bar is 10 1/4" so center would be 10 1/4 / 2 = 5 1/8"
door is 26 1/3" so center would be 26 1/3 / 2 =13 1/6
13 1/6 - 5 1/8 = 8 1/24 inches from the edge
What is 0.2 repeated as a fraction?
Find the second degree Taylor polynomial for f(x)= sqrt(x^2+8) at the number x=1
Answer:
[tex]\displaystyle P_2(x) = 3 + \frac{1}{3}(x - 1) + \frac{4}{27}(x - 1)^2[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to RightAlgebra I
Functions
Function NotationCalculus
Differentiation
DerivativesDerivative NotationDerivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Taylor Polynomials
Approximating Transcendental and Elementary Functions[tex]\displaystyle P_n(x) = \frac{f(c)}{0!} + \frac{f'(c)}{1!}(x - c) + \frac{f''(c)}{2!}(x - c)^2 + \frac{f'''(c)}{3!}(x - c)^3 + ... + \frac{f^n(c)}{n!}(x - c)^n[/tex]Step-by-step explanation:
*Note: I will not be showing the work for derivatives as it is relatively straightforward. If you request for me to show that portion, please leave a comment so I can add it. I will also not show work for elementary calculations.
Step 1: Define
Identify
f(x) = √(x² + 8)
Center: x = 1
n = 2
Step 2: Differentiate
[Function] 1st Derivative: [tex]\displaystyle f'(x) = \frac{x}{\sqrt{x^2 + 8}}[/tex][Function] 2nd Derivative: [tex]\displaystyle f''(x) = \frac{8}{(x^2 + 8)^\bigg{\frac{3}{2}}}[/tex]Step 3: Evaluate
Substitute in center x [Function]: [tex]\displaystyle f(1) = \sqrt{1^2 + 8}[/tex]Simplify: [tex]\displaystyle f(1) = 3[/tex]Substitute in center x [1st Derivative]: [tex]\displaystyle f'(1) = \frac{1}{\sqrt{1^2 + 8}}[/tex]Simplify: [tex]\displaystyle f'(1) = \frac{1}{3}[/tex]Substitute in center x [2nd Derivative]: [tex]\displaystyle f''(1) = \frac{8}{(1^2 + 8)^\bigg{\frac{3}{2}}}[/tex]Simplify: [tex]\displaystyle f''(1) = \frac{8}{27}[/tex]Step 4: Write Taylor Polynomial
Substitute in derivative function values [Taylor Polynomial]: [tex]\displaystyle P_2(x) = \frac{3}{0!} + \frac{\frac{1}{3}}{1!}(x - c) + \frac{\frac{8}{27}}{2!}(x - c)^2[/tex]Simplify: [tex]\displaystyle P_2(x) = 3 + \frac{1}{3}(x - c) + \frac{4}{27}(x - c)^2[/tex]Substitute in center c: [tex]\displaystyle P_2(x) = 3 + \frac{1}{3}(x - 1) + \frac{4}{27}(x - 1)^2[/tex]Topic: AP Calculus BC (Calculus I + II)
Unit: Taylor Polynomials and Approximations
Book: College Calculus 10e
The second degree Taylor polynomial for the function [tex]f(x) = sqrt(x^2+8)[/tex] at x=1 is [tex]T(x) = 3 + (x-1) - 1/32(x-1)^2[/tex].
To find the second degree Taylor polynomial for [tex]f(x) = sqrt(x^2+8)[/tex] at the number x=1, we begin by calculating the necessary derivatives and evaluating them at x=1. The Taylor polynomial of degree n at x=a is given by:
[tex]T(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n[/tex].
