Answer:
To solve the integral, we can use partial fractions and then integrate each term separately. The integrand can be written as:
[tex]\dfrac{x^4(1-x)^4}{1+x^2} = \dfrac{x^4(1-x)^4}{(x+i)(x-i)}[/tex]
Using partial fractions, we can write:
[tex]\dfrac{x^4(1-x)^4}{(x+i)(x-i)} = \dfrac{Ax+B}{x+i} + \dfrac{Cx+D}{x-i}[/tex]
Multiplying both sides by (x+i)(x-i), we get:
[tex]x^4(1-x)^4 = (Ax+B)(x-i) + (Cx+D)(x+i)[/tex]
Substituting x=i, we get:
[tex]i^4(1-i)^4 = (Ai+B)(i-i) + (Ci+D)(i+i)[/tex]
Simplifying, we get:
[tex]16 = 2Ci + 2B[/tex]
Substituting x=-i, we get:
tex^4(1+i)^4 = (Ci+D)(-i-i) + (Ai+B)(-i+i)[/tex]
Simplifying, we get:
[tex]16 = 2Ai + 2D[/tex]
Substituting x=0, we get:
[tex]0 = Bi + Di[/tex]
Substituting x=1, we get:
[tex]0 = A+B+C+D[/tex]
Solving these equations simultaneously, we get:
A = -22/7 + π
B = 0
C = 22/7 - π
D = -2/5
Therefore, the integral can be written as:
[tex]\int_0^1 \dfrac{x^4(1-x)^4}{1+x^2}dx = \int_0^1 \left[\dfrac{-22/7+\pi}{x+i} + \dfrac{22/7-\pi}{x-i} - \dfrac{2/5}{1+x^2}\right]dx[/tex]
Integrating each term separately, we get:
[tex]\int_0^1 \dfrac{-22/7+\pi}{x+i}dx = [-22/7+\pi]\ln(x+i) \bigg|_0^1 = [\pi-22/7]\ln\left(\dfrac{1+i}{i}\right)[/tex]
[tex]\int_0^1 \dfrac{22/7-\pi}{x-i}dx = [22/7-\pi]\ln(x-i) \bigg|_0^1 = [22/7-\pi]\ln\left(\dfrac{1-i}{-i}\right)[/tex]
[tex]\int_0^1 \dfrac{-2/5}{1+x^2}dx = -\frac{2}{5}\tan^{-1}(x)\bigg|_0^1 = -\frac{2}{5}\tan^{-1}(1) + \frac{2}{5}\tan^{-1}(0) = -\frac{2}{5}\tan^{-1}(1)[/tex]
Therefore, the correct options are:
A) [tex]\pi-\frac{22}{7}[/tex]
B) [tex]\frac{2}{105}[/tex]
C) 0
D) [tex]\frac{71}{15}-\frac{3\pi}{2}[/tex]
2.3. Ntando can either walk to school at 5 km/h or ride his bicycle at 15 km/h. If he rides his bicycle, it takes him 10 minutes to get to school. 2.3.1. How long will it take him if he walks to school?
Answer:
30 minutes
Step-by-step explanation:
Use ratios. This is an inverse function, as speeding up makes the time traveling go down. So, when dividing the speed by 3 (done so 15 can get to 5), we multiply the time traveled by 3.
10 minutes * 3 = 30 minutes
x : y = 4:3 andy: z = 5:2
Write the ratio of x:y: z
The combined ratio of x : y : z from x ; y = 4 : 3 and y : z = 5 ; 2 is x:y:z = 20:15:6
How to determine the combined ratioGiven the following ratios
x ; y = 4 : 3
y : z = 5 ; 2
To find the ratio of x:y:z, we need to combine the two given ratios and simplify:
x:y = 4:3
y:z = 5:2
To combine these ratios, we need to have a common term for y.
We can do this by multiplying the first ratio by 5 and the second ratio by 3:
5(x:y) = 5(4:3) = 20:15
3(y:z) = 3(5:2) = 15:6
Now we have a common term of 15 for y, so we can combine the ratios:
x:y:z = 20:15:6
Therefore, the ratio of x:y:z is 20:15:6
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Victor spent $61 on some sandpaper for his model
cars. He bought 2 packages of the smallest-grain
sandpaper and spent the rest on the largest-grain
sandpaper. How many packages of the largest-
grain sandpaper did he buy?
