The value of t from the given equation is 3.11.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
The given equation is 9t=27.99
The solution of an equation is the set of all values that, when substituted for unknowns, make an equation true.
Now, t=27.99/9
t=3.11
Therefore, the value of t is 3.11.
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A political scientist was interested in studying america's voting habits. So, he decided to make a least squares regression equation to predict the percentage of people in a state that would vote for obama in 2012 based on the percentage of people in a state that voted for obama in 2008. The least squares equation is y-hat = -4. 599 +1. 04x. In the state of wyoming, 32. 54% voted for obama in 2008; whereas, 27. 82% voted for him in 2012. What is the value of the residual?.
The residual in this case is 4.72%, which is calculated by taking the difference between the predicted value (32.54%) and the actual value (27.82%).
The political scientist's least squares regression equation predicted that 32.54% of people in Wyoming would vote for Obama in 2012. However, the actual percentage of people who voted for Obama in Wyoming in 2012 was 27.82%, which was lower than the predicted value. This difference between the predicted and actual value is the residual, which in this case is 4.72%. This suggests that the least squares equation was not very accurate in predicting the voting habits in Wyoming for the 2012 election.
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How To Convert 24 °C to °F
Answer: 75.2 °F
Step-by-step explanation: (24°C × 9/5) + 32 = 75.2°F
The calculation would be (24 x 9/5) + 32 = 75.2°F.
How To Convert 24 °C to °F?To convert 24°C to°F, you can use the following formula is°F = (°C x 9/5) + 32Celsius (°C) and Fahrenheit (°F) are both units used to measure temperature.The formula for converting from Celsius to Fahrenheit is based on the relationship between the two temperature scales.To convert from Celsius to Fahrenheit, you need to multiply the temperature in Celsius by 9/5, and then add 32.The result will be the temperature in Fahrenheit.For example, to convert 24°C to°F, the calculation would be (24 x 9/5) + 32 = 75.2°F.This conversion formula is commonly used in various industries, such as meteorology, chemistry, and thermodynamics, to convert temperatures between Celsius and Fahrenheit.It's important to keep in mind that the temperature scales are not linear, meaning that the same difference in temperature will represent a different change in the two temperature scales.To learn more about Celsius to Fahrenheit:
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help me pass alegbra class. the second screenshot is question 6.
The linear system of equations in the question are resolved as follows;
Question 4
The true statements are;
The point satisfies both equation 1 and 2The point is a solution to the systemQuestion 6
Solution: (1.5, -1.375)What is a linear system of equation?A linear system consists of two or more linear equations that operate simultaneously.
Question 4;
The system of equations is presented as follows;
Equation 1: 10·x - 2·y = 15
Equation 2: 4·x + y = 6
Plugging in the values of x = 1.5, y = 0, we get;
10 × 1.5 - 2 × 0 = 15
4 × 1.5 + 0 = 6
The point (1.5, 0), satisfies both equations in the system of equations, therefore, the point (1.5, 0) is a solution of the system of equations.
The true statements are therefore;
The point satisfies both equation 1 and 2The point is a solution of the systemQuestion 6;
The linear system of equation is presented as follows;
-4·x - 8·y = 5
x = 4·y + 7
The substitution method can be used by plugging in the value of x = 4·y + 7, in equation (1) as follows;
-4·x - 8·y = 5
-4·(4·y + 7) - 8·y = -24·y - 28 = 5
Adding 28 to both sides, we get;
-24·y - 28 + 28 = 5 + 28
-24·y + 0 = 5 + 28 = 33
-24·y = 33
y = 33/(-24) = -11/8
y = -11/8 = -1.375
x = 4 × (-11/8) + 7
x = -11/2 + 7 = (-11 + 14)/2 = 3/2
x = 3/2 = 1.5
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A company plans to build houses. The company will use 23
acres of land to build the houses. Each home will be built on acres. How many houses will the company build?
The number of houses that can be built are 94.
What is division?A division is a process of splitting a specific amount into equal parts.
Given that, a construction company plans to build houses. The company will use [tex]23\frac{1}{2}[/tex] acres of land to build houses. Each house will be built on a 1/4 acre lot.
We need to find the number of houses that can be built,
To find the number of houses, we will divide total area by area of one house.
Therefore,
Number of house = [tex]23\frac{1}{2}[/tex] ÷ 1/4
= 47/2 ÷ 1/4
= 47 x 2
= 94
Hence, the number of houses that can be built are 94.
