Answer:
Step-by-step explanation:
To construct the data set, we need to consider the given measures of central tendency:
Mean: 9
Median: 8
Mode: 5
Midrange: 11.5
We can start by finding five values that are less than or equal to the median of 8, and five values that are greater than or equal to the median. Since the mode is 5, we can include it as one of the lower values.
Let's try to find the other four lower values that satisfy the given conditions:
The sum of all values is 10 * 9 = 90 (since there are 10 values and the mean is 9).
The mode is 5, so we need at least two values that are equal to 5.
The midrange is (max + min) / 2 = 11.5, so we need a maximum value that is 3 more than the minimum value.
Here's one possible set of 10 values that satisfy these conditions, listed in ascending order:
{2, 3, 4, 5, 5, 8, 9, 10, 12, 13}
To check that this set of values satisfies the given conditions:
The mean is (2+3+4+5+5+8+9+10+12+13) / 10 = 9.
The median is 8 (the middle value when the values are listed in ascending order).
The mode is 5 (the most common value).
The midrange is (13 + 2) / 2 = 7.5 + 4 = 11.5 (the average of the maximum and minimum values).
Determine the missing description.
The missing description in radical form is ∛64.
What is an exponent?In Mathematics, an exponent can be defined as a mathematical operation that is used in conjunction with an algebraic expression to raise a quantity to the power of another and it is generally written as;
bⁿ
Where:
the variables b and n represent numerical values or an algebraic expression.
Based on the information about the rational exponent form, we have the following:
64^{1/3} = 4
∛64 = 4.
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I want to know what the answer to my equation is
At 4.0 megabits per second, Able Cable provides the fastest average downloading speed.
To find out which company offers the fastest mean downloading speed, we need to calculate the mean download speed for each provider and then compare the results.
The mean download speed for CityNet is:
(3.6 + 3.7 + 3.7 + 3.6 + 3.9) / 5 = 3.7 megabits per second
The mean download speed for Able Cable is:
(3.9 + 3.9 + 4.1 + 4.0 + 4.1) / 5 = 4.0 megabits per second
The mean download speed for Tel-N-Net is:
(3.9 + 3.7 + 4.0 + 3.6 + 3.8) / 5 = 3.8 megabits per second
Therefore, Able Cable offers the fastest mean downloading speed at 4.0 megabits per second.
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a/(a−b)(a−c)+b/(b+c)(b−a)b+c/(a−c)(b−c)
Answer:
a^2-ac/a-b + b^3-b^2*a/b+c + bc-c^2/a-c
Step-by-step explanation:
12. The large triangular figure below is composed of triangles that each have a base and height of 4 cm.
What is the area of the large figure?
A. 64 cm2
B. 72 cm 2
C. 128 cm 2
D. 144 cm2
By answering the above question, we may state that The solution is triangle closest to option d: 58% x, 42% y.
What precisely is a triangle?A triangle is a polygon because it contains four or more parts. It features a simple rectangular shape. A triangle ABC is a rectangle with the edges A, B, and C. When the sides are not collinear, Euclidean geometry produces a single plane and cube. If a triangle contains three components and three angles, it is a polygon. The corners are the points where the three edges of a triangle meet. The sides of a triangle sum up to 180 degrees.
The total sales of product x may be computed as follows: 5,000 units * $110/unit selling price = $550,000
The total sales of product y may be computed as follows: 35,000 units * $70/unit selling price = $2,450,000
The total sales of both goods are as follows: $550,000 + $2,450,000 = $3,000,000
Hence the proportion of revenue provided by product x is: $550,000 / $3,000,000 = 0.1833 or 18.33%
And the proportion of sales produced by product y is: $2,450,000 / $3,000,000 = 0.8167 or 81.67%
As a result, Rusty Co.'s sales mix was 18.33% x and 81.67% y last year.
The solution is closest to option d: 58% x, 42% y.
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Need help with question
A triangular prism is 32 centimeters long and has a triangular face with a base of 30 centimeters and a height of 20 centimeters. The other two sides of the triangle are each 25 centimeters. What is the surface area of the triangular prism?
The surface area of the triangular prism is 1880 square centimeters.
How did we get the value?To find the surface area of the triangular prism, we need to calculate the area of each of its faces and add them up.