Size of
Grain (cm)
0.003
0.011
0.001
Cost per
Package (S)
13
7
20
Answer:
3 packages
Step-by-step explanation:
61 - 20 * 2 (small grain sandpaper) = $21 remain. The largest grain costs $7, so 21/7 = 3 packages of the largest grain sandpaper was bought.
Hope this helped!
Find The surface area of the composite figure
Answer: It should be 470 cm^2
Step-by-step explanation:
a quality-conscious disk manufacturer wishes to know the fraction of disks his company makes which are defective. step 2 of 2 : suppose a sample of 1015 1015 floppy disks is drawn. of these disks, 904 904 were not defective. using the data, construct the 95% 95 % confidence interval for the population proportion of disks which are defective. round your answers to three decimal places.
Therefore, the 95% confidence interval for the population proportion of disks that are defective is (0.8357, 0.9475). This means that there is a 95% probability that the true population proportion of disks that are defective is between 0.8357 and 0.9475.
The quality-conscious disk manufacturer can use a 95% confidence interval to estimate the fraction of defective disks made by the company. The 95% confidence interval is calculated using the sample data to construct an interval estimate of the population proportion.
Using the given data, the sample proportion of disks that are not defective is 904/1015 = 0.8915.
The 95% confidence interval for the population proportion of disks that are defective is then given by:
Lower limit = 0.8915 – (1.96 x √(0.8915 x (1- 0.8915)/1015))
= 0.8915 – (1.96 x 0.0285)
= 0.8915 – 0.0558
= 0.8357
Upper limit = 0.8915 + (1.96 x √(0.8915 x (1- 0.8915)/1015))
= 0.8915 + (1.96 x 0.0285)
= 0.8915 + 0.0558
= 0.9475
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1/1 point (graded) Compute X(), the matrix of predicted rankings UVT given the initial values for U() and V (0). 2 1 (Enter your answer as a matrix, e.g., type [[2,1],[1,0],[3,-1]] for a 3 x 2 matrix 1 0 Note the square brackets, and 3 -1 commas as separators. ) [[24,12,6], [0,0,0], (12,6,3], [24 ✓ 24 12 6 0 0 0 12 6 3 24 12 6
The matrix of predicted rankings UVT is [[48,24,12],[0,0,0],[24,12,6]].
The matrix of predicted rankings UVT can be calculated using the formula UVT = UV.The provided initial values for U() and V(0) are as follows:U() = [[2,1],[1,0],[3,-1]]V(0) = [[24,12,6],[0,0,0],[12,6,3]]Using the above values, the matrix of predicted rankings UVT can be computed as follows:UVT = UVU = [[2,1],[1,0],[3,-1]]V = [[24,12,6],[0,0,0],[12,6,3]]UVT = [[2,1],[1,0],[3,-1]] x [[24,12,6],[0,0,0],[12,6,3]]= [[48,24,12],[0,0,0],[24,12,6]]Therefore, the matrix of predicted rankings UVT is [[48,24,12],[0,0,0],[24,12,6]].
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a teacher monitored the number of people texting during class each day and calculated the corresponding probability distribution. what type of probability distribution did the teacher use?
The given probability distribution "a teacher monitored the number of people texting during class each day and calculated the corresponding probability distribution." is a type of discrete probability distribution.
What is the Probability distribution?The probability distribution is used to describe the probability of each outcome in a series of possible outcomes. It is a mathematical representation of the outcomes of an experiment.
The teacher likely used a discrete probability distribution to calculate the probability of a certain number of people texting during class each day.
A discrete probability distribution is used to analyze data where the outcome is counted in whole numbers, such as the number of people texting in a given class period.
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Suppose Elena has $5 and sells pens for $1.50 each. Her goal is to save $20 to buy a t-shirt. She does not spend any money while she is saving money.
Let p represent the number of pens Elena sells. Write an expression for how much money she saves in total after selling p pens.
Write an equation using your answer from question 1, showing that she saves exactly $20.
Solve the equation you wrote from question 2. What do you notice about your value for p?
What if Elena wants to have some money left over? Write an inequality using your expression from question 1 to show this.
Write down other values for p where Elena would have money left over.
Write an inequality using p, to describe the values of p where she would be able to buy the tshirt and still have money left over.
In conclusion the values of p that satisfy this inequality are p = 11, p = 12, p = 13, etc. The equation showing that Elena saves exactly $20 is:
5 + (1.5)p = 20
Why it is?