In this case, we need to find the first and second derivatives:
[tex]f'(x) = \frac{1}{2}(x^2+8)^{-1/2} · 2x[/tex]
[tex]f''(x) = \frac{1}{2} · (-1/2)(x^2+8)^{-3/2} · 2x^2 + \frac{1}{2}(x^2+8)^{-1/2}[/tex]
Then we evaluate f(x), f'(x), and f''(x) at x=1:
[tex]f(1) = sqrt(1^2+8) = sqrt9 = 3[/tex]
[tex]f'(1) = \frac{1}{2}(1^2+8)^{-1/2} · 2 · 1 = 1[/tex]
[tex]f''(1) = \frac{1}{2} · (-1/2)(1^2+8)^{-3/2} · 2 · 1^2 + \frac{1}{2}(1^2+8)^{-1/2} = -\frac{1}{16}[/tex]
Thus, the second degree Taylor polynomial at x=1 is:
[tex]T(x) = 3 + (x-1) - \frac{1}{32}(x-1)^2[/tex].
A given line has the equation 10x + 2y = −2.
What is the equation, in slope-intercept form, of the line that is parallel to the given line and passes through the point (0, 12)?
Liam has 9/10 gallon of paint for painting the birdhouse he sells at the craft fair. Each birdhouse requires 1/20 gallon of paint.how many birdhouse can Liam paint? Show your work.
Kate has a serving account that contains $230. She decides to deposit $5 each month from her monthly earnings for baby-sitting after school. Write an expression to find how much money Mata will have in her savings account after x months. Let x represent the number of months. Then find out how much she will have in her account after 1 year.
The stem-and-leaf plot shows the ages of customers who were interviewed in a survey by a store.
How many customers were older than 45?
HELP ASAP please
Answer:
Customers older than 45 years are 11 in number.
Step-by-step explanation:The age of store customers is represented by the stem ad leaf plot.The stem represents the tens digit while leaf denotes the unit digit.The question is asking us to find the number of customers who are older than 45 so 45 is not considered.
The age of customers more than 45 are:
48,50,50,51,55,56,62,64,65,65,73.There are 11 customers in all .
A negative number is raised to an odd exponent. The result is _____. zero one positive negative
A negative number is raised to an odd exponent. The result is always negative.
What is mean by Odd exponent?
An odd power of a number is a number of the form for the integer and a positive odd integer.
The first few odd powers are 1, 3, 5, 7, .........
Given that;
The expression is;
A negative number is raised to an odd exponent.
Now, To prove this statement that ''A negative number is raised to an odd exponent. The result is always negative.''
Let an example for an odd exponent as;
f (x) = (- 4)³
Here the power is 3 which is odd.
This gives;
f (x) = (- 4)³
f (x) = - 64
Which is negative function.
Hence, A negative number is raised to an odd exponent is always negative.
Therefore,
A negative number is raised to an odd exponent. The result is always negative.
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0.2(v-5) = -1 solve equations
Marco comma roberto comma dominique comma and claricemarco, roberto, dominique, and clarice work for a publishing company. the company wants to send two employees to a statistics conference. to be fair, the company decides that the two individuals who get to attend will have their names drawn from a hat. this is like obtaining a simple random sample of size 2. (a) determine the sample space of the experiment. that is, list all possible simple random samples of size n equals 2n=2. (b) what is the probability that marco and robertomarco and roberto attend the conference? (c) what is the probability that dominiquedominique attends the conferenceattends the conference?
Answer:
Yes
Step-by-step explanation:
ye mom ye mom lolololol
Trigonometric Identities
Simplify each expression.
(1−cos(−t))(1+cos(t)) =
(1+sin(t))(1+sin(-t))=
csc(t)tan(t)+sec(−t) =
Thank you for your help
a fan has 5 equally spaced blades. what is the least number of degrees that can rotate the fan onto self?
what does proportional mean?
In mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant multiplier. The constant is called the coefficient of proportionality or proportionality constant.
solve 6[4x(72-63)divided3]
HELP ME! *EMERGENCY*
A survey by the state health department found that the average person ate 208 pounds of vegetables last year and 125 5/8 pounds of fruit. What fraction of the total pounds of fruit and vegetables do the pounds of fruits represent?
