Answer:
we need to prove : for every integer n>1, the number [tex]n^{5}-n[/tex] is a multiple of 5.
1) check divisibility for n=1, [tex]f(1)=(1)^{5}-1=0[/tex] (divisible)
2) Assume that [tex]f(k)[/tex] is divisible by 5, [tex]f(k)=(k)^{5}-k[/tex]
3) Induction,
[tex]f(k+1)=(k+1)^{5}-(k+1)[/tex]
[tex]=(k^{5}+5k^{4}+10k^{3}+10k^{2}+5k+1)-k-1[/tex]
[tex]=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k[/tex]
Now, [tex]f(k+1)-f(k)[/tex]
[tex]f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-(k^{5}-k)[/tex]
[tex]f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-k^{5}+k[/tex]
[tex]f(k+1)-f(k)=5k^{4}+10k^{3}+10k^{2}+5k[/tex]
Take out the common factor,
[tex]f(k+1)-f(k)=5(k^{4}+2k^{3}+2k^{2}+k)[/tex] (divisible by 5)
add both the sides by f(k)
[tex]f(k+1)=f(k)+5(k^{4}+2k^{3}+2k^{2}+k)[/tex]
We have proved that difference between [tex]f(k+1)[/tex] and [tex]f(k)[/tex] is divisible by 5.
so, our assumption in step 2 is correct.
Since [tex]f(k)[/tex] is divisible by 5, then [tex]f(k+1)[/tex] must be divisible by 5 since we are taking the sum of 2 terms that are divisible by 5.
Therefore, for every integer n>1, the number [tex]n^{5}-n[/tex] is a multiple of 5.
Final answer:
The statement is true for all integers greater than 1.
Explanation:
The n5 - n is a multiple of 5 for every integer n > 1, we will use proof by induction.
Base Case: For n = 2,
n5 - n = 25 - 2 = 32 - 2 = 30,
which is clearly a multiple of 5. Hence, our base case holds true.
Inductive Step: Assume that for some integer k > 1, the statement holds true, i.e.,
k5 - k is a multiple of 5. We need to show that k5 - k + 5(k4 + k3 + k2 + k + 1) is also a multiple of 5.
If k5 - k is a multiple of 5, then adding a number which is a multiple of 5 (5 times a sum of powers of k) to it will also result in a multiple of 5. This means that (k + 1)5 - (k + 1) will be a multiple of 5, hence the statement holds for k + 1. By the principle of mathematical induction, the statement holds true for all integers n > 1.
Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 200 and 18, respectively. The distribution of the population is unknown. The mean and the standard error of the distribution of the sample mean are
Answer: The mean and the standard error of the distribution of the sample mean are 200 and 2.
Step-by-step explanation:
Given: Sample size : n= 81
Mean of infinite population : [tex]\mu=200[/tex]
We know that the mean of the distribution of the sample mean is same as the mean of the population.
i.e. [tex]\mu_x=\mu=200[/tex]
The standard error of the distribution of the sample mean is given by :-
[tex]S.E.=\dfrac{\sigma}{\sqrt{n}}[/tex]
[tex]\Rightarrow\ S.E.=\dfrac{18}{\sqrt{81}}=\dfrac{18}{9}=2[/tex]
Hence, the mean and the standard error of the distribution of the sample mean are 200 and 2.
The mean of the sample mean distribution for a random sample of size 81 from a population with a mean of 200 and a standard deviation of 18 is 200, and the standard error is 2.
Explanation:The question is about determining the mean and the standard error of the distribution of the sample mean. According to the Central Limit Theorem, regardless of the distribution of the original population, the distribution of the sample mean tends to form a normal distribution as the sample size increases. In this case, the mean of the sample mean distribution is the same as the population mean, which is 200.
The standard error of the mean is calculated as the population standard deviation divided by the square root of the sample size. So, the standard error in this scenario would be 18/√81 = 2.
Therefore, in a random sample of size 81 taken from an infinite population with a mean of 200 and a standard deviation of 18, the mean of the sample mean distribution is 200, and the standard error is 2.