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Find a power series representation for the function.
f(x) = x (1 + 8x)2
f(x) =[infinity] n = 0
Determine the radius of convergence, R.
R =
As a result, the radius of convergence, R, is equal to one. As a result, f(x) = x(1 + 8x)^2 = x(1 + 16x + 64x^2 +...) = x + 16x^2 + 64x^3 +... This is the function f's power series representation (x).
What is function?A function is an equation with just one solution for y for every x. A function produces exactly one output for each input of a certain type. Instead of y, it is usual to call a function f(x) or g(x). f(2) indicates that we should discover our function's value when x equals 2. Example.
Here,
The power series representation for f(x) = x(1 + 8x)^2 can be found by expanding the function (1 + 8x)^2 as a Taylor series centered at x = 0:
(1 + 8x)^2 = 1 + 16x + 64x^2 + ...
So, f(x) = x(1 + 8x)^2 = x(1 + 16x + 64x^2 + ...) = x + 16x^2 + 64x^3 + ...
This is the power series representation for the function f(x).
To determine the radius of convergence, R, we use the formula:
R = 1 / limsup |a_n|^(1/n)
where a_n is the nth coefficient in the power series and limsup refers to the limit superior of the sequence {|a_n|^(1/n)}.
In this case, a_n = 16n x^n, so |a_n|^(1/n) = |16n|^(1/n) * |x|^(1/n). Since |16n|^(1/n) is a constant, the limit superior is just equal to |x|^(1/n). Therefore,
R = 1 / limsup |a_n|^(1/n) = 1 / |x|^(1/n) = 1
So, the radius of convergence, R, is equal to 1. So, f(x) = x(1 + 8x)^2 = x(1 + 16x + 64x^2 + ...) = x + 16x^2 + 64x^3 + ... This is the power series representation for the function f(x).
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an event that has a probality of being 1 is said to be
Answer: Guaranteed
Step-by-step explanation:
Answer: Certain to happen
Hope this helps :)
Step-by-step explanation:
It's important to remember that when regarding probabilities with decimals, 0 is considered least likely, and 1 is most likely.
Determine the intervals on which the following function is continuous. f(x) = x + 6/x^2 - 36 The function is continuous on the interval(s) (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.)
Function is continuous everywhere expect x = 6 and x = -6
What does a math interval mean?
In mathematics, an interval is quantified in terms of numbers. The range of numbers between any two specific numbers is known as the interval.
Every real number between those two values is included in this range. All kinds of numbers, including imaginary ones, are real numbers.
The function will not be continuous if the denominator becomes zero .
f(x) = x+6/(x² - 36 )
to become denominator zero -
x² - 36 = 0
(x + 6) ( x - 6 ) =0
x = 6 or x = -6
so at x = 6 and x = -6 the function is not continuous .
the interval for continuity is -
= ( -∞ , - 6), ( -6 , 6) , ( 6, ∞ )
In other words function is continuous everywhere expect x = 6 and
x = -6
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There i a pattern 1 tree then 2 bead. If there are a total of 42 tree and bead, how many tree and how bead will you need?
If there is a pattern 1 tree then 2 bead. If there are a total of 42 tree and bead, then 84 bead is needed to complete the pattern as 42 tree and 84 bead.
What is pattern?A recurring arrangement of numbers, forms, colours, and other elements constitutes a pattern in mathematics. The Pattern can be connected to any kind of occasion or thing. When a group of numbers are arranged in a particular way, the arrangement is referred to as a pattern. Patterns can also occasionally be referred to as a series.Sequences and patterns. Patterns are common, repeatable sorts of practises or guidelines. Patterns can help us anticipate the future or what has already happened. come following a series of integers that are ordered specifically. A sequence is the name given to this collection of integers.Learn more about patterns refer to :
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how many kilograms are in a pound
0.453592 kilograms are in a pound.
How many kilograms are in a pound?
One pound is approximately equal to 0.45359237 kilograms (kg). The conversion of the pound to kilograms is done by multiplying the given pound value and 0.45359237. Also, one kilogram is approximately equal to 2.2046226218 lbs.One litre of water has a mass of almost exactly one kilogram when measured at its maximal density, which occurs at about 4 °C. It follows, therefore, that 1000th of a litre, known as one millilitre (1 mL), of water has a mass of about 1 g; 1000 litres of water has a mass of about 1000 kg (1 tonne or megagram).Pounds and kilograms are both units for measuring mass. Pound refers to an imperial system of measuring weight or mass. Kilogram, on the other hand, was adopted under the SI or International System of Units in 1959.To learn more about pound refers to:
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Your foot measures 10.8 inches. what proportion can you use to determine how many centimeters, x, it is? :A) 10.8in. 2.54in. xcm 1cm 2.54in. B) 10.8cm xin lcm C) 10.8in lin. хст 2.54cm D) 10.8in 2.54cm XCM lin
Your foot measure in centimeter is 4.25 centimeters and the proportion is 10.8 inches / 2.54 centimeters = x centimeters.