First, let's calculate the area of the triangular base. The formula for the area of a triangle is:
Area = (base x height) / 2
Substituting the given values, we get:
Area = (30 x 20) / 2 = 300 cm²
Since the triangular prism has two identical triangular faces, the total area of the two triangular faces is:
2 x 300 cm² = 600 cm²
Now, let's calculate the area of the rectangular faces. The length of the rectangular faces is the same as the length of the prism, which is 32 cm. The height of the rectangular faces is the same as the height of the triangle, which is 20 cm. The formula for the area of a rectangle is:
Area = length x height
Substituting the given values, we get:
Area = 32 x 20 = 640 cm²
Since the triangular prism has two identical rectangular faces, the total area of the two rectangular faces is:
2 x 640 cm² = 1280 cm²
Finally, to find the total surface area of the triangular prism, we add the areas of the two triangular faces and the two rectangular faces:
600 cm² + 1280 cm² = 1880 cm²
Therefore, the surface area of the triangular prism is 1880 square centimeters.
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Find the average rate of change of the function f(x)
f(x) = 10 from [2, 3].
x-4
Answer:
Step-by-step explanation:
The formula for average rate of change of a function f(x) over an interval [a, b] is given by:
Average rate of change = [f(b) - f(a)] / (b - a)
In this case, we are given the function f(x) = 10 and the interval [2, 3]. So, a = 2 and b = 3. Substituting these values, we get:
Average rate of change = [f(3) - f(2)] / (3 - 2)
= [(10 x 3 - 4) - (10 x 2 - 4)] / (3 - 2)
= (30 - 20) / 1
= 10
Therefore, the average rate of change of the function f(x) = 10 from [2, 3] is 10.
Question 9 Explain work
The condition that proves congruent of two given triangles ΔDEF ∼ ΔJKL would be option (a) DE : JK = 3:1 but ratio value in option is given wrong.
What is the congruent triangle?In this case triangle in congruent when two sides and one angle of first triangle is equal to two sides and one angle of second triangle.
we have given proof two triangles congruent by the postulate of SAS.
Since two corresponding sides of these triangles are congruent and one angle is also congruent. So to prove both the triangles congruent.
∠EDF ≅ ∠KJL will complete the SAS property to prove both the triangles are congruent.
DE : JK
= 21 : 7
= 3 : 1
So, DE: JK = 3:1 is answer.
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349 ×10 to the 5th power =
Answer:
34,900,000
Step-by-step explanation:
349 × [tex]10^{5}[/tex]
[tex]10^{5}[/tex] = 100,000
349 x 100000 = 34,900,000
So, the answer is 34,900,000
One of the requirements for becoming an associate scientist in a certain lab is the ability to accurately process 100 specimen slides per hour. Ibragim can currently process 65 slides per hour and believes that with practice he can increase his processing speed by 2 slides per hour each week. Which of the following represents the number of slides per hour that Ibragim believes he will be able to process w weeks from now?
The number of slides per hour that Ibragim believes he will be able to process w weeks from now is represented by the following calculation: 65 + 2w.
What is number?Number is an abstract concept used to describe a quantity or amount. It can refer to a countable quantity of objects, or it can refer to an abstract concept such as size, direction, or position. In mathematics, number is used as a way of measuring and describing a quantity or amount. It is used to represent quantities in equations and formulas, to compare quantities, and to measure and compare distances. Number is also used to represent relationships between different objects, such as in geometry where angles, lines, and points are all represented by numbers.
This equation shows that Ibragim's slides per hour processing speed will increase by 2 for each week that passes. For example, if Ibragim practises for 4 weeks, his slides per hour processing speed will have increased from 65 to 73. This is because 65 + (2 x 4) = 73. Therefore, if Ibragim practises for w weeks, then the number of slides per hour he believes he will be able to process will be 65 + 2w.
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The expression 65 + 2w represents the number of slides per hour that Ibragim believes he will be able to process w weeks from now. This expression can be used to calculate the number of slides per hour that Ibragim believes he will be able to process after any number of weeks of practice.
What is number?Number is an abstract concept used to describe a quantity or amount. It can refer to a countable quantity of objects, or it can refer to an abstract concept such as size, direction, or position. In mathematics, number is used as a way of measuring and describing a quantity or amount. It is used to represent quantities in equations and formulas, to compare quantities, and to measure and compare distances. Number is also used to represent relationships between different objects, such as in geometry where angles, lines, and points are all represented by numbers.
