a. Using Taylor series d(x³eˣ²)/dx about x = 0 is x⁴.
b. Using Taylor series d⁷(x³eˣ²)/dx⁷ about x = 0 is x¹⁰.
What is a Taylor series expansion?A Taylor series is a polynomial expansion of a function about a given point. It is given by f(x - a) = ∑(x - a)ⁿfⁿ(x - a)/n! where
a = point where f(x) is evaluated fⁿ(a) = nth derivative of f(x) about a and n is a positive integerGiven that the Taylor series of the function f(x) = x³eˣ² about x = 0 is
f(x) = x³ + x⁵ + x⁷/2! + x⁹/3! + x¹¹/4!, (1) we proceed to find the given variables
a. To find d( x³eˣ²)/dx about x = 0, the Taylor series expansion about x = 0 is given by
f(x - a) = ∑(x - a)ⁿfⁿ(a)/n!
f(x - 0) = ∑(x - 0)ⁿf(0)/n!
f(x) = ∑xⁿf(0)/n!
f(x) = x⁰f(x)/0! + xf(x)/1! + x²f(x)/2! + x³f(x)/3! + ....
f(x) = f(x) + xf¹(x) + x²f²(x)/2! + x³f³(x)/3! + ....(2)
Since fⁿ(x) is the nth derivative of f(x), and we desire f¹(x) which is the first derivative of f(x). Comparing equations (1) and (2), we have that
x⁵ = xf¹(x)
f¹(x) = x⁵/x
= x⁴
So, d( x³eˣ²)/dx about x = 0 is x⁴.
b. To find d⁷( x³eˣ²)/dx⁷ about x = 0, the Taylor series expansion about x = 0 is given by
f(x - a) = ∑(x - a)ⁿfⁿ(a)/n!
f(x - 0) = ∑(x - 0)ⁿf(0)/n!
f(x) = ∑xⁿf(0)/n!
f(x) = x⁰f(x)/0! + xf(x)/1! + x²f(x)/2! + x³f(x)/3! + ....
Expanding it up to the 8 th term, we have that
f(x) = f(x) + xf¹(x) + x²f²(x)/2! + x³f³(x)/3! + x⁴f⁴(x)/4! + x⁵f⁵(x)/5! + x⁶f⁶(x)/6! + x⁷f⁷(x)/7!.....(3)
Now expanding equation (1) above to the 8th term by following the pattern, we have that
f(x) = x³ + x⁵ + x⁷/2! + x⁹/3! + x¹¹/4! + x¹³/5! + x¹⁵/6! + x¹⁷/7!.....(4)
Since fⁿ(x) is the nth derivative of f(x), and we desire f⁷(x) which is the seventh derivative of f(x). Comparing equations (3) and (4), we have that
x⁷f⁷(x)/7! = x¹⁷/7!
f⁷(x) = x¹⁷/x⁷
= x¹⁰
So, d⁷( x³eˣ²)/dx⁷ about x = 0 is x¹⁰.
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A company may acquire property, plant, equipment, and intangible assets for cash, in exchange for a deferred payment contract, by exchanging other assets, or by a combination of these methods. Identify six types of costs that should be capitalized as the cost of a parcel of land. For your answer, assume that the land has an existing building that is to be removed in the immediate future so that a new building can be constructed on the site. At what amount should a company record an asset acquired in exchange for a deferred payment contract? In general, at what amount should assets received in exchange for other nonmonetary assets be valued? Specifically, at what amount should a company value a new machine acquired by exchanging an older, similar machine and paying cash?
Six types of costs are:
Closing costsPurchase costBrokerage commissionsSurveying feesDemolition costsSite preparation costsHow the asset is to be recordedFor assets attained in exchange for a deferred payment contract, the asset should be documented at its fair worth, which is the prevailing market rate. If establishing the fair value of the asset proves to be impracticable, the asset should be recorded based on the current value of upcoming payments due under the deferred payment agreement, stated using an apposite interest rate.
