q+3}{5}}[/tex]
Step-by-step explanation:
To find p in terms of q, we have to isolate p.
[tex]\sf \sqrt{(5p^2 - 1)^2}-2 = q\\\\[/tex]
5p² - 1 - 2 = q
5p² - 3 = q
Add 3 to both sides,
5p² = q + 3
Divide both sides by 5,
[tex]\sf p^2 = \dfrac{q+3}{5}\\\\\text{Take square root,}\\\\p =\sqrt{\dfrac{q+3}{5}}[/tex]
14. Jennifer and Francine are running a race where Francine runs the race far faster than Jennifer.
It takes Jennifer just five seconds less than twice Francine's time to finish. If their combined
total time is 86 seconds and frepresents the amount of time it takes Francine to finish, then
which of the following equations models this situation?
(1) ƒ+2(ƒ-5)=86
(3) ƒ+2ƒ-5=86
(2) 2ƒ-5=86
(4) 2f-5=43
The answer is (2) 2f-5=86; I hope- :\
A company is reviewing a batch of 20 products to determine if any are defective. On average, 3.7% of products are defective. Does this situation describe a binomial experiment, and why? Pick What is the probability that the company will find 2 or fewer defective products in this batch? Ex: 1.23 % What is the probability that 4 or more defective products are found in this batch? Ex: 1.28 % If the company finds 5 defective products in this batch, should the company stop production? Pick v
There is a 96.37% probability that the company will find 2 or more defective products.
There is a 2.08% probability that the company will find 4 or more defective products.
There is a 0.02% probability that the company will find 5 or more defective products. So company will give profit.
There are only two possible outcomes for each product: either it is defective or it is not. The binomial distribution is used to answer this question because each product's defect probability is independent of all other products.
Binomial probability distribution
The probability of precisely x successes on n repeated trials is known as the binomial probability.
[tex]P_x = \;^nC_r.p^x.q^{n-x}[/tex]
Where,
Px = binomial probability,
[tex]\;^nC_r =[/tex] number of combination
p = probability of success on a single trial
q = probability of failure on a single trial = (1 - p)
n = number of trials
x = number of times for a specific outcome within n trials
So, from question,
20 products mean n = 203.7% of products are defective mean p = 3.7% = 0.0371) The probability that, 2 or fewer defective products is;
P(X≤2) = P(X=0) + P(X=1)+ P(X=2) -----------------(1)
So,
[tex]P_x = \;^nC_r.p^x.q^{n-x}[/tex]
P(X=0) = [tex]\;^{20}C_0.p^0.q^{20-0}[/tex] = [tex]\; 1.(1-p)^{20-0}[/tex] = (1-0.037)²⁰ = (0.963)²⁰ = 0.4704
P(X=1) = [tex]\;^{20}C_1.p^1.q^{20-1}[/tex] = [tex]\; 20.(0.037).(0.963)^{19}[/tex] = 0.3614
P(X=2) = [tex]\;^{20}C_2.p^2.q^{20-2}[/tex] = [tex]\;190\times(0.037)^2\times(0.963)^{18}[/tex] = 0.1319
Now, putting the all values in equation (1)
P(X≤2) = P(X=0) + P(X=1)+ P(X=2) -----------------(1)
= 0.4704 + 0.3614 + 0.1319
P(X≤2) = 0.9637 = 96.37%
There is a 96.37% probability that the company will find 2 or more defective products.