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The volume of water flowing through a pipe varies directly wlth the square of the radius of the pipe. If the water flows at a rate of 80 liters per minute through a pipe with a radlus of 4 cm, at what rate would water flow through a pipe with a radius of 3 cm? (Rigorous) (Competency 007) 11. A) 45 liters per minute B) 6.67 liters per minute C) 60 liters per minute D) 4.5 liters per minute
Answer:
A
Step-by-step explanation:
Volume varies directly with the square of the radius, so:
V = k r²
When V = 80, r = 4.
80 = k (4)²
k = 5
V = 5r²
When r = 3:
V = 5 (3)²
V = 45
The flow is 45 L/min.
let f(x) = -2x/(x^2-x-5) There are 2 numbers that are not in the domain of f. Give the larger value to 2 decimal places.
Answer:
Step-by-step explanation:
The 2 numbers that are not in the domain of the function are the 2 numbers that cause the denominator of the function to equal 0. In order to find those 2 numbers, we have to factor the quadratic that is in the denominator. When you factor, you get x = 2.79 and x = -1.79
Those are the values of x that cause the denominator to equal 0, which of course is NEVER allowed in math!
If 2000 dollars is invested in a bank account at an interest rate of 8 per cent per year,
Find the amount in the bank after 12 years if interest is compounded annually:
Answer:
$5036.34
Step-by-step explanation:
Each year, 8% of the existing balance is added to the existing balance, effectively multiplying the amount by 1.08. If that is done for 12 years, the effective multiplier is 1.08^12 ≈ 2.51817. The the amount in the bank at the end of that time is ...
$2000×2.51817 = $5036.34
The amount in the bank after 12 years with an annual interest rate of 8% on a principal amount of 2000 dollars, compounded annually, will be approximately $5025.90.
Explanation:This is a compound interest problem. The formula used to solve this type of problem is A = P(1 + r/n)^(nt), where:
P is the principal amount (initial money),r is the annual interest rate,t is the number of years,n is the number of times that interest is compounded per year.In this case, P = $2000, r = 8% or 0.08, t = 12 years and n = 1 (as interest is compounded annually). Substituting these values in the equation, we get:
A = 2000(1 + 0.08/1)^(1*12)
.
The resulting Amount A after 12 years will be approximately $5025.90.
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please help asap!!!!!
Answer:
The volume of the prism is 27√3/2
Step-by-step explanation:
* Lets revise the triangular prism properties
- The triangular prism has five faces
- Two bases and three side faces
- The two bases are triangles
- The three side faces are rectangles
- The rule of its volume is Area of its base × its height
* Lets solve the problem
- The triangular prism has two bases which are equilateral triangles
- The length of each side of the triangular base is 3"
- The height of the prism is 6"
∵ The volume of the prism = area of the base × its height
∵ The base is equilateral triangle of side length 3"
- The area of any equilateral triangle is √3/4 s²
∴ The area of the base of the prism = √3/4 × (3)² = 9√3/4
∵ The length of the height of the prism is 6"
∴ The volume of the prism = 9√3/4 × 6 = 27√3/2
* The volume of the prism is 27√3/2
Use the table above to answer the question.
Ed Employee had $70,000 in taxable income. What was his tax?
Tax = $______
Answer:
$16,479
Step-by-step explanation:
The table tells you Ed's tax is ...
14,138.50 + 0.31×(70,000 -62,450) = 16,479 . . . . dollars
Fewer young people are driving. In 1983, 87% of 19-year-olds had a driver’s license. Twenty-five years later that percentage had dropped to 75% (University of Michigan Transportation Research Institute website, April 7, 2012). Suppose these results are based on a random sample of 1200 19-year-olds in 1983 and again in 2008.
a. At 95% confidence, what is the margin of error and the interval estimate of the number of 19-year-old drivers in 1983?
b. At 95% confidence, what is the margin of error and the interval estimate of the number of 19-year-old drivers in 2008?
c. Is the margin of error the same in parts (a) and (b)? Why or why not?