The expression for how much money Elena saves after selling p pens is:
Total money saved = 5 + (1.5)p
The equation showing that Elena saves exactly $20 is:
5 + (1.5)p = 20
Solving the equation:
5 + (1.5)p = 20
(1.5)p = 15
p = 10
We notice that the value of p is a whole number, which makes sense since Elena cannot sell a fractional number of pens.
If Elena wants to have some money left over, we can write the inequality:
5 + (1.5)p > 20
Some values for p where Elena would have money left over are p = 11, p = 12, p = 13, etc.
To describe the values of p where Elena would be able to buy the t-shirt and still have money left over, we can write the inequality:
5 + (1.5)p > 20
(1.5)p > 15
p > 10
Therefore, the values of p that satisfy this inequality are p = 11, p = 12, p = 13, etc.
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Solve the inequality 12≥ 73x + 2
10/73 is the value of x in inequality.
What does the word "inequality" mean?
In mathematics, inequalities describe the connection between two values that are not equal. Equal does not imply inequality. The "not equal symbol ()" is typically used to indicate that two values are not equal.
However different inequalities are used to compare the values to determine if they are less than or higher than. The term "inequality" refers to a relationship between two expressions or values that is not equal to one another. Inequality originates from an imbalance, thus.
the inequality 12≥ 73x + 2
= 12 - 2 ≥ 73x
= 10 ≥ 73x
= 10/73 ≥ x
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The area of a rectangle is x2 – 6x +8. Find its possible length and breadth:
Answer:
Step-by-step explanation:
To find the possible length and breadth of the rectangle, we need to factor the given expression:
x^2 - 6x + 8 = (x - 4)(x - 2)
Therefore, the length and breadth of the rectangle can be any combination of (x-4) and (x-2).
For example, if we choose (x-4) as the length and (x-2) as the breadth, we have:
Length = x - 4
Breadth = x - 2
Conversely, if we choose (x-2) as the length and (x-4) as the breadth, we have:
Length = x - 2
Breadth = x - 4
So, the possible length and breadth of the rectangle are (x-4) and (x-2), and vice versa.
Answer:(x-4) and (x-2)
Step-by-step explanation:
What is the scale factor of the following pair of similar polygons ?
The scale factor of the following pair of similar polygons after the dilation is 0.7
Calculating the scale factor of the similar polygonsGiven
The pair of similar polygons
From the pair of similar polygons, we have the following corresponding side lengths
Pre-image of the polygon = 30
Image of the polygon = 21
The scale factor of the similar polygons is then calculated as
Scale factor = 21/30
Evaluate the quotient
Scale factor = 0.7
Hence, the scale factor is 0.7
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a cylindrical glass is half full of lemonade. the ratio of lemon juice to water in the lemonade is $1:11$. if the glass is $6$ inches tall and has a diameter of $2$ inches, what is the volume of lemon juice in the glass? express your answer as a decimal to the nearest hundredth.
The volume of lemon juice in the glass is 0.38 cubic inches.
Explanation:
Given,
Let the volume of the lemonade in the glass be V cubic inches
Therefore, the volume of lemon juice in the lemonade is [tex]$\frac{1}{12}$[/tex] V cubic inches
Volume of water in the lemonade is [tex]$\frac{11}{12}$[/tex] V cubic inches
The volume of the cylindrical glass is given by:
[tex]$V_{\text{cylindrical glass}} = \pi r^2h$[/tex]
Here,
Radius r = 1 inch
Height h = 6 inches
[tex]$V_{\text{cylindrical glass}} = \pi r^2h = \pi (1)^2(6) = 6 \pi$[/tex]
Since the glass is half full of lemonade, the volume of lemonade in the glass is:
[tex]$V_{\text{lemonade}} = \frac{1}{2}V_{\text{cylindrical glass}} = \frac{1}{2} 6 \pi = 3\pi$[/tex]
The volume of lemon juice in the lemonade is given by:
[tex]$V_{\text{lemon juice}} = \frac{1}{12}V$[/tex]
Therefore
[tex]$V_{\text{lemon juice}} = \frac{1}{12}3\pi = \frac{1}{4}\pi = 0.7854$[/tex] cubic inches
Hence, the volume of lemon juice in the glass is 0.38 cubic inches (rounded to the nearest hundredth).
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Rob says -3(x-2) is equivalent to -3x-6. What was his error?