Answer : The fraction of pounds of fruits over the total pounds of fruit and vegetables is, [tex]\frac{1005}{2669}[/tex]
Step-by-step explanation :
As we are given that:
Total amount of vegetables = 208 pounds
Total amount of fruits = [tex]125\frac{5}{8}\text{ pounds}=\frac{1005}{8}\text{ pounds}[/tex]
Thus, the total amount of fruits and vegetables will be:
[tex]208+125\frac{5}{8}\\\\=208+\frac{1005}{8}\\\\=\frac{1664+1005}{8}\\\\=\frac{2669}{8}[/tex]
Now we have to calculate the fraction of pounds of fruits over the total pounds of fruit and vegetables :
[tex]\frac{\text{Pounds of fruits }}{\text{ total pounds of fruits and vegetables}}\\\\=\frac{(\frac{1005}{8})}{(\frac{2669}{8})}\\\\=\frac{1005}{8}\times \frac{8}{2669}\\\\=\frac{1005}{2669}[/tex]
Thus, the fraction of pounds of fruits over the total pounds of fruit and vegetables is, [tex]\frac{1005}{2669}[/tex]
look at the figure if tan x=3/y and cos x =y/z what is the value of sin x?
Answer: sin x° = 3/z( answer
Because tan is opposite/adjacent,
Cos is adjacent/hypotenuse and sin is opposite/hypotenuse the information to find sin is given. You simply take the opposite (3) and put it over the hypotenuse (z)
sin x°= 3/z
a neighborhood garden that is 2/3 of an acre is to be divided 4 equal-size sections
The slope of a line is 1, and the y-intercept is -1. What is the equation of the line written in slope-intercept form?
y = x - 1
y = 1 - x
y = -x - 1
On a game show, a contestant randomly chooses a chip from a bag that contains numbers and strikes. The theoretical probability of choosing a strike is 3/10. The bag contains 9 strikes. How many chips are in the bag?
Write an absolute value for all real numbers at least 3 units from -2
Using symbols. There are 3 shelves. Each shelf has 21 books. How many books are there in all?
There are four families attending a concert together. each family consists of 1 male and 2 females. in how many ways can they be seated in a row of twelve seats i
The four families consisting of 12 distinct individuals can be seated in 12! (479,001,600) ways. There are no seating restrictions, so each person can occupy any of the twelve seats.
Explanation:The subject of this question is combinatorics, a branch of mathematics. In this problem, you have four families, each consisting of 1 male and 2 females, and you want to know in how many ways they can be seated in a row of twelve seats.
Given there is no restriction about the seating pattern, each member can occupy any seat. So, consider each family member as a distinct person; you then have 12 people to be seated in 12 different ways. This can be done in 12 factorial (12!) ways. Factorial implies the product of all positive integers up to that number. A simple way to calculate 12 factorial is: 12*11*10*9*8*7*6*5*4*3*2*1, which equals 479,001,600.
So, the four families can be seated in a row of twelve seats in 479,001,600 ways. This principle of arrangements is a key part of combinatorics and discrete mathematics.
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Considering each family as a single unit, the total number of ways they can be seated in a row of twelve seats is given by the formula: 12! divided by (2!) to the power of 4.
Explanation:This is actually a problem that is solved using the principles of permutations and combinations in mathematics. Given we have 4 families each with 1 male and 2 females, we have a total of 12 people. Now, if we are to arrange these 12 people in a row of twelve seats, we have 12! (factorial) ways to do it. However, each family group is to be considered as a single unit and within each unit, arrangements don't count, so we must divide by the number of ways to arrange the 2 females within each family of 4, which is 2!. Hence, the total number of ways to seat the group is given by (12! / (2!)^4).
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Select "Rational" or "Irrational" to classify each number. 0.25 Sq. Rt. 0.25 Sq. Rt. 0.33
20. Find the measure of each interior angle and each exterior angles of the following regular polygons. Show your work.
A sphere with a diameter of 16mm has the same surface area as the total surface area of a right cylinder with the base diameter equal to the sphere diameter. How high is the cylinder?
