After the disk technique and the washer method, the cylindrical shell method of conducting a revolution is the third approach to determine the volume of the solid created by rotating about an axis. Technically, either of the first two approaches can find any revolution, but the cylindrical shell approach may be simpler to compute.
The cylindrical shell method's biggest drawback is that it necessitates more visualization of the regions where the cylinders are formed because you must determine both the cylinder's radius and height, whereas the other two methods only require the radius.
I'm going to start out by presuming that just the area in the first quadrant is rotating about the y-axis. Second, because the shell has a thickness of dx and the rotation revolves around a vertical axis, the integral must be expressed in terms of x. Third, the cylinder's height is and its height has a radius of [tex]36 - x^{2}[/tex]. Last but not least, the bounds are 0≤x≤6 since the area being rotated is constrained by the axes and the function. The integral can now be set up.
[tex]\int\limits {2\pi rh} \, dx = \int\limits^6_0 {2\pi x(36-x^{2} )} \, dx[/tex] [Set Up the Integral]
[tex]= 2\pi \int\limits^6_0 {36x - x^{3} } \, dx[/tex] [Distribute]
[tex]= 2\pi (\frac{36x^{2} }{2} - \frac{x^{4} }{4} )[/tex] = [tex]2\pi ( (\frac{36(6)^{2} }{2} - \frac{6^{4} }{4} ) - (\frac{36(0)^{2} }{2} - \frac{0^{2} }{4} ) )[/tex]
[tex]= 2\pi ( \frac{36(6)^{2} }{2} - \frac{6^{4} }{4} )[/tex] [Evaluate]
[tex]= 1296\pi[/tex] {Answer}
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The right Question is:
Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis.
[tex]y = 36 - x^{2} \\y = 0[/tex]
What are the factors of the polynomial P x x3 5x2 2x 8?.
By usinf Divison method and Prime Factorization, the factors of the polynomial are x=1,2,2
Short explanation of what polynomials are?A polynomial is an expression made up of variables, constants, and exponents that is combined using mathematical operations such as addition, subtraction, multiplication, and division.
cubic polynomial = x³ - 5x² + 8x - 4
suppose x = 1 ,
x - 1 = 0
So, one of the factor = (x - 1)
Now,
By Division method
x³ - 5x² + 8x - 4 = Dividend
x - 1 = Divisor
x² - 4x + 4 = Quotient
Now for x² - 4x + 4
x² - 2x- 2x + 4
(x-2)(x-2)
x=2,2
so, factors are x=1,2,2
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How do you solve a polynomial on a graphing calculator?.
Using mathematical operations like addition, subtraction, multiplication, and division, a polynomial is an expression or equation made up of variables, constants, and exponents.
Find the x-intercepts in step one.
Solve the equation to obtain the x-intercepts.
Find the y-intercepts in step two.
Put x=0 in the equation to determine the y-intercepts.
STEP 3: Determine the final action
A polynomial's final behavior and the final behavior of a leading term are identical.
STEP 4: Locate the pivotal moments
We must locate the first derivative of p in order to identify the turning point (x)
The first derivative's zeros are where the turning point's x coordinate is situated.
Substituting the aforementioned value into p will yield the y coordinate (x).
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Shelly walked 1/3 km. Kelly walked 4/15 km. Who walked farther? How much farther did
one walk than the other?
Answer:
Shelly walked 1/15 km farther
Step-by-step explanation:
We need to get a common denominator.
1/3 * 5/5 =5/15
Shelly walked 5/15 km
Kelly walked 4/15 km
Shelly walked farther
5/15 - 4/15 = 1/15
She walked 1/15 farther
Answer:
Shelly walked the furthest by 0.06 km.
Step-by-step explanation:
On a calculator (you can do it on paper, but it will take longer), take 1/3 and divide 1 by 3:
1 divided by 3 = 0.33
Then, do the same with 4/15:
4 divided by 15 = 0.27
So, Shelly walked the furthest.
To find out how much farther Shelly walked then Kelly, we simply subtract the 2 answers:
0.33 - 0.27 = 0.06
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(1 point) If p(x) and (x) are arbitrary polynomials of degree at most 2, then the mapping
=p(-1)q(-1) + p(070) +p(3)q(3) defines an inner product in P3- Use this inner product to find
||P||||9||, and the angle between p(x) and (x) for p(x) = 2x2 + 6 and 7(x) = 4x2 - 4x. < p, >= ||pl= ||ql| = A = radians.