A) heshould have written -3x-2 which would have given him the right answer.
B) hedid not multiply 3 with x to get 3x, and instead put -3x.
C) he had no error.
D) he did not multiply -3 with -2 to get +6 instead of -6
Answer:
D) he did not multiply -3 with -2 to get +6 instead of -6
Step-by-step explanation:
-3(x - 2)
-3x + 6
Compare the two
-3x + 6 ≠ -3x - 6
They are not equal! Looking at the options, D is the answer.
Answer:
The correct answer is D) he did not multiply -3 with -2 to get +6 instead of -6.
To expand -3(x-2), we need to distribute -3 to both terms inside the parentheses, which gives us:
-3(x-2) = -3x + (-3(-2)) = -3x + 6
Therefore, the correct equivalent expression is -3x + 6, not -3x - 6 as Rob wrote. He made a mistake in multiplying -3 and -2, which should have given him +6 instead of -6.
(please mark my answer as brainliest)
James has 27 metres of red wire and 12 metres
of black wire. He needs to cut both wires into
smaller pieces so that all of the smaller pieces are
the same length and there is no wire left over. The
length of each piece must be a whole number of
metres.
What is the longest he can make each smaller
piece of wire? Give your answer in metres (m).
Answer:
3m
Step-by-step explanation:
red wire = 27
black wire= 12
so, we take HCF (highest common factor)
which would be 3 so all the wires would be cut into 3m long.
I hope it helps.
Answer:2m
Step-by-step explanation:
as the factors of 12 are:1, 2, 3, 4, 6 and 12
and the factors of 26 are:1, 2, 13 and 26
so if you are talking meters 2 would be the longest
i hope you get this right x
if there is a sample, the standard error is used in the denominator of the z stat or t stat formula. group of answer choices true false
It is correct to state that if there is a sample, the standard error is used in the denominator of the z stat or t stat formula.
What is the formula for the test statistic?The t-distribution is used when we have the standard deviation for the sample instead of the population, hence the equation is given as follows:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.[tex]\mu[/tex] is the value tested at the null hypothesis.s is the standard deviation of the sample.n is the sample size.The standard error is modeled as follows:
[tex]s_e = \frac{s}{\sqrt{n}}[/tex]
Which is the denominator of the formula for the test statistic, hence the statement is true.
When we have the standard deviation for the population, we use the z-distribution, however the formula for the standard error is similar.
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what is 2 1/2 + x = 3 1/2. Please answer it quick
Answer:
x=1
Step-by-step explanation:
2.5+x=3.5
3.5-2.5=x
1=x
x=1
The null hypothesis is rejected whenever:A. past studies prove it wrong. B. there is a low probability that the obtained results could be due to random error.C. the independent variable fails to have an effect on the dependent variable.D. the researcher is convinced that the variable is ineffective in causing changes in behavior.
The null hypothesis is rejected whenever "there is a low probability that the obtained results could be due to random error." The correct answer is Option B.
What is the null hypothesis?The null hypothesis is a statistical hypothesis used to test the difference between two sample data groups. The null hypothesis is the hypothesis that the sample statistics are not significantly different. Any significant differences between the sample data are seen as supporting the alternative hypothesis.
A null hypothesis is often expressed as "no difference," "no correlation," or "no significant effect." For example, the null hypothesis for an experiment comparing two groups of people could be "there is no difference between the two groups." When the null hypothesis is rejected, it means that the results of the experiment are statistically significant, and the alternative hypothesis is supported.
Therefore, the null hypothesis is rejected whenever there is a low probability that the obtained results could be due to random error.
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please help with question 6
Answer:
a = -13b = 6f(x) = (2x -1)(x -2)(x +3)Step-by-step explanation:
Given f(x) = 2x³ +x² +ax +b has a factor (x -2) and a remainder of 18 when divided by (x -1), you want to know a, b, and the factored form of f(x).
RemainderIf (x -2) is a factor, then the value of f(2) is zero:
f(2) = 2·2³ +2² +2a +b = 0
2a +b = -20 . . . . . . . subtract 20
If the remainder from division by (x +1) is 18, then f(-1) is 18:
f(-1) = 2·(-1)³ +(-1)² +a·(-1) +b = 18
-a +b = 19 . . . . . . . . . . add 1
Solve for a, bSubtracting the second equation from the first gives ...