A proportion is an equation that states that two ratios are equal. It's often used to convert from one unit of measurement to another.
In your case, you want to convert 10.8 inches to centimeters, so you'll use a proportion to help you do that.
Here the correct proportion to use is 10.8 inches / 2.54 centimeters = x centimeters.
In this proportion, the first ratio on the left side of the equation represents 10.8 inches in terms of centimeters, and the second ratio on the right side of the equation represents x centimeters in terms of inches.
Since we know the conversion factor between inches and centimeters, 2.54 centimeters per inch, we can substitute that into the proportion.
=> 10.8 inches / 2.54 centimeters = x centimeters
=> 10.8 inches / 2.54 centimeters
=> 10.8 / 2.54 centimeters = 4.25 centimeters.
Therefore, 10.8 inches is equal to 4.25 centimeters.
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The variables x
and y
vary directly. Use the values to find the constant of proportionality, k
. Then write an equation that relates x
and y
.
y=72; x=3
Answer: k = 24, y = 24x
Step-by-step explanation:
y = kx (the variables x and y varies directly)
when x = 3, y = 72
k = y/x = 72/3 = 24
y = 24x
A personnel director interviewing 11 senior engineers forfour job openings has scheduled six interviews for the firstday and five for the second day of interviewing. Assumethat the candidates are interviewed in random order.
a. What is the probability that x of the top four candidatesare interviewed on the first day?
b. How many of the top four candidates can be expected tobe interviewed on the first day?
All parts of the probelm given regarding probability have been solved and provided below.
We have been told that a personnel director interviewed 11 senior engineers for four job openings and had scheduled six interviews for the first day and five for the second day of interviewing. Now let us assume that the candidates are interviewed in a random order.
(a) The probability that any quantity, lets say x, of the top four candidates are interviewed on the first day can be calculated using -
= h(x; 6,4,11) = (4;x) (7;6-x) / (11;6)
(b) The expected number of candidates, if top four candidates to be interviewed on the first day can be found out using -
= E[X]=nM/N
= 6*4 / 11
= 2.1812
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Deshawn wrote the following rule for the translation shown. What was his error?
T(-4, -3)(JKL)
The rule for the translation is (-3, -4).
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
A translation is a movement of the graph either horizontally parallel to the -axis or vertically parallel to the -axis.
JKL is the original figure.
J'K'L' is the transformed figure after applying cetain rule.
We obtain J'K'L' by translating the original image left 3 units and down by 4 units.
The rule should be (-3, -4)
Hence, the rule for the translation is (-3, -4).
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How can you add and subtract positive and negative numbers in complicated expressions?
When dealing with complicated expressions that involve the addition and subtraction of positive and negative numbers, it is important to remember the following rules.
What is numbers?Nobody's a mathematical object used to count level and measure it is in the fundamental concept in mathematics and you used in many different way number can be used to represent anything from quantity of object to ideas in the concept additionally number are used to describe relationship between different object or concept.
First, when adding or subtracting two numbers with the same sign, the sum or difference will have the same sign as the numbers. For example, adding 5 + 3 gives 8 and subtracting 5 – 3 gives 2, both of which are positive numbers.
Second, when adding or subtracting two numbers with different signs, the sum or difference will have the sign of the number with the largest absolute value. For example, adding 5 + (-3) gives 2 and subtracting 5 – (-3) gives 8, both of which are positive numbers.
Third, when multiplying and dividing positive and negative numbers, the product or quotient will have the sign of the number with the largest absolute value. For example, multiplying 5 x (-3) gives -15 and dividing 5 / (-3) gives -1.67, both of which are negative numbers.
Finally, when dealing with expressions with multiple operations, it is important to use the correct order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This allows for the correct evaluation of an expression.
By remembering these rules, it is possible to add and subtract positive and negative numbers in complicated expressions.
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I am confused on how to do the probability problem.
The probability that the selected ride will be closed is given as follows:
E. 0.77.
How to calculate a probability?A probability is calculated as the division of the number of desired outcomes by the number of total outcomes.
For this problem, the percentages are given, hence we must just identify the correct percentages.