Answer: the answer to this question is B
Step-by-step explanation: Hope This Helps
Formula to find the margin of error :
[tex]E=z^*\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex] , where n= sample size , [tex]\hat{p}[/tex] is the sample proportion and z*= critical z-value.
Let p be the proportion of 19-year-olds had a driver’s license.
A) As per given , In 1983
[tex]\hat{p}=0.87[/tex]
n= 1200
Critical value for 95% confidence level is 1.96 (By z-table)
So ,Margin of error : [tex]E=(1.96)\sqrt{\dfrac{0.87(1-0.87)}{1200}}\approx0.019[/tex]
Interval : [tex](\hat{p}-E , \ \hat{p}+E)=(0.87-0.019 ,\ 0.87+0.019)[/tex]
[tex]=(0.851,\ 0.889)[/tex]
B) In 2008 ,
[tex]\hat{p}=0.75[/tex]
Margin of error : [tex]E=(1.96)\sqrt{\dfrac{0.75(1-0.75)}{1200}}\approx0.0245[/tex]
Interval : [tex](\hat{p}-E, \hat{p}+E)=(0.75-0.0245,\ 0.75+0.0245)[/tex]
[tex]=(0.7255,\ 0.7745)[/tex]
c. The margin of error is not the same in parts (a) and (b) because the sample proportion of 19-year-olds had a driver’s license are not same in both parts.
Line m is parallel to line n. The measure of angle 2 is 74°. What is the
measure of angle 5?
OA) 74°
O B) 120
OC) 106°
OD) 86°
Answer:
C. 106
Step-by-step explanation:
Angles 2 and 6 are corresponding angles so they're both 74. So you just subtract 74 from 180 to get 106.
Answer:
C. 106 is the answer
Step-by-step explanation:
angle 3 = angle 2 (vertically opp. angle)
angle 3+ angle 5 = 180
74+ angle 5 = 180
angle 5 = 106
The mean number of words per minute (WPM) read by sixth graders is 93 with a standard deviation of 22.If 30 sixth graders are randomly selected, what is the probability that the sample mean would be greater than 97.95 WPM? (Round your answer to 4 decimal places)
Answer: 0.1093
Step-by-step explanation:
Given: Mean : [tex]\mu=93[/tex]
Standard deviation : [tex]\sigma = 22[/tex]
Sample size : [tex]n=30[/tex]
The formula to calculate z-score is given by :_
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x= 97.95, we have
[tex]z=\dfrac{97.95-93}{\dfrac{22}{\sqrt{30}}}\approx1.23[/tex]
The P-value = [tex]P(z>1.23)=1-P(z<1.23)=1-0.8906514=0.1093486\approx0.1093[/tex]
Hence, the probability that the sample mean would be greater than 97.95 WPM =0.1093
PLEASE ANSWER WITH AN EXPLANATION! THANK YOU
Answer:
[tex]\large\boxed{A=153\ cm^2}[/tex]
Step-by-step explanation:
Look at the picture.
We have
square with side length a = 9
trapezoid with base lengths b₁ = 9 and b₂ = 6 and the height length h = 6
right triangle with legs lengths l₁ = 3 + 6 = 9 and l₂ = 6
The formula of an area of a square
[tex]A=a^2[/tex]
Substitute:
[tex]A_I=9^2=81\ cm^2[/tex]
The formula of an area of a trapezoid:
[tex]A=\dfrac{b_1+b_2}{2}\cdot h[/tex]
Substitute:
[tex]A_{II}=\dfrac{9+6}{2}\cdot6=\dfrac{15}{2\!\!\!\!\diagup_1}\cdot6\!\!\!\!\diagup^3=(15)(3)=45\ cm^2[/tex]
The formula of an area of a right triangle:
[tex]A=\dfrac{l_1l_2}{2}[/tex]
Substitute:
[tex]A_{III}=\dfrac{(9)(6)}{2}=\dfrac{54}{2}=27\ cm^2[/tex]
The area of the shape:
[tex]A=A_I+A_{II}+A_{III}\\\\A=81+45+27=153\ cm^2[/tex]
A car was valued at $39,000 in the year 1995. The value depreciated to $11,000 by the year 2003.