The number of slides per hour that Ibragim believes he will be able to process w weeks from now is given by the expression 65 + 2w. In this expression, 65 represents the number of slides per hour that Ibragim can currently process, and 2w represents the number of slides per hour that Ibragim believes he can increase his processing speed by each week.
For example, if Ibragim believes he will be able to process slides at a rate of 100 per hour after 6 weeks of practice, then the expression would be 65 + 2(6) = 77. This means that Ibragim believes he will be able to process 77 slides per hour after 6 weeks of practice. In general, if w represents the number of weeks of practice, then the expression 65 + 2w represents the number of slides per hour that Ibragim believes he will be able to process w weeks from now.
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Complete questions as follows-
Mrs. Juarez graded ten English papers and recorded the scores. 92, 95, 100, 62, 88, 90, 100, 96, 89, 98 Which statements are true? Check all that apply. The range of scores is 38. Without the outlier, the range of scores would be 12. The outlier impacts the range more than it impacts the interquartile range. The interquartile range is 9. The interquartile range is 4. Without the outlier, the interquartile range would be 9.5. Mark this and return Save and Exit Next 4655
The interquartile range, according to the provided assertion, is 9.5.
The interquartile range is what?
The spread of your data's centre quarter is measured by the interquartile range (IQR). It is the limit for your sample's centre 50%. Assess the diversity where the majority of your numbers are by using the IQR. Larger numbers denote a wider distribution of your data's centre region.
The following statements are true:
The range of scores is 38.
The spread of results without the outlier would be 12.
The interquartile range is less affected by the anomaly than the range.
9 is the interquartile number.
The interquartile range would indeed be 9.5 if there were no anomaly.
To calculate the range, we subtract the smallest score from the largest score
Range = 100 - 62 = 38
If we remove the outlier, 62, then the range becomes:
without an anomaly= 100 - 88 = 12
The outlier has a greater impact on the range than on the interquartile range because the interquartile range only considers the middle 50% of the data, whereas the range considers all of the data.
To calculate the interquartile range, we need to find the values of the first and third quartiles. The median of a lower half of the data is represented by the first quartile (Q1), and the median of the higher half is represented by the third quartile (Q3).
Arranging the scores in ascending order:
62, 88, 89, 90, 92, 95, 96, 98, 100, 100
The median is the middle value, which is 93.
Q1 is the median of the lower half of the data, which is 62, 88, 89, 90, 92. The median of this set is (88 + 89) / 2 = 88.5.
Q3 is the median of the upper half of the data, which is 95, 96, 98, 100, 100. The median of this set is (98 + 100) / 2 = 99.
The disparity between Q3 and Q1 is the interquartile range (IQR):
IQR = Q3 - Q1 = 99 - 88.5 = 10.5 ≈ 9
If we remove the outlier, the scores become:
88, 89, 90, 92, 95, 96, 98, 100, 100
Q1 is the median of the lower half of the data, which is 88, 89, 90, 92. The median of this set is (89 + 90) / 2 = 89.5.
Q3 is the median of the upper half of the data, which is 95, 96, 98, 100, 100. The median of this set is (96 + 98) / 2 = 97.
The IQR is:
IQR = Q3 - Q1 = 97 - 89.5 = 7.5 ≈ 9.5
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W(4y-4)
Z
52°
Y
14
3y
X
Check the picture below.
[tex]52=(4y-4)+(3y)\implies 52=76-4\implies 56=7y \\\\\\ \cfrac{56}{7}=y\implies \boxed{8=y} \\\\\\ \stackrel{\measuredangle W}{4y-4}\implies 4(8)-4\implies \text{\LARGE 28}[/tex]
Asher is interviewing for a job as an elementary school teaching assistant. The interviewer notices that he is very friendly, kind, and calm. In terms of context, what might the interviewer think about Asher?
a
Asher comes off as though he is hiding something.
b
Asher has personality traits that are a great fit for this type of job.
c
Asher should be hired as a principal instead of a teaching assistant.
d
Asher would be a better fit as a maintenance worker at the school.
simplify the expression, 23.5 - 2.4 (3.9 - 8.6 ) + 4.4
The temperature of a person has a normal distribution. What is the probability that the temperature of a randomly selected person will be within 2.42 standard deviations of its mean? Provide answer with 4 or more decimal places.
Answer:
Step-by-step explanation:
If the temperature of a person follows a normal distribution, we know that approximately 95% of the observations fall within 2 standard deviations of the mean. Since we are given that we want to find the probability of the temperature being within 2.42 standard deviations of its mean, we can use the standard normal distribution and the z-score formula.