The probability of the park being closed is defined as follows:
75%: closed because of ice weather.8% of 25%: closed because of mechanical problem when the weather is not icy.Hence the probability that the park is closed is obtained as follows:
p = 0.75 + 0.08 x 0.25
p = 0.75 + 0.02
p = 0.77.
Meaning that option E is the correct option.
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The probability of the selected ride being closed is 0.77
How to determine the probabilityLet R be the event that a ride is closed due to icy weather, and
M be the event that a ride is closed due to mechanical problems.
So, we have
The probability of R, P(R) = 0.75.The probability of a ride being closed due to M given R is P(M | R) = 0.90.The probability of a ride being closed due to M given that it is not R is P(M | R') = 0.08.Using the Law of Total Probability, we have:
P(M) = P(R) + P(M | R') * P(R')
So, we have
P(M) = 0.75 + 0.08 * (1 - 0.75)
Evaluate
P(M) = 0.77
Hence, the probability is 0.77
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please help I would really be happy
42 students are picking two activities to do over the weekend.5 picked painting and sport.
8 did not pick painting or sport.
Twice as many students picked sport than painting as one of their activities.Find the amount that picked sport and not painting.
Answer:
25/3
Step-by-step explanation:
Let's call the number of students who picked sport "s" and the number of students who picked painting "p". We know:
p + s = 5 (the number of students who picked painting and sport)
42 - p - s = 8 (the number of students who did not pick either)
s = 2p (twice as many picked sport as painting)
From the first equation, we can solve for p:
[tex]\sf:\implies{p + 2p = 5}[/tex]
[tex]\sf:\implies{3p = 5}[/tex]
[tex]\sf:\implies{p = 5/3}[/tex]
So, 5 students picked painting and 10 students picked sport. And the number of students who picked sport but not painting is:
s - p = 10 - 5/3 = 10 - 5/3 = 25/3.
axial force of 103 kn. if a safety factor against yielding of 1,5 is required, a) what is the lowest necessary yield strength for the axle material?
The lowest necessary yield strength for the axle material is 113.88 M/mm^2.
Torsional moment of Mv = 2×10^7 Nmm.
Bending moment of Mb = 1×10^7 Nmm
Axial force = 10^3 kN
Axial Stress = Axial Force/(πd^2/4)
Axial Stress = 10^3 × 10^3/(π(100)^2/4)
Axial Stress = 4 × 10^6/3.14 × 10^4
Simplifying
Axial Stress = 1.274 × 10^2
Axial Stress = 127.4
The combined tension of a torsional moment and a bending moment is known as shear stress.
τ = 16/πd^3√(Mv^2 + M^2)
τ = 16/π(100)^3√[(2×10^7)^2 + (1×10^7)^2]
τ = 16/π(10)^6√[4×10^14 + 10^14]
τ = 16/π(10)^6√[5×10^14]
After solving, we get
τ = 113.88 M/mm^2
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The complete question is:
The figure shows a circular axial made of steel that is loaded by a torsional moment of Mv=2×10^7 Nmm, a bending moment of Mb = 1×10^7 Nmm and an axial force of 10^3 kN. If a safety factor against yielding of 1,5 is required, what is the lowest necessary yield strength for the axle material?
A pizza place changed the size of their large pizzas from a 28-inch diameter to a 24-inch diameter. What is the percent change?
If we were constructing a box with length l , width w , and height = 8 cm, where the length of the box is twice the width, which could we use as a volume function?A)V = 16W2B)V =8w2C)V =16I2D)V = 16I
We can use (A) V = 16lw as a volume function.
Because the volume of a rectangular box is given by the product of its length, width, and height. In this case, the height is fixed at 8 cm, so the volume would be calculated as V = 16 x l x w, where l = 2w.
A three-dimensional space's volume is measured. t is frequently expressed numerically using units derived from SI (such the cubic Meter and litre) or by different imperial or US customary units (such as the gallon, quart, cubic inch). Volume and length (cubed) are related concepts.
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Which question can be represented by the equation 4 2/7 =
What is 4 groups of 2/7 s?
How many 2/7 s are in 4?
What is 2/7 of 4?
How many 4s are in 2/7?
The equation 4 2/7 = can be represented by the question: "What is 2/7 of 4?"
What is the arithmetic expressions?
An arithmetic expression is a mathematical phrase that can contain numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.