A)What was the annual rate of change between 1995 and 2003? (Round to 4 decimal places)
B)What is the correct answer to part A written in percentage form?
C)Assume that the car value continues to drop by the same percentage. What will the value be in the year 2007?
Please help ASAP the homework is due Monday!!! :(
Answer:
14.6328% , $5836.03
Step-by-step explanation:
Here we are going to use the formula
[tex]A_{0}(1-r)^n = A_{n}[/tex]
[tex]A_{0}[/tex] = 39000
r=?
[tex]A_{8}[/tex] = 11000
n=8
Hence
[tex]39000(1-r)^8 = 11000[/tex]
[tex](1-r)^8 = \frac{11000}{39000}[/tex]
[tex](1-r)^8 = 0.2820[/tex]
[tex](1-r) = 0.2820^{\frac{1}{8}[/tex]
[tex](1-r) = 0.2820^{0.125}[/tex]
[tex](1-r) = 0.8536[/tex]
[tex](1-0.8536=r[/tex]
[tex]r = 0.1463[/tex]
Hence r= 0.1463
In percentage form r = 14.63%
Now let us see calculate the value of car in 2003 that is after 12 years
we use the main formula again
[tex]A_{0}(1-r)^n = A_{n}[/tex]
[tex]A_{0}[/tex] = 39000
r=0.1463
[tex]A_{12}[/tex] = ?
n=12
[tex]39000(1-0.14634)^{(12} = A_{12}[/tex]
[tex]39000(0.8536)^{12} = A_{12}[/tex]
[tex]39000*0.1497 = A_{12}[/tex]
[tex]A_{12}=5840.34[/tex]
Hence the car's value will be depreciated to $5840.34 (approx) by 2003.
The annual rate of change between 1995 and 2003 is -0.1463
The annual rate of change between 1995 and 2003 is -14.63%
The value of the car in 2007 would be $5,844.24
The value of the car decreases as the years go by. This is referred to as depreciation. Depreciation is the decline in value of an asset as a result of wear and tear.
In order to determine the annual rate of change, use this formula:
g = [tex](FV / PV) ^{\frac{1}{n} } - 1[/tex]
Where:
g = depreciation rate
FV = value of the car in 2003 = $11,000
PV = value of the car in 1995 = $39,000
n = number of years = 2003 - 1995 = 8
[tex](11,000 / 39,00)^{\frac{1}{8} } - 1[/tex] = -0.1463 = -14.63%
The value of car in 7 years can be determined using this formula:
FV = P (1 + g)^n
$39,000 x (1 - 0.1463)^12
$39,000 x 0.8537^12 = $5,844.24
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x^2+y^2=25 Find the distance of point (x,y) from origin.
Answer:
5 units
Step-by-step explanation:
This is a circle with a center of (0, 0). The square root of 25 represents the radius of the circle which is 5. The radius represents the distance that the outside of the circle is from the center.
A large school district in southern California asked all of its eighth-graders to measure the length of their right foot at the beginning of the school year, as part of a science project. The data show that foot length is approximately Normally distributed, with a mean of 23.4 cm and a standard deviation of 1.7 cm. Suppose that 25 eighth-graders from this population are randomly selected. Approximately what is probability that the sample mean foot length is less than 23 cm?