Generally, assets acquired in exchange of other non-monetary commodities should be appraised at their fair worth. If the truthful appraisal of the exchanged assets is uncertain, the assets should be noted down according to the seized amount of the items provided, including any received or paid cash during the transaction.
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The scale of this blueprint of an art gallery is 1 in 48 ft. find the actual lengths of the following walls.
The calculated values of the actual lengths of the walls are AB = 144 ft, CD = 288 ft, EF = 528 ft and EF = 240 ft
Finding the actual lengths of the walls.From the question, we have the following parameters that can be used in our computation:
Scale of blueprint = 1 in : 48 ft
This means that
Scale = 48 ft/1in
Calculating the actual lengths of the walls, we have
AB = 3in * 48 ft/1in
AB = 144 ft
CD = 6in * 48 ft/1in
CD = 288 ft
EF = 11in * 48 ft/1in
EF = 528 ft
FG = 5in * 48 ft/1in
EF = 240 ft
Hence, the actual lengths of the walls are AB = 144 ft, CD = 288 ft, EF = 528 ft and EF = 240 ft
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 how many kilograms are in 14500 grams
Answer: 14.5 kg
Step-by-step explanation:
Helpppppppp pleaseeeeee
The value of the product of -4 row one and row 3 is [90 -9 -1]
What is the product of a matrix?A matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or expression.
The given matrix is
[tex]\left[\begin{array}{ccc}1&2&1\\0&4&-2\\4&-1&6\end{array}\right] \left[\begin{array}{ccc}-5&\\3\\-8&\end{array}\right][/tex]
the product is given as
-4[1 2 1] + [4 -1 3]
= [ -4 -8 -4] + [ 4 -1 3]
= [90 -9 -1]
The sum of the product is [90 -9 -1]
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if (-8,3) lies on the circle and its Center is (-4,3) find the radius
If (-8,3) lies on the circle and its Center is (-4,3) then, the radius of the circle is 4.
We know that the center of the circle is (-4, 3). Let's call the radius of the circle "r".
The distance between the center of the circle and the point (-8, 3) on the circle is equal to the radius "r".
Using the distance formula, we can find the distance between these two points;
d =√[(x2 - x1)² + (y2 - y1)²]
d = √[(-8 - (-4))² + (3 - 3)²]
d = √[(-8 + 4)² + 0²]
d = √[4²]
d = 4
Since the distance between the center of the circle and the point on the circle is equal to the radius, we have;
r = d = 4
Therefore, the radius of the circle is 4.
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1 Soo divides a square into sixths. She colors one of the
equal parts red. What fraction of the square is red?
The fraction of the square that is red is 1/6.
Define fractionA fraction is a number that represents a part or parts of a whole. It consists of two numbers separated by a horizontal or diagonal line, with the number above the line (numerator) representing the part and the number below the line (denominator) representing the whole. For example, the fraction 3/4 represents three parts out of a whole that is divided into four equal parts.
A square divided into six equal parts has six equal sixths.
If Soo colors one of these equal parts red, then she has colored one-sixth of the square red.
Therefore, the fraction of the square that is red is 1/6.
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You are making scarves for presents. Each scarf needs 5/6 yd of fabric. How many yards of fabric do you need for 6 scarves?
What is the value of sine at 330
An angle measures 125°. Through what fraction of a circle does the angle turn?
If an angle measure 125° then the angle measures a fraction of 25/72 of a full circle.
What is a circle?A circle is a two-dimensional geometric shape that is defined as the set of all points that are equidistant from a single point, called the center. The distance from the center to any point on the circle is called the radius, and the distance across the circle, passing through the center, is called the diameter. The circumference of a circle is the distance around the edge or perimeter of the circle. The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle, and π (pi) is a mathematical constant that represents the ratio of the circumference to the diameter of a circle, approximately equal to 3.14159.
According to the given informationA circle has 360 degrees. To find what fraction of a circle an angle measures, we can divide the angle by 360. In this case, the angle measures 125 degrees, so the fraction of a circle it turns can be calculated as:
125/360 = 0.347222222...