If p(x) and q(x) are arbitrary polynomials of degree at most 2 then
||p||||q|| = 26([tex]\sqrt{640}[/tex]) and angle between p(x) and q(x) is 0.233.
Given that
<p,q> = p(-1)q(-1) + p(0) q(0) + p(3)q(3)
and p(x) = 2x²+ 6 , q(x)= 4x²-4x
then the values of p and q at x = -1,0,3 are given as;
x = -1,
p(-1) = 2(-1)² + 6 = 8 , q(-1) = 4(-1)² - 4(-1) = 8
x = 0,
p(0) = 2(0)² + 6 = 6 , q(0) = 4(0)² - 4(0) = 0
x = 3,
p(3) = 2(3)² + 6 = 24 , q(3) = 4(3)²- 4(3) = 24.
<p,q> = p(-1)q(-1) + p(0)q(0) + p(3)q(3)
= (8)(8) + (6)(0) + 24(24)
= 64 + 0 + 576
<p,q> = 640
Now we have to find ||p|| ||q||, for this we'll find ||p|| and ||q||
||p|| = [tex]\sqrt{ < p,p > }[/tex]
= [tex]\sqrt{8(8) + 6(6) + 24(24)}[/tex]
= [tex]\sqrt{676}[/tex]
||p|| = 26
and
||q|| = [tex]\sqrt{ < q,q > }[/tex]
=[tex]\sqrt{8(8) + 0(0) + 24(24)}[/tex]
||q|| =[tex]\sqrt{640}[/tex]
∴||p||||q|| = 26([tex]\sqrt{640\\[/tex])
Now we have to find angle between p(x) and q(x),
∴ α = cos⁻¹[tex]\frac{ < p,q > }{||p||||q||}[/tex]
= cos ⁻¹ [tex]\frac{640}{26(\sqrt{640}) }[/tex]
= cos ⁻¹ [tex]\frac{4\sqrt{10} }{13}[/tex]
α = 13.34°
In radian
α = 0.233.
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there are 7 children in a family. the number of children defines a population. the number of simple random samples of size 5 (without replacement) that are possible equals . a. 6 b. 21 c. 35 d. 32
The number of ways to select a simple random sample of size 5 from a population of 7 children in a family is equal to 21ways.
As given in the question,
Number of children in the family 'n' = 7
Number of simple random sample of size = 5 ( without replacement )
Number of ways to select the children defines a population
= ⁷C₅
= ( 7! ) / ( 7 - 5 )! ( 5! )
= ( 7 × 6 × 5! )/ ( 2! ) ( 5! )
= 7 × 3
= 21 ways
Therefore, the number of ways to select children of simple random sample of size 5 is equal to 21ways.
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identify the expression as a numerical expression or a variable expression. for a variable expression, name the variable. 9 10 it is a variable expression, and 9 and 10 are the variables. it is a numerical expression. it is a variable expression, and is the variable. it is a variable expression, and there is no variable.
This expression (9+10) is a numerical expression. And this is not a variable expression.
An expression made up only of numbers and operations is termed as numerical expression. While an expression written with a variable in it, is termed as variable expression. But this Both expression, that is a numerical expression and a variable expression can include powers. For example, from the given question 9 + 10 is a numerical expression, 9 and 10 are not variables. And in the expression, for example: 9a + 10b is a variable expression in which 9a and 10b are variable terms.
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Complete question- Identify the expression as a numerical expression or a variable expression. For a variable expression, name the variable.
9 + 10
A) It is a variable expression, and + is the variable.
B) It is a variable expression, and there is no variable.
C) It is a variable expression, and 9 and 10 are the variables.
D) It is a numerical expression.
Answer:
This expression (9+10) is a numerical expression. And this is not a variable expression.
Step-by-step explanation:
use the given data to find the equation of the regression line. round the final values to three significant digits, if necessary. managers rate employees according to job performance and attitude. the results for several randomly selected employees are given below.