The z-score formula is given by:
z = (x - μ) / σ
where x is the observed value, μ is the mean, and σ is the standard deviation. In this case, we want to find the probability that the temperature is within 2.42 standard deviations of the mean, so we can set:
z = 2.42
Since the normal distribution is symmetric, we can find the area to the right of the mean (z = 0) and double it to get the total probability. Using a standard normal distribution table or calculator, we find that the area to the right of z = 2.42 is approximately 0.0074. So the area to the left of z = 2.42 is approximately 0.9926.
Doubling this area gives us the total probability:
P(z < 2.42 or z > -2.42) = 2 * P(z < 2.42) = 2 * 0.9926 = 0.9852
Therefore, the probability that the temperature of a randomly selected person will be within 2.42 standard deviations of its mean is 0.9852, or approximately 0.9852 with four decimal places.
whats the answer please
Therefore, y = 7x + 2/5, 7x + 5 = y, and y = 7x + 4 are the equations that depict a line parallel to y = 7x + 3.
what is equation ?A mathematical statement known as an equation demonstrates the equality of two expressions or values by separating them with an equal symbol. The values on either side of the equal symbol in an equation are equal and can be used to find the value of an unknown variable. For instance, the equation "2x + 3 = 7" states that "2x + 3" equals the number "7".
given
Since parallel lines have identical slopes, the equation of a line that is perpendicular to y = 7x + 3 will have the same slope as 7.
Any equation of the form y = 7x + b, where b is a constant, will therefore indicate a line parallel to y = 7x + 3.
y = 3x + 7: This line's slope, 3, is less than 7, so it is not parallel to the line y = 7x Plus 3.
y = 7x + 2/5: This line is parallel to y = 7x + 3 because it has the same slope as that equation.
7x + 5 = y: This equation is parallel to y = 7x + 3 because y = 7x + 5 has the same slope as it.
y = 7 + 3x is not parallel to y = 7x + 3 because it has the same slope as the solution y = 3x + 7, which does not equal 7.
Therefore, y = 7x + 2/5, 7x + 5 = y, and y = 7x + 4 are the equations that depict a line parallel to y = 7x + 3.
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The complete question is:- Choose all the equations below that represent a line parallel to
y=7x+3
y=3x+7 y=7x+2/5
7x+5=y
y=2x+3
y=4+7x y=3-7x
y+7x=3 y=7+3x
The equation, A=P(1+0.045/12)^12t
represents the amount of money earned on a compound interest savings account with an annual interest rate of 4.5% compounded monthly. if after 20
years the amount in the account is $18,539.38, what is the value of the principal investment? Round the answer to the nearest hundredths place.
$6,872.98
$7,550.25
$10,989.13
$17,202.73
Therefοre , the sοlutiοn οf the given prοblem οf amοunt cοmes οut tο be Optiοn B ($7,550.25) is the cοrrect chοice.
What is an amοunt?Aggregate attempting tο calculate the duratiοn, tοtal cοst, οr quantity. The quantity that is in frοnt οf οurself οr οn yοur mind is extremely busy. the result, its impοrtance, οr its relevance. Principal, interest, and third bοοkkeeping make up the tοtal. Amοunted, amοunts, and amοunting are sοme οf the wοrd variants. flexible term Its quantity refers tο hοw much that anything is.
Here,
The cοmpοund interest expressiοn is as fοllοws:
=> [tex]A = P(1 + r/n)^{(nt)[/tex]
Here are the facts:
=> A = $18,539.38 (the sum after 20 years) (the amοunt after 20 years)
The yearly interest rate, which is 4.5 percent, is r = 0.045.
n = 12 (the interest is cοmpοunded mοnthly, sο there are 12 cοmpοunding times per year) (the interest is cοmpοunded mοnthly, sο there are 12 cοmpοunding periοds per year)
t = 20 (the periοd in years) (the time in years)
By rearranging the sοlutiοn, we can find P:
=>[tex]A = P(1 + r/n)^{(nt)[/tex]
Adding the numbers we are familiar with
=>[tex]P = \$18,539.38 / (1 + 0.045/12)^{(12*20)[/tex]
=> P ≈ $7,550.25
Cοnsequently, the initial investment is wοrth rοughly $7,550.25. The number is $7,550.25, rοunded tο the nearest hundredth.