To simplify this fraction, we can multiply both the numerator and denominator by 2:
125/360 = (1252)/(3602) = 250/720
We can further simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 10:
250/720 = (2510)/(7210) = 25/72
Therefore, the angle measures a fraction of 25/72 of a full circle.
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0.6/(-0.9- -0.5 x -0.3)
The value of the expression is -0.8
What is PEDMAS?PEDMAS can be described as a mathematical acronym that is used for the representation of arithmetic operations on the basis of how they are to be carried out sequentially in an equation.
The alphabet represents different operations. They are;
P represents parenthesesE represents exponentsD represents divisionM represents multiplicationA represents additionS represents subtractionFrom the information given, we have that;
0.6/(-0.9- -0.5 x -0.3)
Multiply the values
0.6/(-0.9 +0. 15)
add the values
0.6/0. 75
Now, divide the values
-0.8
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-3k - 7 ≤ 17 help!!!!!!!!!!!!!!!
In AJKL, m/J = (8x - 7), m/K = (x + 7)°, and m/L= (2x + 15)°. What
is the value of x?
Therefore, the equation for that value x = 6, and m(A) = 90.
What is a formula or equation?A mathematical equation expresses two things as being equal to one another, or as having the same value and worth. A specific equation that expresses a significant link between variables and numbers is called a formula.
The fact that the sum of a triangle's angles is 180 degrees must be used to determine the value of x. Angles J, A, and K in the triangle AJK allow us to write:
m(J) + m(A) + m(K) = 180
We can substitute the given angle measures into this equation:
(8x - 7) + m(A) + (x + 7) = 180
Simplifying this equation, we get:
9x + m(A) = 180 - 7 - 7
9x + m(A) = 166
We can use the same reasoning for triangle AJL:
m(J) + m(A) + m(L) = 180
Substituting the given angle measures, we get:
(8x - 7) + m(A) + (2x + 15) = 180
Simplifying this equation, we get:
10x + m(A) = 172
The two unknowns in our current set of equations are x and m(A). By removing the first equation from the second, we can find m(A):
10x + m(A) = 172
(9x + m(A) = 166)
x = 6
Now that we know x, we can substitute it into either equation to find m(A):
8x - 7 + m(A) + x + 7 = 180
15x + m(A) = 180
m(A) = 180 - 15x
Substituting x = 6, we get:
m(A) = 180 - 15(6) = 90
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a 10 x 10 x 10 cub is painter red on all faces and then cut into ten 10 x 10 x 1 slices each slice is then cut into twenty five 2 x 2 x 1 blocks how many of the 250 blocks have exactly on face painted red
There are exactly 250 blocks with exactly one face painted red.
What is cube ?
A cube is a three-dimensional solid shape with six square faces of equal size that meet at 90-degree angles. It is a special case of a rectangular prism, where all of the edges have the same length.
When the 10 x 10 x 10 cube is painted red on all faces, it means that all of the 1000 unit cubes inside the large cube are red.
When the cube is cut into ten 10 x 10 x 1 slices, each slice will contain 100 unit cubes. When each slice is cut into twenty-five 2 x 2 x 1 blocks, each block will contain 4 unit cubes.
Therefore, each 2 x 2 x 1 block will come from 4 unit cubes, and each unit cube will contribute to 4 different blocks.
To count the number of blocks with exactly one face painted red, we can focus on the blocks that have a red face on the bottom (or top) and three unpainted faces on the other sides. Each such block will correspond to one of the 100 unit cubes in a slice, and there are 10 slices, so there are 1000 such blocks in total.
However, each of these blocks has four unit cubes, so we need to divide by 4 to get the number of blocks. Therefore, the answer is:
1000 ÷ 4 = 250
Therefore, there are exactly 250 blocks with exactly one face painted red.
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I would love some help please 11-13
(11) The value of expression 3/y + 2y/4 when y = 4 is [tex]2\frac{3}{4}[/tex].