The results for several randomly selected employees are given below [tex]Y = 11.7 + 1.02x.[/tex]
Below, we will discover the road equation, dissociation for each the management's conduct and fulfillment.
Here, attitudes are the reaction variable even as overall performance appears to be the predictor as mindset is fulfillment dependant.If Y shows the mindset even as X pertains to workers' overall performance.Let's describe the information like: [tex]y = a + bx + e[/tex]Now we get the estimates of e through the least quadratic method, – i.e., through decreasing the quantity of segments.S= sum e [tex]i^{2}[/tex] = [tex]sum (y_ - a - b x x)/ ^ 2[/tex]limit S to gain the estimates of a and b as:hat b = sum(y i - overline y )(x i - overline x ) sum(x i - overline x )^ 2hat a = overline y - hat b overline xSo,a = [tex]11.7b = 1.02[/tex] Y = [tex]11.7 + 1.02x[/tex].Read more about regression line:
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Is it possible to have a triangle with 3cm 4cm 9cm?.
Answer:
No, it is not possible to create a triangle with these measurements.
Step-by-step explanation:
It must follow these rules:
a + b > c
b + c > a
a + c > b
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Using compound interest formula [tex]A = p(1 +\frac{r}{n} )^{nt}[/tex] , the the amount after 5 years is $15,095.39
How to find the amount Eric owes for 6 years?The total amount a college student has in a saving account if 13000 dollars was invested and earned 3% compounded quarterly for 5 years can be calculated as follows:
Therefore, it is compounded quarterly which means 4 times.
[tex]A = p(1 +\frac{r}{n} )^{nt}[/tex]
where
p = principalr = rate t = timen = number of time compoundedTherefore,
p = 13000 dollars
t = 5 years
n = 4
r = 3% = 0.03
Therefore,
[tex]A = p(1 +\frac{r}{n} )^{nt}[/tex]
[tex]A = 13000(1+ \frac{0.03}{4} )^{(4)(5)}[/tex]
A = 13,000.00(1 + 0.0075)²⁰
A = $15,095.39
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What is a 2-digit product?.
When the original two-digit number is equal to the sum of the number's digits plus the product of its digits, the number is said to be special two-digit number.
We are required to describe the 2-digit special number in the provided question.
When the original two-digit number is equal to the sum of the number's digits plus the product of its digits, the number is said to be special.
Remove the first and last digits of the number, then add and multiply each digit separately. The two-digit number should then be multiplied by the sum and product of its digits, and the output should be compared to the initial value. It is a special two-digit number if they coincide; if not, it is not.
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Given a polynomial function f(x) = 2x2 – 3x 5 and an exponential function g(x) = 2x – 5, what key feature do f(x) and g(x) have in common? Both f(x) and g(x) have the ame domain of ([infinity], -[infinity]). Both f(x) and g(x) have the ame range of [0, -[infinity]). Both f(x) and g(x) have the ame x-intercept of (2, 0). Both f(x) and g(x) increae over the interval of [-4 , [infinity])
The key feature f(x) and g(x) have in common is both f(x) and g(x) have the same domain of ([infinity], -[infinity]). Option A is correct.
We are given the functions;
Polynomial function; f(x) = 2x² - 3x + 5
Exponential function; g(x) = 2ˣ - 5
In this question we must analyze the domain, range, x-intercepts, and changing intervals:
The domain of both functions is a set of all real numbers with both functions continuous.
The range of both functions is;
Range of f(x) = (3.875, +∞)
Range of g(x) = (-5, +∞)
The x-intercepts of both functions are;
x-intercept for f(x) is non-existent.
x-intercept for g(x) is (2.322, 0).
Thus, the key feature f(x) and g(x) have in common is both f(x) and g(x) have the same domain of ([infinity], -[infinity]). Option A is correct.
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complete the coordinate proof of the theorem.given: a b c d is a rectangle. prove: the diagonals of a b c d are congruent. art: a rectangle is graphed on a coordinate plane. the horizontal x-axis and vertical y-axis are solid. the vertex labeled as a lies on begin ordered pair 0 comma 0 end ordered pair. the vertex labeled as b lies on begin ordered pair a comma 0 end ordered pair. the vertex labeled as d lies on begin ordered pair 0 comma b end ordered pair. the vertex c is unlabeled. diagonals a c and b d are drawn by dotted lines.enter your answers in the boxes.