2)
1.23% are defective means p = 0.0123
The probability that find 4 or more defective products are;
P(X≤4) = 1 - P(X< 4) -----------(2)
P(X< 4) = P(X=0) + P(X=1)+ P(X=2) + P(X=3)-----------------(3)
So,
P(X=0) = = [tex]\;^{20}C_0.p^0.q^{20-0}[/tex] = [tex]\; (0.9877)^{20}[/tex] = 0.7807
P(X=1) = [tex]\;^{20}C_1.p^1.(1-p)^{20-1}[/tex] = [tex]20\times (0.0123)\times (0.9877)^{19}[/tex] = 0.1944
P(X=2) = [tex]\;^{20}C_2.p^2.(1-p)^{20-2}[/tex] = [tex]190\times (0.0123)^2\times (0.9877)^{18}[/tex] = 0.0024
P(X=3) = [tex]\;^{20}C_3.p^3.(1-p)^{20-3}[/tex] = [tex]1140\times (0.0123)^3\times (0.9877)^{18}[/tex] = 0.0017
So,
P(X< 4) = 0.7807+0.1944+0.0024+0.0017 =0.9792
P(X≤4) = 1 - P(X< 4) = 1 - 0.9792 = 0.0208 = 2.08%
There is a 2.08% probability that the company will find 4 or more defective products.
3)
1.28% are defective means p = 0.0128
The probability that find 5 or more defective products are;
P(X≤5) = 1 - P(X< 5) -----------(4)
P(X< 5) = P(X=0) + P(X=1)+ P(X=2) + P(X=3)+ P(X=4)-----------------(5)
So,
P(X=0) = = [tex]\;^{20}C_0.p^0.q^{20-0}[/tex] = [tex]\; (0.9872)^{20}[/tex] = 0.7728
P(X=1) = [tex]\;^{20}C_1.p^1.(1-p)^{20-1}[/tex] = [tex]20\times (0.0128)\times (0.9872)^{19}[/tex] = 0.2004
P(X=2) = [tex]\;^{20}C_2.p^2.(1-p)^{20-2}[/tex] = [tex]190\times (0.0128)^2\times (0.9872)^{18}[/tex] = 0.0246
P(X=3) = [tex]\;^{20}C_3.p^3.(1-p)^{20-3}[/tex] = [tex]1140\times (0.0128)^3\times (0.9872)^{18}[/tex] = 0.0019
P(X=4) = [tex]\;^{20}C_4.p^4.(1-p)^{20-4}[/tex] = [tex]4845 \times (0.0128)^3\times (0.9872)^{18}[/tex] = 0.0001
So,
P(X< 5) = 0.7728+0.2004+0.0246+0.0019+0.0001 = 0.9998
P(X≤5) = 1 - P(X< 5) = 1 - 0.9998 = 0.0002 = 0.02%
There is a 0.02% probability that the company will find 5 or more defective products. So company will give profit.
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Find the midpoint of a line segment with endpoints A(-5, 5) and B(2, -5).
Please help show your own work
Answer: (-1.5, 0)
Step-by-step explanation:
Use the midpoint formula, [tex](\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})[/tex]. Plug in the points into the equation to get [tex](\frac{-5+2}{2},\frac{5-5}{2})[/tex] = (-1.5, 0)
Jill has 4 one-dollar bills, 3 quarters, 4 dimes, and 3 pennies. Mark has 3 one-dollar bills, 4 dimes, and 2 pennies. What is the difference between the amount of money Jill has and the amount of money Mark has? A B C $1.01 $1.76 $7.85 D $8.60.
Answer: Answer choice 'B'
Step-by-step explanation:
4.00 + 0.75 + 0.40 + 0.03 = $5.18. This is Jill's amount.
3.00 + 0.40 + 0.02 = $3.42. This is Mark's amount.
To find out how much more money Jill has in comparison to how much mark has, we need to subtract Jill's money by Mark's money ($5.18 - $3.42)
if done correctly, your answer should be $1.76 (answer 'B')
f(x)=x 3 −9xf, Over which interval does f have a positive average rate of change?
The interval in which the function has positive average rate of change is x > √3
How to determine the interval which the function has positive average rate of change?The function is given as
f(x) = x^3 - 9x
Differentiate the above function to determine the average rate of change
So, we have
f'(x) = 3x^2 - 9
Set the function to greater than 0
So, we have
3x^2 - 9 > 0
Add 9 to both sides
3x^2 > 9
Divide by 3
x^2 > 3
Take the square root of both sides
x > √3
Hence, the interval in which the function has positive average rate of change is x > √3
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PLEASE HELP FAST I ONLY HAVE ONE HOUR!!!!!!