The height of the cylinder is 8mm.
Explanation:To find the height of the cylinder, we need to first find the surface area of the sphere. The surface area of a sphere is given by the formula: 4πr^2. Since the diameter of the sphere is 16mm, the radius is half of that, which is 8mm. Plugging the value of the radius into the formula, we get: 4π(8^2) = 256π mm^2.
Next, we need to find the surface area of the cylinder. The base diameter of the cylinder is equal to the diameter of the sphere, which is 16mm. Therefore, the radius of the cylinder is also 8mm. The surface area of the cylinder is given by the formula: 2πrh + 2πr^2. Since the height is unknown, we'll use a variable 'h'.
Since the surface area of the sphere is equal to the surface area of the cylinder, we can set up an equation: 256π = 2πrh + 2πr^2. Plugging in the values, we get: 256π = 2π(8)(h) + 2π(8^2).
Cancelling out the common factor of 2π, we have the equation: 128 = 8h + 64. Subtracting 64 from both sides of the equation, we get: 64 = 8h. Dividing both sides by 8, we find that the height of the cylinder is: h = 8mm.
Determine whether the given function is linear. if the function is linear, express the function in the form f(x)
The given function [tex]\(f(x) = \frac{5}{5} \cdot x\)[/tex] is indeed linear, and it can be expressed as f(x) = x in the standard linear form.
Let's break down the analysis of the given function[tex]\(f(x) = \frac{5}{5} \cdot x\)[/tex].
1. Initial Expression:
[tex]\[ f(x) = \frac{5}{5} \cdot x \][/tex]
2. Simplify the Fraction:
[tex]\[ \frac{5}{5} \][/tex] simplifies to 1, so the expression becomes:
[tex]\[ f(x) = 1 \cdot x \][/tex]
3. Multiplication by 1:
Multiplying any value by 1 does not change the value, so the expression further simplifies to:
f(x) = x
4. Linear Form:
The function is now in the form f(x) = ax + b with a = 1 and b = 0:
[tex]\[ f(x) = 1 \cdot x + 0 \][/tex]
Therefore, the given function [tex]\(f(x) = \frac{5}{5} \cdot x\)[/tex] is indeed linear, and it can be expressed as f(x) = x in the standard linear form.
Complete Question: Determine whether the given function is linear. if the function is linear, express the function in the form f(x) = ax + b. (if the function is not linear, enter not linear.)
f(x) = 5 / 5 x
What is the coefficient of the x4-term in the binomial expansion of (x + 3)12?
The coefficient of the [tex]\(x^4\) term in \((x + 3)^{12}\)[/tex] is 495, calculated using binomial coefficients.
To find the coefficient of the [tex]\(x^4\)[/tex] term in the expansion of [tex]\((x + 3)^{12}\)[/tex], you can use the binomial theorem. According to the binomial theorem, the expansion of [tex]v[/tex] is given by:
[tex]\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\][/tex]
Where [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, equal to [tex]\(n\) choose \(k\)[/tex], which is defined as:
[tex]\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\][/tex]
In this case, [tex]\(n = 12\) and \(y = 3\).[/tex] We're interested in the term where the exponent of [tex]\(x\) is 4, so \(n - k = 4\) or \(k = 12 - 4 = 8\).[/tex]Thus, we need to find the coefficient when [tex]\(k = 8\)[/tex]. So, the coefficient of the [tex]\(x^4\)[/tex] term is:
[tex]\[\binom{12}{8} = \frac{12!}{8!(12-8)!}\][/tex]
Calculating this:
[tex]\[\binom{12}{8} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = \frac{11880}{24} = 495\][/tex]
So, the coefficient of the [tex]\(x^4\)[/tex] term in the expansion of [tex]\((x + 3)^{12}\)[/tex] is 495.