(2a +b) -(-a +b) = (-20) -(19)
3a = -39
a = -13
b = 19 +a = 6
The values of 'a' and 'b' are -13 and 6, respectively.
Factored formWe can find the quadratic factor using synthetic division, given one root is x=2. The tableau for that is ...
[tex]\begin{array}{c|cccc}2&2&1&-13&6\\&&4&10&-6\\\cline{1-5}&2&5&-3&0\end{array}[/tex]
The remainder is 0, as expected, and the quadratic factor of f(x) is 2x² +5x -3. Now, we know f(x) = (x -2)(2x² +5x -3).
To factor the quadratic, we need to find factors of (2)(-3) = -6 that have a sum of 5. Those would be 6 and -1. This lets us factor the quadratic as ...
2x² +5x -3 = (2x +6)(2x -1)/2 = (x +3)(2x -1)
The factored form of f(x) is ...
f(x) = (2x -1)(x -2)(x +3)
NEED HELP ASAP
This composite figure is created by placing a sector of a circle on a rectangle. What is the area of this composite figure? Show you work
Thus, the total area of composite figure (circle on a rectangle) is found as 29.91 m².
Explain about the sector of the circle?A sector is a portion of the circle that is formed by two distinct radii. somewhat of like a slice of pie or pizza. A line segment known as a chord connects two points on a circle. A unique kind of chord called the diameter passes through the circle's focal point.Area of sector of the circle = (Ф/360°) *π*r²
r is the radius = 5.5 m
π = 3.14
Area = (30/360°) *3.14 * 5.5²
Area = 7.91 m²
Area of rectangle = width × length
Area = 4 × 5.5 = 22 m²
Total area of composite figure = area of the circle's sector + area of the rectangle
Total area of composite figure = 7.91 m² + 22 m² = 29.91 m²
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Find the value of x.
22
39
X
The value of x in the right triangle when calculated is approximately 13.8 units
Calculating the value of x in the triangleGiven the right-angled triangle
The side length x can be calculated using the following sine ratio
So, we have
sin(39) = x/22
To find x, we can use the fact that sin(39 degrees) = x/22 and solve for x.
First, we can use a calculator to find the value of sin(39 degrees), which is approximately 0.6293.
Then, we can set up the equation:
0.6293 = x/22
To solve for x, we can multiply both sides by 22:
0.6293 * 22 = x
13.8446 = x
Rewrite as
x = 13.8446
Approximate the value of x
x = 13.8
Therefore, x is approximately 13.8 in the triangle
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Solve 2log 12 (-8x)=6
The solution to the logarithmic equation [tex]2log12(-8x) = 6[/tex] is [tex]x = -9/32[/tex] .
What are logarithmic properties ?
Logarithmic properties are the rules that govern the behavior of logarithmic functions. These properties are important in simplifying logarithmic expressions and solving logarithmic equations. Some of the commonly used logarithmic properties include:
Product property: [tex]logb(xy) = logb(x) + logb(y)[/tex]
This property allows us to simplify the logarithm of a product of two numbers into the sum of logarithms of the individual numbers.
Quotient property: [tex]logb(x/y) = logb(x) - logb(y)[/tex]
This property allows us to simplify the logarithm of a quotient of two numbers into the difference of logarithms of the individual numbers.
Power property:[tex]logb(x^y) = ylogb(x)[/tex]
This property allows us to simplify the logarithm of a power of a number by bringing the exponent outside of the logarithm and multiplying it with the logarithm of the base.
Change of base formula: [tex]logb(x) = logc(x) / logc(b)[/tex]
This property allows us to change the base of a logarithm by dividing the logarithm of the number by the logarithm of the base in a different base.
Solving the given logarithmic equation :
The equation can be solved by using logarithmic properties and basic algebraic manipulation.
We can begin by using the property that states [tex]loga(b^n) = nloga(b)[/tex] for any base a and any positive real number b. Applying this property, we can rewrite the left side of the equation as:
[tex]log12((-8x)^2) = log12(64x^2)[/tex]
Next, we can use the property that states [tex]loga(b) = c[/tex] is equivalent to [tex]a^c = b[/tex]. Applying this property, we can rewrite the equation as:
[tex]12^{2log12(64x^2)} = 12^6[/tex]
Simplifying the left side, we get:
[tex]64x^2 = 12^6 / 12^2[/tex]
[tex]64x^2 = 144[/tex]
Dividing both sides by 64, we get:
[tex]x^2 = 144/64[/tex]
[tex]x^2 = 9/4[/tex]
Taking the square root of both sides, we get:
[tex]x=\pm 3/2[/tex]
However, we need to check the solutions for extraneous roots since the original equation has a logarithm with a negative argument. We can see that the solution x = 3/2 is extraneous since it results in a negative argument for the logarithm. Therefore, the only valid solution is x = -9/32.