Answer:
The probability of the sample mean foot length less than 23 cm is 0.120
Step-by-step explanation:
* Lets explain the information in the problem
- The eighth-graders asked to measure the length of their right foot at
the beginning of the school year, as part of a science project
- The foot length is approximately Normally distributed, with a mean of
23.4 cm
∴ μ = 23.4 cm
- The standard deviation of 1.7
∴ σ = 1.7 cm
- 25 eighth-graders from this population are randomly selected
∴ n = 25
- To find the probability of the sample mean foot length less than 23
∴ The sample mean x = 23, find the standard deviation σx
- The rule to find σx is σx = σ/√n
∵ σ = 1.7 and n = 25
∴ σx = 1.7/√25 = 1.7/5 = 0.34
- Now lets find the z-score using the rule z-score = (x - μ)/σx
∵ x = 23 , μ = 23.4 , σx = 0.34
∴ z-score = (23 - 23.4)/0.34 = -1.17647 ≅ -1.18
- Use the table of the normal distribution to find P(x < 23)
- We will search in the raw of -1.1 and look to the column of 0.08
∴ P(X < 23) = 0.119 ≅ 0.120
* The probability of the sample mean foot length less than 23 cm is 0.120
The speed of cars on a stretch of road is normally distributed with an average 48 miles per hour with a standard deviation of 5.9 miles per hour. What is the probability that a randomly selected car is violating the speed limit of 50 miles per hour? (a) 0.37 (b) 0.48 (c) 0.21 (d) 0.63
Answer: (a) 0.37
Step-by-step explanation:
Given: The speed of cars on a stretch of road is normally distributed with an average 48 miles per hour with a standard deviation of 5.9 miles per hour.
i.e. Mean : [tex]\mu = 48\text{ miles per hour} [/tex]
Standard deviation : [tex]\sigma = 5.9\text{ miles per hour}[/tex]
The formula to calculate z is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For the probability that a randomly selected car is violating the speed limit of 50 miles per hour (X≥ 50).
For x= 80
[tex]z=\dfrac{50-48}{5.9}=0.338983050847\approx0.34[/tex]
The P Value =[tex]P(z>0.34)=1-P(z<0.34)=1-0.6330717\approx0.3669283\approx0.37[/tex]
Hence, the probability that a randomly selected car is violating the speed limit of 50 miles per hour =0.37
........Help Please.......
Answer:
b = 1.098
Step-by-step explanation:
Each year, the GDP is 9.8% higher than the year before, so the multiplier each year is 1 + 9.8% = 1.098. This is the value of b.
b = 1.098
Write the standard equation of a circle with center (-4 0) and radius 3 brainly
a figure has a vertex at (-1,-3). if the figure has a line symmetry about x-axis , what are the coordinates of another vertex of the figure?
a. (3,1)
b. (-1,3)
c. (-3,-1)
d. (1,-3)
Answer:
b. (-1,3)
Step-by-step explanation:
the image of the point (x ; y) by symmetry about x-axis is : ( x ;- y)
so the answer "b" : (-1,3)
The conversion factor relating feet to meters is 1 ft=0.305 m. Keep in mind that when using conversion factors, you want to make sure that like units cancel leaving you with the units you need. You have been told that a certain house is 164 m2 in area. How much is this in square feet?
Answer:
1763 ft²
Step-by-step explanation:
Using the given conversion factor, ...
(164 m²)(1 ft/(.305 m))² = 165/.093025 ft² ≈ 1763 ft²
_____
The exact conversion factor is 1/0.3048, so the area is closer to 1765 ft². For a 4-significant digit answer, you need to use a conversion factor accurate to 4 significant digits.
To convert square meters to square feet, you must square the feet to meter conversion factor, resulting in approximately 10.764 sq ft/sq m. You then multiply this by the square meter measurement to get the equivalent in square feet. Therefore, the house's area, which was provided as 164 square meters, translates to approximately 1765.736 square feet.
Explanation:The measure of the area in square feet can be derived using the conversion factor for feet to meters, 1 ft = 0.305 m. However, when you deal with areas, you must square the conversion factor. We then apply the conversion factor to the known area in reference, which in our case is 164 square meters.
So our conversion factor becomes (1/0.305)² sq ft/sq m = 10.764 sq ft/sq m.
To use the conversion factor, we multiply it by the metric unit measurement, like this: 164 m²*(10.764 ft²/m²) = 1765.736 square feet.
So, the house's area is approximately 1765.736 square feet.
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I need help with math work!!