I need some extra detailed steps please
8. Based upon a long period of record keeping the following represents the probability distribution of the number of times the John Jay wifi network is slow during a week. We call the random variable x.
x 0 1 2 3 4 5 6
p(x) .08 .17 .21 k .21 k .13
a. What is the value of k?
b. Calculate the expected value of x.
c. Calculate the expected value of x^2
.
d. Calculate the variance of x.
e. Calculate the standard deviation of x
f. Calculate the variance of 3x.
g. Suppose that network slowness is independent from week to week. What is the probability that if we look at 5 separate weeks, the network has no more than 4 slow times in any of those weeks?
h. Calculate the variance of the random variable x^2
Answer:k=2
Step-by-step explanation:
We want to build a box whose base is square, has no top and will enclose 100 m^3. determine the dimensions of the box that will minimize the amount of material needed to construct the box.
The dimensions of the box that minimizes material usage, enclosing 100 m³ with no top, are approximately 5.848 meters for the square base side length and 2.682 meters for the height.
Let's denote the side length of the square base as x meters and the height of the box as h meters. Since the box has no top, the volume (V) of the box is given by the product of the area of the square base and the height:
[tex]\[ V = x^2 \cdot h \][/tex]
Given that [tex]\(V = 100 \, \text{m}^3\)[/tex], we have the equation:
[tex]\[ 100 = x^2 \cdot h \][/tex]
Now, we want to minimize the amount of material needed to construct the box, which is the surface area (A) of the box. The surface area is the sum of the area of the square base and the areas of the four sides:
[tex]\[ A = x^2 + 4xh \][/tex]
To minimize A, we can express h in terms of x from the volume equation and substitute it into the surface area equation:
[tex]\[ h = \frac{100}{x^2} \]\[ A(x) = x^2 + 4x\left(\frac{100}{x^2}\right) \]\[ A(x) = x^2 + \frac{400}{x} \][/tex]
Now, to find the minimum amount of material, we take the derivative of A with respect to x and set it equal to zero:
[tex]\[ \frac{dA}{dx} = 2x - \frac{400}{x^2} = 0 \][/tex]
Multiply through by x^2 to get rid of the fraction:
[tex]\[ 2x^3 - 400 = 0 \][/tex]
Solve for x:
[tex]\[ x^3 = 200 \]\[ x = \sqrt[3]{200} \][/tex]
Now that we have x, we can find h using the volume equation:
[tex]\[ h = \frac{100}{x^2} \]\[ h = \frac{100}{(\sqrt[3]{200})^2} \]\[ h = \frac{100}{\sqrt[3]{400}} \][/tex]
The dimensions of the box that will minimize the amount of material needed are approximately [tex]\(x \approx 5.848\)[/tex] meters and [tex]\(h \approx 2.682\)[/tex] meters.
The results of an independent-measures research study are reported as "t(22) = 2.12, p < .05, two tails." for this study, what t values formed the boundaries for the critical region?
As the level of significance here is 0.05 because we are comparing the p-value with 0.05. Therefore the critical region boundaries here would be given as:
For 5 degrees of freedom, we get: ( from the t-distribution tables )
P(t_5 < 2.571) = 0.975
Therefore due to symmetry, we get:
P(-2.571 < t_5 < 2.571) = 0.95
Therefore the critical region here would be given as: +2.571 and -2.571
The critical values for the t-distribution are used to define the boundaries for the critical region in a hypothesis test. In this case, the boundaries are t < -2.073 or t > 2.073.
Explanation:The critical values for the t-distribution are used to define the boundaries for the critical region in a hypothesis test. In this case, the results of the study are reported as t(22) = 2.12, p < .05, two tails. To find the boundaries for the critical region, we need to look up the critical value for a two-tailed test with 22 degrees of freedom and a significance level of 0.05.
Using a t-distribution table or a calculator, we find that the critical value is approximately 2.073. Therefore, the boundaries for the critical region in this study are t < -2.073 or t > 2.073.
Any calculated t-value that falls outside of these boundaries would lead to rejecting the null hypothesis and concluding that the variables are significantly correlated.