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a) What is the mean proportional between 3 and 12?
b) What is the geometric mean of 3 and 12?
Answer:
a. 6
b. 6
Step-by-step explanation:
A. mean proportion
3:x::x:12
product of mean = product of extreme
3 x 12 = x^2
x^2= 36
now take square root on both sides
x = 6
as sqaure root will cancel the square on x and square root of 36 is 6.
B. the geometric mean is the product of all
the numbers in a set, with the root of how many numbers there are. here, there are two numbers, so a square root is used. the value of the geometric mean of 3 and 12 is 6.
Hope's it helps...
camilia saved 4/5% of her allowance. What is this percent expressed as a fraction and as a decimal
b) Father is 30 years older than his son. After five years, he will be three times as old as his son will be. Find their present age.
Step-by-step explanation:
40 Years is the right answer
Answer:
son is 10 , father is 40
Step-by-step explanation:
let x be the sons age then father is x + 30
in 5 years
son is x + 5 and father is x + 30 + 5 = x + 35
at this time the father is three times as old as his son , then
x + 35 = 3(x + 5)
x + 35 = 3x + 15 ( subtract x from both sides )
35 = x + 15 ( subtract 15 from both sides )
20 = x
then sons age = x = 10 and fathers age = x + 30 = 10 + 30 = 40
What is the y-intercept of the line
with the equation y = - 4x - 12
Answer:
-12 is the y intercept while your slope is -4
Step-by-step explanation:
What is 252 divided by 9
Answer:
28 is the ans of 252÷9here we go
252 divided by 9 equals 28.
To divide 252 by 9, you can use long division, which involves dividing the number in steps until there is no remainder left.
Here's the step-by-step process:
Write down the dividend (252) and the divisor (9), and set up the long division format:
9 | 252
Look at the leftmost digit of the dividend (2) and see if it's divisible by the divisor (9). Since 2 is less than 9, we bring down the next digit (5) to the right of 2, making it 25.
9 | 252
2
Divide the new number (25) by the divisor (9). The result is 2, which is the first digit of the quotient. Multiply this result by the divisor (2 x 9 = 18) and write it below the 25, then subtract it from 25:
9 | 252
25
18
--
7
Bring down the next digit (2) from the dividend to the right of the remainder (7), making it 72. Now, divide 72 by 9, which gives you 8. Multiply this result by the divisor (8 x 9 = 72) and write it below the 72, then subtract it from 72:
9 | 252
25
18
--
72
72
---
0
There is no remainder left, and the dividend has been completely divided. The quotient is the result of the division, which is 28.
Therefore, 252 divided by 9 equals 28.
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The speed s (in miles per hour) of a car can be given by s = √(30fg), where fis the
coefficient of friction and d is the stopping distance (in feet). The table shows the
coefficient of friction for different surfaces. You are driving 35 miles per hour on an
icy road when a deer jumps in front of your car. How far away must you begin to
brake to avoid hitting the deer? Round your answer to the nearest whole integer. NO
UNITS NEEDED
Answer:
Step-by-step explanation:
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pihpin
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oiy69877t7gbkjb
Answer:
s= 6
Step-by-step explanation:
The speed s (in miles per hour) of a car can be given by s = √(30fg), where fis the
coefficient of friction and d is the stopping distance (in feet). The table shows the
coefficient of friction for different surfaces. You are driving 35 miles per hour on an
icy road when a deer jumps in front of your car. How far away must you begin to
brake to avoid hitting the deer? Round your answer to the nearest whole integer. NO
UNITS NEEDED
Un número tiene 8 divisores. Además, cada uno de la mitad y la tercera parte de él tienen cuatro divisores. Si la suma de todos los divisores del número es 216, obtén tal número
The number we are looking for is N = 2 × 2^3 × 3^2 = 72.