The equation 4 2/7 = is asking for the value of 2/7 times 4. The right side of the equation is a mathematical expression that represents the product of 2/7 and 4. The value of this expression can be found by performing the calculation and replacing the expression with the resulting numerical value. In this case, 2/7 of 4 can be found by dividing 2 by 7 and then multiplying that result by 4. The answer to this question would give the value of 2/7 times 4.
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what is a quadrilateral symmertrical across both diagonals
A rhombus has reflectional symmetry over either of its diagonals.
Define quadrilateral?In geometry, a quadrilateral is a four-sided polygon with four edges and four corners. The word is derived from the Latin words quadri (a variant of four) and latus (a side).Quadrilaterals are shapes with four sides and four corners, or vertices, and two of the sides are often parallel, which means they are always the same distance apart no matter how long they stretch. Squares and rectangles are the most common quadrilaterals, but there are also trapezoids, parallelograms, and kites.A quadrilateral is a closed shape and type of polygon with four sides, four vertices, and four angles. It is formed by connecting four non-collinear points.The sum of the interior angles of quadrilaterals is always 360 degrees.To learn more about quadrilateral refer to:
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how to determine how many rectangles we should use in order for our estimate to be accurate within some range.
To determine the number of rectangles needed to achieve a desired accuracy, one could start with a small number of rectangles, calculate the approximation error, and then increase the number of rectangles until the error is within the desired range.
How determine the number of rectangles?The number of rectangles to use in a rectangle approximation of a definite integral depends on several factors:
The desired accuracy: The smaller the desired accuracy, the more rectangles will be required.
The shape of the function: If the function is highly curved, more rectangles will be required to capture the shape of the function than if the function is more linear.
The range of integration: The larger the range of integration, the more rectangles will be required.
In general, to determine the number of rectangles needed to achieve a desired accuracy, one could start with a small number of rectangles, calculate the approximation error, and then increase the number of rectangles until the error is within the desired range. Alternatively, one can use mathematical techniques such as adaptive quadrature methods that automatically adjust the number of rectangles to achieve a desired accuracy.
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WILL GIVE BRAINLIEST TO FIRST ANSWER, EASY POINTS
Which of the following tables represents a linear relationship that is also proportional?
x −1 2 5
y −1 0 1
x −2 −1 0
y −1 0 1
x −2 −1 0
y −4 −2 0
x −4 −2 1
y −2 0 3
Answer:
x −2 −1 0
y −1 0 1
Step-by-step explanation:
Condition:
A linear relationship is proportional if the ratio of the change in the dependent variable to the change in the independent variable is constant. This constant is known as the proportionality constant and can be represented by "k" in the equation y = kx. When a linear relationship is proportional, there is a direct relationship between the variables and as one increases, the other increases and vice versa.
The second table (x = -2, -1, 0 and y = -1, 0, 1) represents a linear relationship that is also proportional. A proportional relationship means that the ratio of y to x remains constant, or in other words, y is a constant multiple of x. In this table, for every 1 unit increase in x, y increases by 1 unit, making it proportional.
The third table represents a linear relationship that is also proportional.
3.) Katy Perry's bank granted her a single-payment loan of $4,400 at an interest rate of 6% exact interest to buy
a new sound system. The term of the loan is 172 days. What is the exact interest and what is the maturity value
of the loan?
The interest on the loan is: $375.44
The maturity value is: $4,775.44.
How to Calculate Maturity Value and Interest on a Loan?Maturity value and interest on a loan can be calculated as follows:
Maturity value: it is the amount of the loan plus the interest charged on it. It can be calculated using the formula:
MV = P + I
where MV is the maturity value, P is the principal (original loan amount), and I is the interest charged on the loan.
Interest: it is the amount charged by the lender for the use of their money. It can be calculated using the formula:
I = P * r * t
where I is the interest, P is the principal, r is the interest rate, and t is the term of the loan (in years or days).
Therefore:
Exact interest: 4,400 * 6% * 172/365 = $375.44
Maturity value: 4,400 + 375.44 = $4,775.44
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Monthly maintenance costs are distributednormally with a μ=$250 and σ=$50. What percent of months have maintenancecosts in the range of $200 to $300?
68.26% percent of months have maintenance costs in the range of $200 to $300
How do you calculate maintenance costs?The monthly maintenance cost per square foot is calculated by dividing the total cost by the property's total area. It is multiplied by the unit area of each home to determine the individual maintenance contributions.
Your home's square footage can also be used to estimate your yearly maintenance expenses. If you follow this guideline, you should begin by figuring annual maintenance expenditures at $1 per square foot. Your estimated annual maintenance cost for a 2,000 square foot home is $2,000.