Hence the correct option is (b).
(12) The value of expression 12/y + 3y/4 when y = 8 is [tex]7\frac{1}{2}[/tex].
Hence the correct option is (c).
(13) 2x/3 + 4 = 10 equation does x = 9.
Hence the correct option is (b).
(11) The given expression is,
3/y + 2y/4 = 3/y + y/2
Substituting the value of y = 4 we get,
3/4 + 4/2 = 3/4 + 2 = [tex]2\frac{3}{4}[/tex].
Hence the correct option is (b).
(12) The given expression is,
12/y + 3y/4
Substituting y = 8 in the given equation we get,
12/8 + (3*8)/4 = 3/2 + 6 = 1 + 1/2 + 6 = [tex]7\frac{1}{2}[/tex].
Hence the correct option is (c).
(13) Given the solution is x =9.
For first option: (2/3)*9 - 6 = 6 - 6 = 0
2x/3 - 6 = 12 does not satisfied by x = 9.
For second option: (2/3)*9 + 4 = 6 + 4 = 10
So, x = 9 satisfies 2x/3 + 4 = 10.
For third option: 3*9 - 12 = 27 - 12 = 15
So x = 9 does not satisfy 3x - 12 = 21.
For fourth option: 3*9 + 12 = 27 + 12 = 39
So x = 9 does not satisfy 3x + 12 = 19.
Hence the correct option is (b).
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Find the critical points of the function f (x) = x − 1 /x^2 + 3
there is no real number that can be raised to the third power to give a negative number, there are no real solutions to this equation. Therefore, there are no critical points of the function f(x) = x - 1/x² + 3.
what is critical points ?
In calculus, critical points are points on a function where its derivative is either zero or undefined. These points are important because they can indicate the location of maximum or minimum values of the function or points of inflection where the concavity of the function changes.
In the given question,
To find the critical points of the function f(x) = x - 1/x² + 3, we need to find the values of x where the derivative of the function is equal to zero or undefined.
First, we need to find the derivative of the function:
f'(x) = 1 + 2/x³
Next, we set the derivative equal to zero and solve for x:
1 + 2/x³ = 0
2/x³ = -1
x³ = -2
Since there is no real number that can be raised to the third power to give a negative number, there are no real solutions to this equation. Therefore, there are no critical points of the function f(x) = x - 1/x² + 3.
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On average, a front desk operator receives 2 calls per 30 seconds.
(1)Find the probability that at most 1 call is received in 10 seconds
The probability that at most 1 call is received in 10 seconds is given as follows:
0.8557 = 85.57%.
What is the Poisson distribution?In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following mass probability function:
[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]
The parameters are listed and explained as follows:
x is the number of successes that we want to find the probability of.e = 2.71828 is the Euler number[tex]\mu[/tex] is the mean in the given interval or range of values of the input parameter.On average, a front desk operator receives 2 calls per 30 seconds, hence the mean for 10 seconds, which is one third of the time, is given as follows:
[tex]\mu = \frac{2}{3}[/tex]
The probability of at most one call is given as follows:
P(X <= 1) = P(X = 0) + P(X = 1)
Hence:
[tex]P(X = 0) = \frac{e^{-\frac{2}{3}}\frac{2}{3}^{0}}{(0)!} = 0.5134[/tex][tex]P(X = 1) = \frac{e^{-\frac{2}{3}}\frac{2}{3}^{1}}{(1)!} = 0.3423[/tex]Hence:
P(X <= 1) = P(X = 0) + P(X = 1) = 0.5134 + 0.3423 = 0.8557 = 85.57%.
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Find the measure of the angle between the two vectors. u = -7i and v = -4i + 4j
6. Ari, a gardener, estimates the cost of seeding a 150 m by 210 m area with grass seed. He needs 3 pounds of seed per 100 000 square feet. How many pounds of seed will Ari need?
Ari will need 3.6 pound of seed.