Answer:
(g-h)=x^2+5-8-x
=x^2-x-3
(g-h)(-9) means x= -9
x^2-x-3
=(-9)^2-(-9)-3
=81+9-3
=90-3
=87
Find the point, P, at which the line intersects the plane. x equals 7 plus 9 t, y equals 3 minus 7 t, z equals 7 minus 5 t; 5 x minus 6 y minus 9 z equals negative 1 \
Answer:
The point of intersection [tex]P\left(\dfrac{1133}{122},\dfrac{149}{122},\dfrac{699}{122}\right)[/tex]
Step-by-step explanation:
Equation of line:
[tex]x=7+9t[/tex]
[tex]y=3-7t[/tex]
[tex]z=7-5t[/tex]
Equation of plane:
[tex]5x-6y-7z=-1[/tex]
We need to find the point of intersection of line and plane.
Point of intersection: When both line and plane meet at single point.
So, put the value of x, y and z into plane.
[tex]5(7+9t)-6(3-7t)-7(7-5t)=-1[/tex]
[tex]35+45t-18+42t-49+35t=-1[/tex]
[tex]122t=-1+32[/tex]
[tex]t=\dfrac{31}{122}[/tex]
Substitute the value of t into x, y and z
[tex]x=7+9\cdot \dfrac{31}{122}=\dfrac{1133}{122}[/tex]
[tex]y=3-7\cdot \dfrac{31}{122}=\dfrac{149}{122}[/tex]
[tex]z=7-5\cdot \dfrac{31}{122}=\dfrac{699}{122}[/tex]
Point of intersection:
[tex]\left(\dfrac{1133}{122},\dfrac{149}{122},\dfrac{699}{122}\right)[/tex]
Hence, The point of intersection [tex]P\left(\dfrac{1133}{122},\dfrac{149}{122},\dfrac{699}{122}\right)[/tex]
Solve the following using the crossing-graphs method. (Round your answer to two decimal places.) 5 + 69 × 0.96t = 32
Answer:
The solution is 22.98.
Step-by-step explanation:
Here, the given equation,
[tex]5 + 69\times 0.96^t = 32[/tex],
Let [tex]f(t) = 5 + 69\times 0.96^t[/tex]
And, [tex]f(t) = 32[/tex]
Where, t represents x-axis and f(t) represents y-axis,
Since, [tex]f(t) = 5 + 69\times 0.96^t[/tex] is an exponential decay function having y-intercept (0,74).
Also, f(t) = 32 is the line, parallel to x-axis,
Thus, after plotting the graph of the above functions,
We found that they are intersecting at (22.984, 32)
Hence, the solution of the given equation = x-coordinate of the intersecting point = 22.984 ≈ 22.98
To solve the given equation, 5 + 69 × 0.96t = 32, you start by subtracting 5 from both sides, then divide by 69. Then, divide both sides by 0.96 to solve for t. The solution is t ≈ 0.41 (rounded to two decimal places).
Explanation:To solve the equation 5 + 69 × 0.96t = 32 using the crossing-graphs method, we first simplify the equation:
Start by subtracting 5 from both sides of the equation: 69 × 0.96t = 32 - 5.This results in: 69 × 0.96t = 27.Next, divide both sides by 69: 0.96t = 27/69.Which simplifies to 0.96t ≈ 0.391 (rounded to three decimal places).Finally, divide both sides by 0.96 to solve for t: t ≈ 0.391/0.96 ≈ 0.41 (rounded to two decimal places).Learn more about Crossing-graphs method here:https://brainly.com/question/34386833
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Solve for x in the equation
Answer:
[tex]\large\boxed{x=6\pm3\sqrt{10}}[/tex]
Step-by-step explanation:
[tex]x^2-12x+36=90\\\\x^2-2(x)(6)+6^2=90\qquad\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\(x-6)^2=90\iff x-6=\pm\sqrt{90}\\\\x-6=\pm\sqrt{9\cdot10}\\\\x-6=\pm\sqrt9\cdot\sqrt{10}\\\\x-6=\pm3\sqrt{10}\qquad\text{add 6 to both sides}\\\\x=6\pm3\sqrt{10}[/tex]
An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 120 lb and 161 lb. The new population of pilots has normally distributed weights with a mean of 125 lb and a standard deviation of 28.1 lb.