Since the lengths of both diagonals are given by the square root of the sum of the squares of their coordinates, they have the same length. Therefore, the diagonals of a b c d are congruent.
What are congruency?Triangles with the same size, form, and arrangement are said to be congruent. This indicates that all three of the triangle' side lengths are congruent, or equal in size. Congruent triangles can be rotated or reversed, yet their congruence remains constant.
The Side-Side-Side (SSS) Congruence Postulate argues that if three sides of one triangle are identical in length to the corresponding sides of another triangle, the halves of are congruent.
If two angles and a side of one triangle are congruent to the corresponding angles and sides of another triangle, then the two triangles are congruent. As an alternative, one can apply the Angle-Side-Angle (ASA) Congruence Postulate.
How to solve?
The distance formula is given by d = √((x2 - x1)^2 + (y2 - y1)^2), where d is the distance between the two points, (x1, y1) and (x2, y2).
To find the length of diagonal a c, we plug in the coordinates of points a and c into the distance formula: d = √((c - 0)^2 + (d - 0)^2) = √(c^2 + d^2)
To find the length of diagonal b d, we plug in the coordinates of points b and d into the distance formula: d = √((a - 0)^2 + (b - 0)^2) = √(a^2 + b^2)
Since the lengths of both diagonals are given by the square root of the sum of the squares of their coordinates, they have the same length. Therefore, the diagonals of a b c d are congruent.
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A factory produced 475812 pen of one kind and 651652 pen of another kind. All the pen are mixed thoroughly and packed equally in 296 boxe. How many pen are packed in a box
The total number of pens packed in a box is 3809 pens.
What is an arithmetic operation?
The branch of mathematics known as arithmetic deals with the study of numbers and the numerous operations that can be performed on them. Addition, subtraction, multiplication, and division are fundamental mathematical operations.
Given, A factory produced 475812 pens of one kind
And 651652 pens of another kind.
So, total pens = 475812 + 651652 = 1127464 pens
Total number of boxes = 296 boxes
Now,
Total number of pens packed in a single box = 1127464/296
= 3809 pens per box
Hence, the total number of pens packed in a box is 3809 pens.
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Write an equivalent expression for 8(1/2y - 7)
4y - 56 is the equivalent expression for 8(1/2y - 7)
How to write equivalent expressions?Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we substitute the same value(s) for the variable(s).
Given: 8(1/2y - 7)
In order to write an equivalent expression, clear the parenthesis:
8(1/2y - 7) = 8(1/2y ) - 8(7)
= 4y - 56
Therefore, the equivalent expression for 8(1/2y - 7) is 4y - 56
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When you want a method to be able to change the value of a variable that is passed to it as an argument, the variable must be .
passed by reference
If you want a method to be able to alter the value of a variable that is supplied to it as an argument. The variable must be pass by reference.
Given that,
If you want a method to be able to alter the value of a variable that is supplied to it as an argument. The variable must be------.
We have to fill the blank.
We know that,
What is a variable?A variable is an alphabet used to represent an unknown quantity in algebra. It embodies the value. In mathematics, a variable is a value that can be changed depending on the problem. Numerous algebraic expressions and equations use the generic letters x, y, and z. Or to put it another way, a variable is a symbol for a number whose value is unknown.
Therefore, If you want a method to be able to alter the value of a variable that is supplied to it as an argument. The variable must be pass by reference.
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What are the 3 properties of parallel lines?.
The three properties of parallel lines are transitive property of parallel lines, symmetric properties of parallel lines and properties cut by transversal.
Properties of parallel lines when cut by a transversal
Two lines when cut by a transversal line are parallel if the corresponding angles so formed are equal. Generally, the corresponding angles are in relative positions and lie along the same side of the transversal.
Two lines cut by a transversal line are said to be parallel if the alternate interior angles so formed are equal. In general, the pairs of the alternate interior are found in the inner side but lie on the opposite sides of the transversal.
Two lines cut by a transversal line are said to be parallell if the alternate exterior angles so formed are equal. In general, the pairs of alternate exterior angles are found on the outer side but lie opposite each other.
The above mentioned are 3 properties of parallel lines.