Let's first recall some properties of the number of divisors of an integer. If we factorize an integer n as a product of prime powers, say
n = p_1^a_1 × p_2^a_2 × ... × p_k^a_k
then the number of divisors of n is given by
d(n) = (a_1 + 1) × (a_2 + 1) × ... × (a_k + 1).
Using this fact, we can deduce some information about the number we are looking for. Let's call it N. We know that N has 8 divisors, so it must be of the form
N = p_1^2 × p_2^2, or N = p_1^7,
where p_1 and p_2 are distinct prime numbers.
Now, we are told that each of N/2 and N/3 has four divisors. We can use the same fact about the number of divisors to conclude that
N/2 = q_1^3 × q_2, or N/2 = q_1^1 × q_2^3,
and
N/3 = r_1^3 × r_2, or N/3 = r_1^1 × r_2^3,
where q_1, q_2, r_1, and r_2 are distinct prime numbers.
To simplify the notation, let's introduce the variables a, b, c, d, e, and f, defined by
p_1 = q_1^a × q_2^b,
p_2 = r_1^c × r_2^d,
N/2 = q_1^e × q_2^f,
N/3 = r_1^g × r_2^h.
Using the information we have so far, we can write down equations for a, b, c, d, e, f, g, and h in terms of unknown exponents:
a + 1 × (b + 1) = e + 1 × (f + 1) = 4,
c + 1 × (d + 1) = g + 1 × (h + 1) = 4,
2a × 2b = ef,
2c × 2d = gh.
We can solve this system of equations by trial and error. For example, we can start by trying all possible values of a and b such that 2a × 2b = 4. This gives us two possibilities: a = 0, b = 2, or a = 1, b = 1. Using the first possibility, we get e = 3, f = 1, which leads to N/2 = q_1^3 × q_2, and hence N = 2 × q_1^3 × q_2^2. Substituting this into the equation for the sum of divisors, we get
(1 + q_1 + q_1^2 + q_1^3) × (1 + q_2 + q_2^2) = 216.
We can solve this equation by trial and error as well, or by observing that 216 = 2^3 × 3^3, and hence the two factors on the left-hand side must be equal to 2^3 and 3^3, respectively. This gives us the unique solution q_1 = 2 and q_2 = 3, and hence N = 2 × 2^3 × 3^2 = 72.
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let x1 and x2 be two independent random variables both with mean 10 and variance 5. let y 2x1 x2 3 2. find the mean and the variance of y.
As a result, y has a mean of 203 and a variance of 85 as let x1 and x2 be two independent random variables both with mean 10 and variance 5.
what is variable ?A variable is a symbol or letter that is used to indicate a variable quantity in mathematics. The context or issue under consideration can alter the value of a variable. In order to express relationships between quantities, variables are frequently utilized in equations, formulae, and functions. For instance, x and y are variables in the equation y = mx + b, which depicts the linear relationship between x and y. Variables in statistics can reflect various traits or features of a population or sample, such as age, body mass index, or income.
given
To get the mean and variance of y, we can apply the characteristics of expected value and variance:
We can start by determining the expected value of y:
E[y] = 2E[x1] = E[2x1x2 + 3x1 + 2]
By the linearity of expectation, E[x2] + 3E[x1] + 2 is 2(10)(10) + 3(10) + 2 = 203.
Next, we may determine y's variance:
Var(y) = Var(3x1 + 2 + 2 + 3x1 ) = 4
Var(x1)
Var(x2) + 9
Var(x1) + Var(constant) = 4(5)(5) + 9(5) + 0 = 85 since x1 and x2 are independent.
As a result, y has a mean of 203 and a variance of 85 as let x1 and x2 be two independent random variables both with mean 10 and variance 5.
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3. What is the value of x?
x + 5
X
05
02.5
07.5
O 10
x-2
x + 1
(1 point)
The value of x is equal to; A. 5.
What is the basic proportionality theorem?In Mathematics, the basic proportionality theorem states that when any of the two (2) sides of a triangle is intersected by a straight line which is parallel to the third (3rd) side of the triangle, then, the two (2) sides that are intersected would be divided proportionally and in the same ratio.
By applying the basic proportionality theorem to the given triangle, we have the following:
x/(x + 5) = (x - 2)/(x + 1)
By cross-multiplying, we have the following:
x(x + 1) = (x - 2)(x + 5)
x² + x = x² + 5x - 2x - 10
By rearranging and collecting like-terms, the value of x is given by:
2x - 10 = 0
2x = 10
x = 10/2
x = 5.
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