a)If a pilot is randomly selected, find the probability that his weight is between 120 lb and 161 lb.The probability is approximately?
b. If 36 different pilots are randomly selected, find the probability that their mean weight is between 120 lb and 161 lb. The probability is approximately?
c. When redesigning the ejection seat, which probability is more relevant? . Part (b) because the seat performance for a single pilot is more important. B. Part (b) because the seat performance for a sample of pilots is more important. C. Part (a) because the seat performance for a sample of pilots is more important. D. Part (a) because the seat performance for a single pilot is more important.
Solve the following system of equations
3x - 2y =5
-2x - 3y = 14
Answer:
x = -1 and y = -4
Step-by-step explanation:
It is given that,
3x - 2y = 5 ----(1)
-2x - 3y = 14 ------(2)
To find the solution of equations
(1) * 2 ⇒
6x - 4y = 10 -----(3)
(2) * 3 ⇒
-6x - 9y = 42 ----(4)
eq(3) + eq(4) ⇒
6x - 4y = 10 -----(3)
-6x - 9y = 42 ----(4)
0 - 13y = 52
y = 52/(-13) = -4
Substitute the value of y in eq(1)
3x - 2y = 5 ----(1)
3x - (2 * -4) = 5
3x +8 = 5
3x = 5 - 8 = -3
x = -3/3 = -1
Therefore x = -1 and y = -4
Answer:
The solution is:
[tex](-1, -4)[/tex]
Step-by-step explanation:
We have the following equations
[tex]3x - 2y =5[/tex]
[tex]-2x - 3y = 14[/tex]
To solve the system multiply by [tex]\frac{3}{2}[/tex] the second equation and add it to the first equation
[tex]-2*\frac{3}{2}x - 3\frac{3}{2}y = 14\frac{3}{2}[/tex]
[tex]-3x - \frac{9}{2}y = 21[/tex]
[tex]3x - 2y =5[/tex]
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[tex]-\frac{13}{2}y=26[/tex]
[tex]y=-26*\frac{2}{13}[/tex]
[tex]y=-4[/tex]
Now substitute the value of y in any of the two equations and solve for x
[tex]-2x - 3(-4) = 14[/tex]
[tex]-2x +12 = 14[/tex]
[tex]-2x= 14-12[/tex]
[tex]-2x=2[/tex]
[tex]x=-1[/tex]
The solution is:
[tex](-1, -4)[/tex]
A club has 50 members, 10 belonging to the ruling clique and 40 second-class members. Six members are randomly selected for free movie tickets. What is the probability that 3 or more belong to the ruling clique?
Answer: The probability that 3 or more belong to the ruling clique is 0.34.
Step-by-step explanation:
Since we have given that
Number of total members = 50
Number of belonging to ruling clique = 10
Number of belonging to second class member = 40
We need to find the probability that 3 or more belong to the ruling clique.
Let X be the number of outcomes belong to ruling clique.
So, it becomes,
P(X≥3)=1-P(X<3)
[tex]P(X\geq 3)=1-P(X=1)-P(X=2)\\\\P(X\geq 3)=1-\dfrac{^{10}C_1\times ^{40}C_5}{^{50}C_6}-\dfrac{^{10}C_2\times ^{40}C_4}{^{50}C_6}\\\\P(X\geq 3)=1-0.41-0.25\\\\P(X\geq 3)=0.34[/tex]
Hence, the probability that 3 or more belong to the ruling clique is 0.34.
The probability of selecting 3 or more members from the ruling clique when choosing 6 members randomly from a club of 50 members (10 in ruling clique, 40 second-class) is 8.56%.
Explanation:This probability problem can be solved using the concepts of Combinations and Binomial Theorem. You need to determine the number of ways to choose 3, 4, 5, or 6 members from the ruling clique (10 members) and the remaining from the second-class members (40 members). For each case, divide by the total number of ways to choose 6 members from all 50 members to get the probability. Sum up all the probabilities for each case to get the total probability of having 3 or more from the ruling clique.