We are Given that total number of pounds of grass seed needed
9
Let number of pounds of Bermuda seeds is x
Let number of pounds of Fescue seeds is y
Then sum of both gives equation:
x+y=9
y=9-x...(i)
Given that cost of 1 pound of Bermuda seed = $4.80
Then cost of x pound of Bermuda seed = $4.80x
The cost of 1 pound of Fescue seed = $3.50
Then cost of y pound of Fescue seed = $3.50y
The total cost will be 4.80x + 3.50y
Now combining all three parts, we get equation:
4.80x+3.50y=4.02 x 9
4.80x+3.50y=36.18...(ii)
4.80x+3.50y=36.18
4.80x+3.50(9-x)=36.18
4.80x+31.5-3.50x=36.18
4.80x-3.50x=36.18-31.5
1.3x=4.68
x=3.6
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Find the missing side lengths. Leave your answers as radicals in simplest form.
The value of m and n are 2√3 and 4√3/3 respectively.
What is trigonometric ratio?The trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.
Sin(tetha) = opp/hyp
cos(tetha) = adj/hyp
tan(tetha) = opp/adj
There are special angles in trigonometry, examples are; 60° , 30° and 90°. This angles have exact values and can be calculated without using calculator.
Tan 60 = n/2
√3 = n/2
n = 2√3
cos 60 = 2/m
√3/2 = 2/m
m√3 = 4
m = 4/√3
= 4√3/3
Therefore the value of m and n are 2√3 and 4√3/3 respectively.
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4. What is the equation of the given graph?
1
A) y=-2x-3
B) y = -1/2x +3
C) y=1/2x-3
D)y=1/2x-3
Answer: D
Step-by-step explanation:
The equation for a linear function is y = mx + b
We first need to find the slope of the object, we can see here than it perfectly lands on the point, (0, -3) and (6,0). We can use this formula to determine the slope:
(y2 - y1)/(x2 - x1)
Now let's plug in the numbers.
(0 + 3) / (6 - 0)
3/6
1/2
Now we know the slope is 1/2 and the y-intercept is -3, so let's make it into a slope-intercept form.
y = 1/2x - 3
Please help with these two equations, and please show work as well, thank you!
The simplified polynomial for the area and perimeter of the rectangles are:
13). Area = 5x² + 40, Perimeter = 12x + 16
14). Area = x² + 10x + 12, Perimeter = 4x + 20
How to evaluate for the area and perimeter of the rectanglesArea of rectangle = Length × Width
Perimeter of rectangle = 2(Length + Width)
13). Area of the rectangle = 5x × (x + 8)
Area of the rectangle = 5x² + 40
Perimeter of the rectangle = 2[5x + (x + 8)]
Perimeter of the rectangle = 2(6x + 8)
Perimeter of the rectangle = 12x + 16
12). Area of the rectangle = (x + 3)(x + 7)
Area of the rectangle = x² + 7x + 3x + 21
Area of the rectangle = x² + 10x + 21
Perimeter of the rectangle = 2[(x + 3) + (x + 7)]
Perimeter of the rectangle = 2(2x + 10)
Perimeter of the rectangle = 4x + 20.
Therefore, the simplified polynomial for the area and perimeter of the rectangles are:
13). Area = 5x² + 40, Perimeter = 12x + 16
14). Area = x² + 10x + 12, Perimeter = 4x + 20
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Suppose a person were to breath an average of 13 times per minute. How many days would it take for them to breathe one billion times? Round your answer to the nearest single days.
Answer:
12 / min
720 / hr
17,280 / day
6,307,200 / yr
Now divide 1,000,000,000 by 6,307,200 (darn close to your 158!)
Solve the following system of equations.
The solutions for the system of equations " 3x²-2y²+5=0 and 2x²-y²+2=0" are :
First Solution : x = 1, y = 2
Second Solution : x = 1, y = -2
Third Solution : x = -1, y = 2
Fourth Solution : x = -1 , y = 2.