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One circle ha a radiu of 1/4 and another circle ha a radiu of 2. What i the ratio of the circumference of the larger circle to the circumference of the maller circle?
The ratio of the larger circle's circumference to the smaller circle's circumference is 8π : π.
Given info;
There are two circles, one with a radius of 1/4 and the other with a radius of 2.
To know the ratio of the larger circle's circumference to the smaller circle's circumference,
Let,
Circle's circumference = π * diameter
The circumference of the smaller circle = π/2
The diameter of the larger circle = 4π
Now,
The ratio of the circumference of the larger circle to the circumference of the smaller circle is,
⇒ 4π : π/2
8π : π
Hence, the ratio is 8π : π.
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at a high school administrators wish to collect a sample of 80 students. they decide to randomly sample 20 freshmen, 20 sophomores, 20 juniors, and 20 seniors and ask them how much time they spent on their math homework last night. which of the following methods was use to select the sample?
For the given a sample of 80 students further divided in subgroups like 20 freshman, 20 juniors ,20 sophomores and 20 senior on the basis of how much time they spent to complete their maths homework last night. This is random sampling, the method which used here is Stratified Sampling.
A sample is a specific group of people from whom we collect data.
Stratified Sampling:
In statistics, a simple random sample is a subset of people selected from a larger set in which subsets of people are randomly selected, all with equal probability. This is the process of randomly selecting samples.
In stratified sampling, a population is divided into subpopulations that may differ in important ways. By ensuring that each subgroup is well represented in your sample, you can draw more precise conclusions.
To use this sampling method, divide the population into subgroups (called strata) based on relevant characteristics (gender identity, age group, income class, occupational role, etc.).
Calculate how many people to sample from each subgroup based on the total population percentage. Then use random or systematic sampling to select one sample from each subgroup.
Using the same sampling technique, subsample of students based on the amount of time they spent completing their math homework.
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Consider the number of blocked intrusions below:
Before firewall change: 56, 47, 49, 37, 38, 60, 50, 43, 43, 59, 50, 56, 54, 58
After firewall change: 53, 21, 32, 49, 45, 38, 44, 33, 32, 43, 53, 46, 36, 48, 39, 35, 37, 36, 39, 45
(a) Construct a 95% confidence interval for the difference between the average number of intrusion attempts per day before and after the change of firewall setting (assume equal variance)
(b) Can we claim significant reduction in the rate of intrusion attempts? The number of intrusion each day has approximately normal distribution. Compute P-values and state your conclusions under the assumption of equal variances and without it. Does this assumption make a difference?
a)The 95% confidence interval are
0-44,44-88
b))Yes, we can claim significant reduction in the rate of intrusion attempts
The p-value is 0.067No, the given assumption does not make a difference.We have firewall change of 14 people.
We need to compute total sum before firewall change which can be done by
(56+47+49+37+38+60+50+43+59+50+56+54+58)/14
=657/14
=46.92
Now, similarly, after firewall change total sum is
(53+21+32+49+45+38+44+33+32+43+53+46+36+48+39+35+37+36+39+45)/20
=756/20
=37.8
Now the difference between average number of intrusion attempts before and after firewall change is
(46.92-37.8)
=9.12
Now, we need to make confidence interval to only 95% of this difference=
(95×9.12)/100
=44.196=44(approx)
It means our interval range varies from 0-44,44-88 with a difference of 44.
b)Yes, we can easily claim that rate of intrusion attempts decreases.
Before firewall change, we can see average number of intrusion attempt is =46.92
After firewall change, we can see average number of intrusion attempt is
37.8
So, we can see rate of intrusion decreases on increasing the number of blocked intrusions.
For computing z-value, we just need to take square root of the difference of average number of intrusions attempst
=>Z-value= √9.12=3.01
Using the z-table, we get that for the above z-value, corresponding p-value is 0.067
Hence, the interval range is 0-44,44-88 and b)Yes we can claim, p-value is 0.067.No, this assumption does not make a difference.
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Line segment wx is the radius of circle x, and line segment zy is the radius of circle y. Points w, x, c, y, and z are all on line segment wz. What is the area of circle c, which passes though points w and z?.
The area of circle c, which passes though points w and z is 324[tex]\pi[/tex].