Calculations:1. Number of ways of choosing 3 from the ruling clique and 3 from the second class: C(10,3)*C(40,3) = 120*9880 = 1,185,600 ways
2. Number of ways of choosing 4 from the ruling clique and 2 from the second class: C(10,4)*C(40,2) = 210*780 = 163,800 ways
3. Number of ways of choosing 5 from the ruling clique and 1 from the second class: C(10,5)*C(40,1) = 252*40 = 10,080 ways
4. Number of ways of choosing 6 from the ruling clique and 0 from the second class: C(10,6)*C(40,0) = 210*1 = 210 ways
Total ways to choose 3 or more from the ruling clique: 1,185,600 + 163,800 + 10,080 +210 = 1,359,690 ways
From 50 members, the total ways to choose 6: C(50,6) = 15,890,700 ways
The Probability of 3 or more from the ruling clique = 1,359,690 / 15,890,700 = 0.0856 or 8.56%
Learn more about Probability here:https://brainly.com/question/32117953
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a theater has two screens and shows its movies continuously. a 30 minute documentary is shown on one. a 120 minute film is shown on the other. If both showings start at noon, how many minutes will pass before both movies begin again at the same time?
8. Write three other proportions for each given proportion. 35 miles/2 hours=87.5 miles/5 hours
Answer:
218.75 miles / 12.5 hours
437.5 miles / 25 hours
656.25 miles / 37.5 hours
Step-by-step explanation:
35 miles / 2 hours = 87.5 miles / 5 hours
This is the ratio of 2.5. So, the other proportions are
87.5 x 2.5 miles / 5 x 2.5 hours = 218.75 miles / 12.5 hours
87.5 x 5 miles / 5 x 5 hours = 437.5 miles / 25 hours
87.5 x 7.5 miles / 5 x 7.5 hours = 656.25 miles / 37.5 hours
A coin is tossed 30 times it lands 12 times on heads and 18 times on tails what is experimental probability of the coin landing on tails?
Answer:
3/5
Step-by-step explanation:
Total tossed : 30
# of times landed on tails : 18
Experimental probability of tails = 18/30 = 3/5
if cos θ = -0.6, and 180° < θ < 270°, find the exact value of sin 2θ.
Answer:
sin(2θ) = 0.96
Step-by-step explanation:
In the third quadrant, both sin(θ) and cos(θ) are negative. Then the double-angle trig identity tells us ...
sin(2θ) = 2·sin(θ)·cos(θ) = -2cos(θ)√(1 -cos(θ)²) . . . . using the negative root
Filling in the given value, we have
sin(2θ) = -2·(-0.6)(√(1-(-0.6)²) = 2·0.6·0.8 = 0.96
State the linear programming problem in mathematical terms, identifying the objective function and the constraints. A firm makes products A and B. Product A takes 2 hours each on machine M; product B takes 4 hours on L and 3 hours on M. Machine L can be used for 8 hours and M for 6 hours. Profit on product A is $9 and $7 on B. Maximize profit.
Final answer:
The problem requires formulating a linear programming model to maximize the profit function Z = 9x + 7y with constraints on machine time for product A and B (2x + 3y ≤ 6 for machine M and 4y ≤ 8 for machine L) and the non-negativity restrictions (x, y ≥ 0).
Explanation:
The linear programming problem can be stated in mathematical terms with an objective function and constraints for a firm making products A and B. The objective is to maximize profit, which is the sum of 9 dollars per unit of product A and 7 dollars per unit of product B. Let the number of products A and B produced be represented by variables x and y, respectively.
The objective function to maximize is Z = 9x + 7y.
Constraints:
Machine M's availability limits product A to 2 hours each, and product B to 3 hours each, with a total available time of 6 hours: 2x + 3y ≤ 6.Machine L can be used for 8 hours and is only required for product B, which takes 4 hours: 4y ≤ 8.Non-negativity constraints: x ≥ 0 and y ≥ 0, since the number of products cannot be negative.