Optiοn B ($7,550.25) is the cοrrect chοice.
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17. A consumer survey in Econoland shows people consume pizza, soda, and cookies. The consumer spending is
listed below for the years 2019 and 2020. The base year is 2019.
Item
Pizza
Soda
Cookies
2019 Quantities
100
50
200
2019 Prices
$10.00
$3.00
$2.00
2020 Quantities
150
100
250
2020 Prices
$15.00
$3.25
$2.50
A. What and how much is in the market basket?
B. What did the market basket cost in 2019?
C. What did the market basket cost in 2020?
D. What was the inflation rate between 2019 and 2020?
Respond to each of the following using the data provided. Show all caculations where appropriate.
a. The market basket fοr 2020 is: $3200
b. The market basket cοst $1,550.00 in 2019.
c. The market basket cοst $3,200.50 in 2020.
d.The inflatiοn rate between 2019 and 2020 is 106.45%.
Hοw can we calculate inflatiοn?Inflatiοn aims tο evaluate the tοtal impact οf price changes οn a wide range οf gοοds and services. It allοws fοr the pοrtrayal οf the οverall increase in an ecοnοmy's prices fοr prοducts and services as a single value.
A. Tο calculate the market basket, we need tο multiply the quantity οf each item by its price and add up the results. Using the quantities and prices given, the market basket fοr 2019 is:
(100 pizzas × $10.00/pizza) + (50 sοdas × $3.00/sοda) + (200 cοοkies × $2.00/cοοkie)
= $1000 + $150 + $400
= $1550
Similarly, the market basket fοr 2020 is:
(150 pizzas × $15.00/pizza) + (100 sοdas × $3.25/sοda) + (250 cοοkies × $2.50/cοοkie)
= $2250 + $325 + $625
= $3200
B. Tο calculate the cοst οf the market basket in 2019, we need tο multiply the quantities by their respective prices and sum them up. The cοst in 2019 is:
(100 x $10.00) + (50 x $3.00) + (200 x $2.00) = $1,550.00
Therefοre, the market basket cοst $1,550.00 in 2019.
C. Tο calculate the cοst οf the market basket in 2020, we use the same methοd. The cοst in 2020 is:
(150 x $15.00) + (100 x $3.25) + (250 x $2.50) = $3,200.50
Therefοre, the market basket cοst $3,200.50 in 2020.
D. Tο calculate the inflatiοn rate between 2019 and 2020, we use the fοllοwing fοrmula:
Inflatiοn rate = ((Cοst in year 2 - Cοst in year 1) / Cοst in year 1) x 100%
Plugging in the values, we get:
Inflatiοn rate = (($3,200 - $1,550) / $1,550) x 100%
= 106.45%
Therefοre, the inflatiοn rate between 2019 and 2020 is 106.45%.
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Find the present value of $80,000 due in 4 years at the given rate of interest. (Use a 365-day year. Round your answer to the nearest cent.)
4%/year compounded quarterly
$73,394.49 is the present value of $80,000 that is due in 4 years at a 4% interest rate.
What is compound interest?Compound interest is interest that is accrued on both the principal and the prior interest. It is frequently referred to as "interest over the interest" as a result. Here, the interest that has already accrued is added to the principal, and the resulting amount acts as the new principal for the following period.
Hence, compound interest is the sum of interest on the principal and interest on earlier interest.
Using the formula below, it is possible to determine the present value of $80,000 payable in 4 years at a 4% rate of interest compounded quarterly:
PV = FV / (1 + r/n) ^(nt)
such that:
Present Value, or PV, is the term.
FV stands for future value.
The annual number of compounding periods is n, and the interest rate is r.
The number of years is t.
To solve this problem, we have:
FV = $80,000
r = 4% = 0.04
n = 12 (as compounding is done monthly)
t = 4
Adding these values to the formula provides the following results:
PV = 80,000 / (1 + 0.04/12) ^ (12 × 4)
PV = 80,000 / (1.003333) ^48
PV = 80,000 / 1.090777
PV = 73,394.49
As a result, $73,394.49 is the present value of $80,000 that is due in 4 years at a 4% interest rate.
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Using Formulas
Ava's minimum payment is 2.5% of her new balance. What is her minimum
payment if her new balance is $760.00?