Point D here is the midpoint of the outer circle that we aim to find the area of. Here, the circle C has a diameter of WZ and radii of WC and CZ
We know that YZ = YD = 10 cm is as measured
Let DC be denoted by [tex]x[/tex] and CY be denoted by [tex]10-x[/tex].
The radius of the outer circle can be written as [tex]8+8+x[/tex] or [tex]10+10-x[/tex] which we can equate by following to find the value of [tex]x[/tex]
By equating both sides we get
⇒ 8+8+x = 10+10-x
⇒ 16+x = 20-x
⇒ 2x = 4
⇒ x = 2
Putting the value of x =2 in 8+8+x we get 18 as a radius.
Thus, area of circle =[tex]\pi 18^{2}[/tex] = 324[tex]\pi[/tex]
Hence, the area of circle c, which passes though points w and z is 324[tex]\pi[/tex].
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one of the digits marcel writes down is chosen at random. what is the probability that this digit is a ?
The probability of randomly selecting alphabet a is 1/26.
What is probability?Simply put, the probability is the likelihood that something will occur. When we don't know how an event will turn out, we can discuss the likelihood or likelihood of several outcomes.
Statistics is the study of events that follow a probability distribution.
An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.
So, the probability of the digit a:
We know that there are 26 digits in the alphabet.
Marcel selects 1 digit at random.
The probability of digit a will be:
The probability formula: P(E) = Favourable events/Total events
Put values and calculate:
P(E) = Favourable events/Total events
P(E) = 1/26
Therefore, the probability of randomly selecting alphabet a is 1/26.
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Choose five other iterated integrals that are equal to the given iterated integral. integral^2_0 integral^2_y integral^y_0 f(x, y, z) dz dx dy f(x, y, z) dz dy dx f(x, y, z) dz dy dx f(x, y, z) dz dy dx f(x, y, z) dz dy dx f(x, y, z) dz dy dx
By Changing the order of integral we can find five iterated integral that are equal to [tex]\int\limits^2_0 \int\limits^2_y \int\limits^y_0 {f(x,y,z)}\, dz dx dy[/tex]
Changing the order of integral means switching the dx dy order of integration into dy dx .
According to the question ,
The given integral is
=> [tex]\int\limits^2_0 \int\limits^2_y \int\limits^y_0 {f(x,y,z)}\, dz dx dy[/tex]
where;
the region is bounded by 0 ≤ z ≤ y, y≤ x ≤ 2, 0 ≤ y ≤ 2
(a) Changing the order of integral from dzdxdy to dzdydx
When y= 0 => x = y
and y = 2 => x = 2
[tex]\int\limits^2_0\int\limits^x_0 \int\limits^y_0 {f(x, y ,z )} \, dz dy dx[/tex]
(b) Now , changing the order of integral from dzdxdy to dxdzdy
When x = 0 => z = 0
x = 2 => z = y
[tex]\int\limits^2_0\int\limits^y_0 \int\limits^2_y {f(x, y ,z )} \, dx dz dy[/tex]
(c) Now ,Changing the order of integral from dzdxdy to dxdydz
[tex]\int\limits^2_0\int\limits^2_z \int\limits^2_y {f(x, y ,z )} \, dxdydz[/tex]
(d) Now ,Changing the order of integral from dzdxdy to dydzdx
[tex]\int\limits^2_0\int\limits^x_0 \int\limits^x_z {f(x, y ,z )} \, dydzdx[/tex]
(e) Now ,Changing the order of integral from dzdxdy to dydxdxz
[tex]\int\limits^2_0\int\limits^2_0 \int\limits^x_0 {f(x, y ,z )} \, dydxdz[/tex]
Are the five iterated integral that are equal to the integral [tex]\int\limits^2_0 \int\limits^2_y \int\limits^y_0 {f(x,y,z)}\, dz dx dy[/tex].
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Solving by factoring which equation is equivalent to f(x) = 16x4 – 81 = 0?
(4x² - 9)(4x² + 9) = 0 equation is equivalent to the given equation.
What is factorization?
Factorization in mathematics is the process of dividing a large number into smaller ones that, when multiplied together, give you the original number.