In order to solve the system of equations 3x²-2y²+5=0 and 2x²-y²+2=0, we use substitution method;
From the second equation, we solve for y² in terms of x²:
The second equation is : 2x² - y² + 2 = 0;
We get, y² = 2x² + 2
Substitute this expression for y² into the first equation:
We get,
3x² - 2(2x² + 2) + 5 = 0
3x² - 4x² - 4 + 5 = 0
-x² + 1 = 0
x² = 1
We get, x = ±1
Substitute these values of x into the equation for y²:
y² = 2x² + 2
For x = 1, y² = 4,
Which means the solution set is (1,2) and (1,-2),
For x = -1, y² = 4
the solution set is (-1,2) and (-1,-2).
Therefore, the solutions to the system of equations are (1, 2), (1,-2), (-1,2) and (-1, 2).
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using calculations show that the height of the barrel of oil is 96.82cm
Answer:
Step-by-step explanation:
V = πr²h
h = V/(πr²)
V = 42 gal
42 gal x 3.7854 l/gal = 158.987 l
158.987 l = 158,987 ml = 158,987 cm³
r = 18/2 = 9 in
9 in x 2.54 cm/in = 22.86 cm
h = (158987 cm³) / π(22.86 cm)² ≈ 96.82 cm
depending on how you round, the more precise answer is 96.8285 ≈ 96.83
Man I don’t even know how to solve this, can y’all help me???
Answer:
3
Step-by-step explanation:
The inverse function returns the original value for which a function gave the output.
This means if there is a function f(x) and at x = a, f(a) = k then f⁻¹(k) = a
This may be difficult to grasp with variables but essentially returning to the specific question
if f(3) = 17
then
f⁻¹(17) = 3
And that is the answer
100 POINTS!!!!!! please answer will give brainliest If the length of each side of the right triangle shown on the grid is measured in cm; find the area of the triangle: 25 cm2 50 cm2 100 cm? 225 cm?'
Answer:
B) 50cm^2
Step-by-step explanation:
Pyramid A and Pyramid B are similar. Pyramid A has a volume of 648m° and Pyramid B has a volume of 1029m?. What is the ratio of the surface areas of Pyramid A to Pyramid B?
The ratio of the surface area of Pyramid A to Pyramid B is: 36/49
We have the information from the question is:
Pyramid A and Pyramid B are similar.
The volume of Pyramid A is 648 [tex]m^3[/tex]
The volume of Pyramid B is 1029 [tex]m^3[/tex]
To find the ratio of the surface areas of Pyramid A to Pyramid B.
Now, As we know that:
If two solids are similar, then the ratio of their volumes is equal to the cube
of the ratio of their corresponding linear measures.
[tex]\frac{Vol of A}{Vol of B} =(\frac{a}{b} )^3 =\frac{684}{1029}=\frac{216}{343}\\ \\ \frac{a}{b}=\frac{6}{7}[/tex]
If two solids are similar, then the n ratio of their surface areas is equal to the square of the ratio of their corresponding linear measures.
Surface area of A/ Surface area of B [tex]=(\frac{a}{b} )^2=\frac{36}{49}[/tex]
So, the ratio of the surface area of Pyramid A to Pyramid B is: 36/49
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At a discount store,all DVDs have the same price. Chan bought 6 DVDs and paid a total of $62.87,which includes $2.99 in sales tax.What is the first step in finding the cost of one DVD? Complete the drip diagram to solve the first step. Please help!!!
Step-by-step explanation:
Answer
The price of 6 DVDs, including sales tax = $62.87
Sales tax = $2.99
Required:
The cost of one DVD.
Explanation:
Step 1: Price of 6 DVDs excluding sales tax.
The price of 6 DVDs excluding sales tax = $62.87 - $2.99 = $59.88
Step 2: Use the Unitary Method:
According to the question,
6 DVDs price = $59.88
1 DVD price excluding sales tax = $ 59.88/6
What integer values of x make the statement -3/x lesser than -x/3 true?
x=0
x=2
x=3
Any interger greater than -3
Any integer less than -3
x=1
You need to choose more than 1 answer
The integer values that make the statement true are x = -3, -2, -1, 1, 2, 3
We have,
We can start by multiplying both sides of the inequality by -3x to get rid of the denominators:
-3/x < -x/3
Multiplying by -3x on both sides.