S
Ben went to buy some brushes and oaints to finish a project.he bought 25 brushes and paints in total.if each brush cost IDR 1.500 , each paint IDR 800 and the total pruchase cost was IDR 27.000. how many brushes and paints did he buy
The number of brushes and number of paint bought by Ben are 7 and 18 respectively.
What is an equation system, and how can one be utilised to address issues?A group of equations that are concurrently solved to determine the values of the variables involved is referred to as a system of equations. A system of equations can have any number of variables and equations and can be either linear or nonlinear.
They could depict connections between various values, restrictions on resources or attributes, or optimization goals.
Let us suppose the number of brushes = b.
Let us suppose the number of paints = p.
Thus,
b + p = 25
1500b + 800p = 27000
Multiplying the first equation by 1500 and subtracting it from the second equation, we get:
800p - 1500b = -7500
p = (1500b + 7500) / 800
Substituting the value of p in first equation:
b + (1500b + 7500) / 800 = 25
Multiplying both sides by 800 and simplifying, we get:
2000b + 7500 = 20000
b = 6.25 = 7
Substitute the value of b in first equation:
7 + p = 25
p = 18
Hence, the number of brushes and number of paint bought by Ben are 7 and 18 respectively.
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Answer this question please
The coordinates of the vertices from of the image of triangle ABC are A'(x, y) = (0, - 1), B'(x, y) = (- 1, - 1) and C'(x, y) = (- 0.5, - 4).
How to determine the image of a triangle by rotation
Herein we find the representation of a triangle on a Cartesian plane, which is generated by three non-colinear points set on the plane mentioned above. The image of the triangle can be found by applying the following rigid transformation formulas:
x' = Ox + (x - Ox) · cos θ - (y - Oy) · sin θ
y' = Oy + (x - Ox) · sin θ + (y - Oy) · cos θ
Where:
(Ox, Oy) - Coordinates of the center of rotation.θ - Angle of rotation, in degrees.(x, y) - Coordinates of the vertex.If we know that (Ox, Oy) = (- 2, 0), θ = 180°, A(x, y) = (- 4, 1), B(x, y) = (- 3, 1) and C(x, y) = (- 3.5, 4), then the resulting coordinates of the triangle are, respectively:
xA' = Ox + (xA - Ox) · cos θ - (yA - Oy) · sin θ
xA' = - 2 + [- 4 - (- 2)] · cos 180° - (1 - 0) · sin 180°
xA' = 0
yA' = Oy + (xA - Ox) · sin θ + (yA - Oy) · cos θ
yA' = 0 + [- 4 - (- 2)] · sin 180° + (1 - 0) · cos 180°
yA' = - 1
xB' = Ox + (xB - Ox) · cos θ - (yB - Oy) · sin θ
xB' = - 2 + [- 3 - (- 2)] · cos 180° - (1 - 0) · sin 180°
xB' = - 1
yB' = Oy + (xB - Ox) · sin θ + (yB - Oy) · cos θ
yB' = 0 + [- 3 - (- 2)] · sin 180° + (1 - 0) · cos 180°
yB' = - 1
xC' = Ox + (xC - Ox) · cos θ - (yC - Oy) · sin θ
xC' = - 2 + [- 3.5 - (- 2)] · cos 180° - (4 - 0) · sin 180°
xC' = - 0.5
yC' = Oy + (xC - Ox) · sin θ + (yC - Oy) · cos θ
yC' = 0 + [- 3.5 - (- 2)] · sin 180° + (4 - 0) · cos 180°
yC' = - 4
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Differentiate in respect to yS(siny + y cosy)dy
Evaluate the following using suitable identities.
a) 972
b) 102 X 105
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Using suitable identities, the following mathematical expressions are evaluated:
a) 972 is evaluated as 1000 - 28
b) 102 x 105 is evaluated as = 10710
How did we evaluate?Mathematical evaluation refers to the process of finding the numerical value of a mathematical expression or equation using mathematical operations and rules. The evaluation involves substituting values for variables and simplifying the expression or equation until a final answer is obtained.
In more complex cases, mathematical evaluation may involve multiple steps and the use of various identities and formulas to simplify the expression or equation before arriving at the final answer
The given values are evaluated as follows:
a) 972 can be evaluated using the following identity:
972 = 1000 - 28
b) 102 x 105 can be evaluated using the following identity:
102 x 105 = (100 + 2) x 105 = 100 x 105 + 2 x 105 = 10500 + 210 = 10710
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Find the equation of the graphed line.