Costs incurred during normal measures to maintain an asset's original state are referred to as maintenance costs. Simple electrical repairs, bulb replacement, paint touch-ups, pool cleaning, etc. are a few examples of maintenance charges.
Remember that, [tex]P(X < A) = P(Z < (A - \mu )/\sigma)[/tex]
and given that \mu = [tex]$250\ and \ \sigma = \$50[/tex]
So, P(maintenance cost is in the range $200 to $300) = P(200 < X < 300)
= P(X < 300) - P(X < 200)
= P(Z < (300 - 250)/50) - P(Z < (200 - 250)/50)
= P(Z < 1) - P(Z < -1)
= 0.8413 - 0.1587
= 0.6826
= 68.26%
Thus, 68.26% percent of months have maintenance costs in the range of $200 to $300
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Pls Help me on this problem it is number 19 I have no idea what to do
Answer:
$81.15
Step-by-step explanation:
So to find the total amount of money, you take the original price and add the sales tax.
We need to find the sales tax first, which is telling us to multiply the original price by 0.065.. so lets do that
(79.95)(0.065) = 5.19675
Since we got that answer there, we add that to the original price
$79.95 + $5.19675 = $81.14675
Which rounds to $81.15
A sequence can be generated by using an=an−1+17, where a1=15 and n is a whole number greater than 1.
What are the first 5 terms in the sequence?
32, 49, 66, 83, 100
17, 32, 47, 62, 77
15, 32, 49, 66, 83
15, 30, 45, 60, 75
Answer:
Step-by-step explanation:
A sequence can be generated by using an=an−1+17, where a1=15 and n is a whole number greater than 1. The first 5 terms in the sequence are:
a2 = a1 + 17 = 15 + 17 = 32
a3 = a2 + 17 = 32 + 17 = 49
a4 = a3 + 17 = 49 + 17 = 66
a5 = a4 + 17 = 66 + 17 = 83
So the first 5 terms in the sequence are:
15, 32, 49, 66, 83
some of our farm fields are being left unused. does this have any implications for the economy's ppf diagram (with agricultural products on one axis and all other products on the other axis)?
Yes, the implications of leaving farm fields unused for the economy's PPF diagram can be seen in the output of agricultural products.
The PPF diagram shows the maximum output of two products that an economy can produce at a given time, and it is shaped like a curve. When farm fields are left unused, the amount of agricultural products that can be produced is decreased, resulting in a shift in the PPF curve inwards. This means that the economy has to sacrifice other goods to maintain the same level of production of agricultural products, leading to an opportunity cost. Thus, leaving farm fields unused has implications for the economy's PPF diagram, as it reduces the output of agricultural products, resulting in a shift in the PPF curve inwards.
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Consider the given matrix A and right-hand sides 6, and b2: [o.6 A = ~0.8 [ijp]. 6, = [o.8 [4 6= [1] 0.6 Find the pseudoinverse A+ Compute the projection matrices AtA and AA+ _ Find the ~best" solution z+ = A+b for each right-hand side. Which solutions fromn (c) arC exact? Which solutions are unique?
The "best" solution z+ = A^+b is the one that minimizes the 2-norm of the residual vector r = b - Az. In general, not all right-hand sides b will have a unique solution z+, and even if there is a solution, it may not be exact (i.e., Az ≠ b).
To find the pseudoinverse of a matrix A, the first step is to take the singular value decomposition (SVD) of A, which can be written as A = UΣV^T, where U and V are orthogonal matrices and Σ is a diagonal matrix with nonnegative singular values. The pseudoinverse of A is given by A^+ = VΣ^+U^T, where Σ^+ is the same as Σ except that the nonzero elements of Σ are replaced by their inverses.
Next, the projection matrices are given by AtA and AA+, which are used to project a vector onto the column space and the row space of A, respectively.
The "best" solution z+ = A^+b is the one that minimizes the 2-norm of the residual vector r = b - Az. In general, not all right-hand sides b will have a unique solution z+, and even if there is a solution, it may not be exact (i.e., Az ≠ b).
Whether a solution is exact or unique depends on the rank of A and the number of linearly independent columns and rows. If the rank of A is equal to the number of columns, then the solution is unique, otherwise, the solution may not be unique.
Therefore, The "best" solution z+ = A^+b is the one that minimizes the 2-norm of the residual vector r = b - Az. In general, not all right-hand sides b will have a unique solution z+, and even if there is a solution, it may not be exact (i.e., Az ≠ b).
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