-9 < -x²
Rearranging.
x² < 9
Taking the square root of both sides.
|x| < 3
This means that x can take any integer value between -3 and 3, excluding 0.
Thus,
The integer values are x = -3, -2, -1, 1, 2, 3
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Hypothesis testing question.
From a sample of 30 patients, Divoc Health Group found that the mean days for a patient to discharge is 25 days. The hospitalization duration was assumed to follow a normal distribution. Last year, the mean days for a patient to discharge was 20 days with a standard deviation of 5 days. Divoc Health Group suspects that the mean days for a patient to discharge has increased because of Covid-19 cases and other health-related issues. Divoc Health Group decides to carry out a hypothesis test.
a) State the null and alternative hypotheses.
b) At a 5% significance level, is there sufficient evidence to conclude that the mean days for a patient to discharge has increased? Show all steps clearly. [Express your answers up to 3 decimal places]
Based on the information, Divoc Health Group now samples an additional 10 clients. The mean days for a patient to discharge were 22 days.
c) Using the new information, construct a 95% confidence interval for the mean days for a patient to discharge. [Express your answers up to 3 decimal places]
d) What is the minimum sample size required if Divoc Health Group wants to estimate the mean days for a patient to discharge to within 2 days of error with a 99% confidence?
a) Null: Mean = 20 days; Alt: Mean > 20 days. b) supported by t-value 6.708 > critical value at α=0.05. c) 95% confidence interval for mean patient discharge time is between 20.528 and 23.472 days, d) 99% confidence, a minimum sample size of 67 patients is required.
a) The null hypothesis is that the mean days for a patient to discharge is still 20 days, while the alternative hypothesis is that the mean days for a patient to discharge has increased and is now greater than 20 days.
Null hypothesis: 0: = 20
Alternative hypothesis: 1: > 20
b) We can use a one-sample t-test to test the hypothesis. We will use a significance level of 0.05.
The test statistic is given by:
t = (x - ) / (s / √n)
where x is the sample mean, is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
Here, x = 25, = 20, s = 5, and n = 30. Plugging these values into the formula, we get:
t = (25 - 20) / (5 / √30) = 6.708
The degrees of freedom (df) for the t-distribution is n - 1 = 29. Using a t-table or calculator, the p-value for a one-tailed test with df = 29 and t = 6.708 is < 0.0001.
Since the p-value is less than the significance level of 0.05, we reject the null hypothesis. There is sufficient evidence to conclude that the mean days for a patient to discharge has increased.
c) To construct a 95% confidence interval for the mean days for a patient to discharge, we use the formula:
CI = x ± tα/2 * (s/√n)
where x is the sample mean, s is the sample standard deviation, n is the sample size, and tα/2 is the critical value for a t-distribution with df = n-1 and α = 0.05/2 = 0.025.
Here, x = 22, s = 5, and n = 40 (30 from the first sample + 10 from the second sample). The critical value for tα/2 with df = 39 is 2.022.
Plugging these values into the formula, we get:
CI = 22 ± 2.022 * (5/√40) = (20.528, 23.472)
Therefore, we can be 95% confident that the true mean days for a patient to discharge is between 20.528 and 23.472 days.
d) To find the minimum sample size required to estimate the mean days for a patient to discharge to within 2 days of error with a 99% confidence, we use the formula:
n = (zα/2 * s / E)²
where zα/2 is the critical value for a z-distribution with α = 0.01/2 = 0.005, s is the sample standard deviation (which we assume is the same as before, i.e., s = 5), and E is the desired margin of error (which is 2 days).
Plugging in the values, we get:
n = (2.576 * 5 / 2)² = 66.18
Therefore, the minimum sample size required is 67 patients.
Read more about the hypothesis on:
https://brainly.com/question/29576929
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