Answer:
A
Step-by-step explanation:
find the volume of each sphere in terms of π with a radius that equals 3ft
Answer:
36π
Step-by-step explanation:
To find the volume of the sphere we have to use the equation: [tex]\frac{4}{3}[/tex]πr³
To work our answer out we have to distribute the values we are given into the question...
[tex]\frac{4}{3}[/tex] × 3³We can ignore π for now as we will add it at the end
Now we have to solve what we are given...
[tex]\frac{4}{3}[/tex] × 3³3³ = 27[tex]\frac{4}{3}[/tex] × 27 = 36Now we can put π into our answer...
36πHope this helps, have a lovely day! :)
Simplify to a single trig function with no fractions.
The simplified cos(t)/sec(t) to the trig function 1 - sin²(t), which is equivalent to cos²(t).
Recall that the secant function is defined as the reciprocal of the cosine function. In other words, sec(t) = 1/cos(t). Therefore, we can rewrite cos(t)/sec(t) as cos(t)/(1/cos(t)).
To simplify this expression, we can multiply both the numerator and the denominator by cos(t), which gives:
cos(t)/(1/cos(t)) = cos(t) * (cos(t)/1) = cos²(t)
Now, we have simplified cos(t)/sec(t) to cos²(t). Alternatively, we could have used the identity cos²(t) = 1 - sin²(t) to simplify the expression. This identity follows directly from the Pythagorean identity cos²(t) + sin²(t) = 1.
Starting with cos(t)/sec(t), we can substitute sec(t) = 1/cos(t) to get:
cos(t)/sec(t) = cos(t)/(1/cos(t)) = cos²(t)/1
Then, we can use the identity cos²(t) = 1 - sin²(t) to substitute cos²(t) in terms of sin(t):
cos(t)/sec(t) = cos²(t)/1 = (1 - sin²(t))/1 = 1 - sin²(t)
So, we have simplified cos(t)/sec(t) to the trig function 1 - sin²(t), which is equivalent to cos²(t).
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barrowed 2,500 2 year loan intrest 155what was intres rate charged when opened account
If one borrowed $2,500 for 2 years and paid interest of $155, the interest rate charged was 3.053%.
How is the interest rate determined?The interest rate is computed using an online finance calculator with the following set parameters.
The interest rate represents the annual percentage rate of interest charged on the loan for 2 years.
N (# of periods) = 2 years
PV (Present Value) = $2,500
PMT (Periodic Payment) = $-0
FV (Future Value) = $-2,655
Results:
I/Y = 3.053%
Total Interest = $155.00
Thus, for this loan, the interest rate was 3.053%.
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given a population with a normal distribution, a mean of 40, and a standard deviation of 15, find the probability of a value between 50 and 70
Answer:
To find the probability of a value between 50 and 70 in a normal distribution with mean 40 and standard deviation 15, we need to first standardize the values using the z-score formula:
z = (x - μ) / σ
where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
For the lower bound of 50:
z = (50 - 40) / 15 = 0.67
For the upper bound of 70:
z = (70 - 40) / 15 = 2
Using a standard normal distribution table or a calculator with a built-in normal distribution function, we can find the probabilities corresponding to these z-scores:
P(0 < z < 0.67) = 0.2514
P(0 < z < 2) = 0.4772
To find the probability of a value between 50 and 70, we can subtract the probability of the lower bound from the probability of the upper bound:
P(50 < x < 70) = P(0 < z < 2) - P(0 < z < 0.67)
P(50 < x < 70) = 0.4772 - 0.2514
P(50 < x < 70) = 0.2258
Therefore, the probability of a value between 50 and 70 in this normal distribution is 0.2258 or about 22.58%.
Find all critical points for the function
4x + 6
x² + x + 1
on (-∞, ∞) and then list them (separated by commas) in the box below.
List of critical points:
f(x) =
Answer: [tex]\frac{-3+\sqrt{7}}{2}, \ \frac{-3-\sqrt{7}}{2}[/tex]
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Explanation:
Let
g(x) = 4x+6h(x) = x^2+x+1Each derivative is,
g ' (x) = 4h ' (x) = 2x+1which will be useful in the next section.
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[tex]f(\text{x}) = \frac{4\text{x}+6}{\text{x}^2+\text{x}+1} = \frac{g(\text{x})}{h(\text{x})}\\\\f(\text{x}) = \frac{g(\text{x})}{h(\text{x})}\\\\[/tex]
Apply the derivative with respect to x. We'll use the quotient rule.