Here, we have
Given equation,
f(x) = 16x⁴ – 81 = 0
The difference of squares is used to solve this equation. There are three perfect squares: 16, x⁴, and 81. ( which means that it is a number multiplied by itself, 16 is 4 x4 for example).
We solve this equation by formula: (a² - b²) = (a - b)(a + b)
= 16x⁴ – 81 = 0
Now, we determine the square root of 16 which is 4, and of x⁴ which is x², and 81 which is 9.
Now we write the equation in the (a - b)(a + b) form :
⇒ (4x² - 9)(4x² + 9) = 0
Hence, the (4x² - 9)(4x² + 9) = 0 equation is equivalent to the given equation.
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If the nth term of a number sequence is 5n? - 2n,
find the first 3 terms and the 10th term.
The first three terms are 3,16,39 and the 10th term is 480.
What is Arithmetic progression?
The difference between any two consecutive integers in an arithmetic progression (AP) sequence of numbers is always the same amount. It also goes by the name Arithmetic Sequence.For instance, the natural number sequence 1, 2, 3, 4, 5, 6,... is an example of an arithmetic progression. It has a common difference of 1 between two succeeding terms (let's say 1 and 2). (2 -1). We can see that the common difference between two subsequent words will be equal to 2 in both the case of odd and even numbers.tₙ = 5n² - 2nnow,
t₁ = 5(1)² - 2(1)
t₁ = 5 - 2
t₁ = 3
t₂ = 5(2)² - 2(2)
t₂ = 5(4) - 4
t₂ = 16
d = t₂ - t₁
d = 16 - 3
d = 13
t₃ = 5(3)² - 2(3)
t₃ = 45 - 6
t₃ = 39
t₁₀ = 5(10)² - 2(10)
= 500 - 20
t₁₀ = 480
Hence, the first three terms are 3,16,39 and the 10th term is 480.
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jenny has a box of candies with different flavors. she will randomly choose one piece of candy. the probability that she chooses a strawberry-flavored piece is 611. what are the odds against her choosing a strawberry-flavored piece? 116
The odds against her choosing a strawberry-flavored piece is 5/11.
What is the probability?Probability is the odds that a random event would happen. The probability the event occurs is 1 and the probability that the event does not occur is 0.
The odds against her choosing a strawberry-flavored piece = 1 - probability that she chooses a strawberry-flavored piece
The odds against her choosing a strawberry-flavored piece = 1 - 6/11
The odds against her choosing a strawberry-flavored piece = 5/11
Here is the complete question:
jenny has a box of candies with different flavors. she will randomly choose one piece of candy. the probability that she chooses a strawberry-flavored piece is 6/11. what are the odds against her choosing a strawberry-flavored piece?
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What are the solutions of the equation x4 + 6x2 + 5 = 0? use u substitution to solve. X = i and x = i i and x = i and x = and x = -.
The given equation's solution is x = 5i or x = i.
An equation has been provided.
x⁴+6x²+5 = 0
We to settle it utilizing replacement.
If u = x2, then u2 = x4 In the above equation, we find that u2+6u+5 = 0. Now, use the factorization method.
Divide the middle term of the previous equation as follows: u2+5u+u+5 = 0; u(u+5)+1(u+5) = 0; u(u+5)(u+1) = 0 Using the zero-product property, we can determine that u+5 or u+1 equals 0; u = -5 or u = -1; x2 equals -5 or x2 equals -1; taking the square root of both sides of.
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The cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed. A certain ferry boat uses $100 worth of fuel per hour when cruising at 10 miles per hour. Apart from fuel, the cost of running this ferry (labor, maintenance, and so on) is$675 per hour. At what speed should it travel so as to minimize the cost per mile traveled?
The cost is minimum per mile traveled "V = 15 mph".
Given,
In the question:
The cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed. A certain ferry boat uses $100 worth of fuel per hour when cruising at 10 miles per hour.
and, the cost of running this ferry is $675.
To find the speed should it travel so as to minimize the cost per mile traveled.