[tex]f(\text{x}) = \frac{g(\text{x})}{h(\text{x})}\\\\\\f'(\text{x}) = \frac{g'(\text{x})h(\text{x})-g(\text{x})h'(\text{x})}{\big[h(\text{x})\big]^2}\\\\\\f'(\text{x}) = \frac{4(\text{x}^2+\text{x}+1)-(4\text{x}+6)(2\text{x}+1)}{(\text{x}^2+\text{x}+1)^2}\\\\[/tex]
The critical value(s) occur when either...
f ' (x) = 0f ' (x) doesn't exist, when x is in the domain of f(x)The first criteria will be handled in the next section.
The second criteria is handled in the section after that.
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f ' (x) is in the format A/B. It means f ' (x) = 0 leads to A/B = 0 and A = 0.
We set the numerator equal to zero and solve for x.
[tex]4(\text{x}^2+\text{x}+1)-(4\text{x}+6)(2\text{x}+1) = 0\\\\4(\text{x}^2+\text{x}+1)-(8\text{x}^2+16\text{x}+6) = 0\\\\4\text{x}^2+4\text{x}+4-8\text{x}^2-16\text{x}-6 = 0\\\\-4\text{x}^2-12\text{x}-2 = 0\\\\-2(2\text{x}^2+6\text{x}+1) = 0\\\\2\text{x}^2+6\text{x}+1 = 0\\\\[/tex]
From here we use the quadratic formula.
Plug in a = 2, b = 6, c = 1.
[tex]\text{x} = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\\text{x} = \frac{-6\pm\sqrt{(6)^2-4(2)(1)}}{2(2)}\\\\\text{x} = \frac{-6\pm\sqrt{28}}{4}\\\\\text{x} = \frac{-6\pm2\sqrt{7}}{4}\\\\\text{x} = \frac{2(-3\pm\sqrt{7})}{4}\\\\\text{x} = \frac{-3\pm\sqrt{7}}{2}\\\\\text{x} = \frac{-3+\sqrt{7}}{2} \text{ or } \text{x} = \frac{-3-\sqrt{7}}{2}\\\\[/tex]
If x is equal to either of those values, then f ' (x) = 0 would be the case. Therefore, these are the critical points of f(x).
There may be other critical values. We'll still need to check the second criteria.
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f ' (x) doesn't exist when we divide by zero.
Set the denominator equal to 0 and solve for x.
[tex](\text{x}^2+\text{x}+1)^2 = 0\\\\\text{x}^2+\text{x}+1 = 0\\\\[/tex]
Plug a = 1, b = 1, c = 1 into the quadratic formula.
[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-1\pm\sqrt{(1)^2-4(1)(1)}}{2(1)}\\\\x = \frac{-1\pm\sqrt{-3}}{2}\\\\[/tex]
We have a negative number as the discriminant, which leads to complex number solutions in the form a+bi where [tex]i = \sqrt{-1}[/tex]
Therefore, there aren't any real number values for x that lead [tex](\text{x}^2+\text{x}+1)^2[/tex] to be zero.
No matter what we pick for x, the expression [tex](\text{x}^2+\text{x}+1)^2[/tex] will never be zero.
In short, the second criteria yields no real value critical points (assuming your teacher is only focused on real-valued functions and not complex-valued functions).
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Summary:
The first criteria f ' (x) = 0 led to [tex]\text{x} = \frac{-3+\sqrt{7}}{2} \text{ or } \text{x} = \frac{-3-\sqrt{7}}{2}[/tex] as the critical valuesThe second criteria, f ' (x) doesn't exist where x is in the domain of f(x), leads to no critical values (assuming your teacher is not focusing on complex-valued functions).Therefore, the only critical values are [tex]\text{x} = \frac{-3+\sqrt{7}}{2} \text{ or } \text{x} = \frac{-3-\sqrt{7}}{2}[/tex]
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Extra info:
A critical value is where a local/absolute min, a local/absolute max, or a saddle point would be at this x value. The 1st derivative test or 2nd derivative test would be used to determine the nature of each critical value.
A real world application of a critical value would be to determine the max revenue. Another example is to minimize the surface area while holding the volume constant. Linear regression relies on a similar concept.
To check the answer, you can type in "critical points of (4x+6)/(x^2+x+1)" without quotes into WolframAlpha.