Now, According to the question:
C(V) ∝ V³
then,
C(V) = KV³
When, C = 100 and V = 10
then, 100 = k[tex](10)^3[/tex]
=> k = 1/10
∴ [C(V) = [tex]\frac{V^3}{10}[/tex]]
As we know,
Total cost per hour,
[tex]C_1[/tex] = [tex]\frac{V^3}{10}[/tex] + 675
Cost per mile = C/V
Now,
C₂ = [tex]\frac{\frac{V^3}{10}+675 }{V}[/tex]
=> [tex]\frac{V^2}{10} +\frac{675}{V}[/tex]
Then we'll need to discover their derivative as well as set that to zero (0) to minimize it.
[tex]C_2'[/tex],
=> [tex]\frac{2V}{10} - \frac{675}{V^2}[/tex] = 0
=> [tex]\frac{V}{5}[/tex] = [tex]\frac{675}{V^2}[/tex]
On applying cross-multiplication, we get
[tex]V^3[/tex] = 3375
V = 15
So that the cost is minimum at "V = 15 mph".
Hence, The cost is minimum per mile traveled "V = 15 mph".
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the variance inflationary factor (vif) measures the group of answer choices contribution of each x variable with the y variable after all other x variables are included in the model. correlation of the x variables with the y variable. correlation of the x variables with each other. standard deviation of the slope. next
The correct statement that describes variance inflationary factor (VIF) is option (a) the correlation of the x variable with each other
The variance inflationary factor ( VIF ) defined as the measure of the amount of multicollinearity in regression analysis.
When there is a correlation between multiple independent variable in multiple regression model, then there will be multicollinearity, therefore it will adversely affect the regression result
Correlation is defined as the extend to which the two variables are linearly connected. Therefore it will change at a constant rate together
Hence, the variance inflationary factor( VIF ) measures option (a) the correlation of the x variable with each other
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What is the purpose of making a geological timeline Quora?.
Geological timeline is based on the presence of fossils on the rocks of various sequences, and also the classification of rock types and the correlation of rocks of different locations.
The purpose that geological timeline serve and reason scientist do make it, is portray the history of the Earth.
A geological timeline can be regarded as a system which gives the relationship between geological strata as well as events based on chronological time.This helps the scientist in studying events as well as frequency of events to Know if there are chances of recurrence of that particular event.Examples of this event could be eclipse of sun that usually occur at some point in time.The geologic time scale also helps in portraying the history of the Earth such as rocks and fossils.Therefore, geologic time scale serves as events as regards Earth history.
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The classification of rock types, the correlation of rocks from diverse locations, and the presence of fossils on the rocks of distinct sequences are all used to create the geological timeline.
What is a Geological timeline?Geological timelines have a purpose, and scientists create them in order to depict Earth's history.
(A) The relationship between geological strata and events based on chronological time can be shown using a geological timeline.
(B) The scientist can use this information to investigate events and their frequency and determine whether there is a possibility that a specific event will occur again.
(C) A sun eclipse, which often happens at some point in time, could serve as an illustration of this occurrence.
(D) Rocks and fossils that depict Earth's past can be found thanks to the geologic time scale.
(E) As a result, the geologic time scale provides information about historical events on Earth.
Therefore, the classification of rock types, the correlation of rocks from diverse locations, and the presence of fossils on the rocks of distinct sequences are all used to create the geological timeline.
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an inventory consists of 89 items each marked as available or unavailable. there are 50 available item. show that there are at least 2 available items in the list with exactly 9 items apart.
If n pigeons are divided among k pigeonholes and k<n, then some pigeonholes contain at least two pigeons.
Pigeonhole principle- It's given a list of 89 items, 50 are available. we have to show there are at least 2 available items in the list with exactly 9 items apart.
Let's number the items in the list from 1 to 115.
Let "Ai" represents the i th available number.
Let the pigeons be the numbers in the two sequences: a1, a2, a3,...... a50 and a1 + 9, a2 + 9, ...... a50 +9.
Now, pigeonholes be the possible numbers of the items in the sequences, there are k = 89+9 = 98 possible numbers for the items in the sequence and n= 50+50=100 numbers in the two sequences.
n= 100, k=98
Since, k<n ,there are at least 2 numbers in the two sequences that are the same number by the pigeonhole principle. Then need to be some number aj in the sequence a1, a2, a3,...... a50 and some number ai + 9 in the sequence a1 + 9, a2 + 9, ...... a50 +9.
ai + 9 = aj
Thus, there are at least 2 available items ( ai and aj ) that are exactly 9 items apart.
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