(a) The null hypothesis is that the true average compressive strength of the mixture is less than or equal to 1300 KN/m2.
The alternative hypothesis is that the true average compressive strength of the mixture is greater than 1300 KN/m2.
(b) To find the P-value, we need to calculate the probability of getting a sample mean of 1340 or greater, assuming the null hypothesis is true. Using the sample size n=12 and sample standard deviation σ=67, the test statistic is (1340-1300)/(67/sqrt(12)) = 2.523.
The P-value is the probability of getting a value of 2.523 or greater from a t-distribution with 11 degrees of freedom, which is 0.0193.
(c) The test statistic when μ = 1350 follows a t-distribution with 11 degrees of freedom.
The mean of the test statistic is equal to the true value of the mean, which is 1350. The standard deviation of the test statistic can be calculated as σ/sqrt(n) = 67/sqrt(12) = 19.341.
To find the probability of a type II error with α = 0.01, we need to calculate the probability of accepting the null hypothesis when the true value of the mean is 1350.
This can be calculated using the t-distribution with 11 degrees of freedom and the critical value corresponding to α=0.01, which is 2.718. The probability of a type II error is the probability that the test statistic is less than 2.718 when the true value of the mean is 1350, which is 0.2157 (rounded to four decimal places).
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how to solve 2400 at 10.5% for 5 years
Answer: 1260
Step-by-step explanation:
To solve for simple interest, we can use the formula:
I = P * r * t
where:
I = interest
P = principal amount
r = interest rate
t = time (in years)
In this case:
P = 2400
r = 10.5% = 0.105 (note that we convert the percentage to a decimal)
t = 5
Substituting these values into the formula, we get:
I = 2400 * 0.105 * 5
I = 1260
Therefore, the simple interest on a principal amount of 2400 at 10.5% for 5 years is 1260.
If s(x)=2-x^2 and t(x)=3x, which value is equivalent to (s•t)(-7)?
A(-439)
B(-141)
C(153)
D(443)
Answer:
A (-439)
Step-by-step explanation:
To find the value of (s•t)(-7), we need to first evaluate s(t(-7)) which means we substitute -7 for x in the function t, and then substitute the resulting value in the function s.
So, we have:
t(-7) = 3(-7) = -21
s(-21) = 2 - (-21)^2 = 2 - 441 = -439
Therefore, the value of (s•t)(-7) is -439, which corresponds to option A.
So, the correct answer is A(-439).
Answer:
A
Step-by-step explanation:
I believe you mean
(s ￮ t)(- 7)
to evaluate, first evaluate t(- 7) then substitute the value obtained into s(x)
t(- 7) = 3(- 7) = - 21 , then
s(- 21) = 2 - (- 21)² = 2 - 441 = - 439
in triangle abc, the length of side ab is 16 inches and the length of side bc is 24 inches. which of the following could be the length of side ac?
We have triangle ABC, the length of side AB is 16 inches and the length of side BC is 24 inches, the possible length of the side AC is less than 40 inches
To find the possible length of side AC, we need to use the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side, that is, a + b > c, b + c > a, c + a > b.
Now, for the given triangle ABC:
a = AB = 16 inches
b = BC = 24 inches
c = AC
We know that a + b > c (using the triangle inequality theorem)
16 + 24 > cc < 40
Therefore, By using the triangle inequality theorem, the possible length of the side AC is less than 40 inches.
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A grocer has two kinds of candies, one selling for 90 cents a pound and the other for $1.40 per pound. How many pounds of each kind must he use to make 100 pounds worth 85 cents a pound?
? pounds of 40 − cent candies, ? pounds of candies that cost $1.40 per pound
Using equations we know that 45 pounds are included in the $1.40 worth of groceries and 55 pounds worth of groceries are being purchased for 40 cents.
What are equations?An equation is a mathematical statement that contains the symbol "equal to" between two expressions with identical values.
As in 3x + 5 = 15, for example.
There are many different types of equations, including linear, quadratic, cubic, and others.
So, we have:
x + y = 100 .........A
40 × x + 140 y = 85 × 100
40 x + 140 y = 8500 ........B
Solving (A) and (B) as follows:
(40 x + 140 y) - 40 × ( x + y ) = 8500 - 40 × 100
(40 x - 40 x) + (140 y - 40 y) = 8500 - 4000
0 + 100 y = 4500
y = 4500/100
Hence, the price per unit of grocery is $1.40 = y = 45 pounds.
Now, put the value of y in equation (A) as follows:
x + y = 100
x = 100 - y
x = 100 - 45
x = 55 pounds
The number of groceries at the 40-cent price is x = 55 pounds.
Therefore, using equations we know that 45 pounds are included in the $1.40 worth of groceries and 55 pounds worth of groceries are being purchased for 40 cents.
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Correct question:
A grocer has two kinds of candies, one selling for 40 cents a pound and the other for $1.40 per pound. How many pounds of each kind must he use to make 100 pounds worth 85 cents a pound?
Question 4(Multiple Choice Worth 2 points)
(Irrational Numbers MC)
Order √50,-7.1.3-7 from least to greatest.
0 -7.1.-7. √50,23
O
0-71.-7.7.23,√50
O
0 -7.1.-723√50
0-7-7.1,√50,23
Answer:
D
Step-by-step explanation:
The square root of 50 is approximately equal to 7.07
-7.1111… can be rounded to -7.11
23/3 is equal to approximately 7.67
-7 1/5 is equal to -7.2
match the following. 1. an angle whose vertex is the center of the polygon and whose sides contain consecutive vertices of the polygon center of a regular polygon 2. the common center of its inscribed and circumscribed circles apothem of a regular polygon 3. the distance from the center of the polygon to a vertex radius of a regular polygon 4. the perpendicular distance from the center of the polygon to a side central angle of a regular polygon
Match the following with the given options:
1. An angle whose vertex is the center of the polygon and whose sides contain consecutive vertices of the polygon - Central angle of a regular polygon2. The common center of its inscribed and circumscribed circles - Center of a regular polygon3. The distance from the center of the polygon to a vertex - Radius of a regular polygon4. The perpendicular distance from the center of the polygon to a side - Apothem of a regular polygonA polygon is a two-dimensional object that has three or more straight sides and angles. A regular polygon is a polygon with equal sides and equal angles.
A regular polygon is a shape that has all its sides the same length and all of its angles the same degree. Below are the matched answers for the given options.1. An angle whose vertex is the center of the polygon and whose sides contain consecutive vertices of the polygon - Central angle of a regular polygonThe central angle of a regular polygon is defined as the angle whose vertex is the center of the polygon and whose sides contain consecutive vertices of the polygon.
The angle measures (360 / n) degrees, where n is the number of sides of the polygon.2. The common center of its inscribed and circumscribed circles - Center of a regular polygonThe center of a regular polygon is the point at which its axes of symmetry intersect. The center is also the common center of the circumscribed and inscribed circles.3. The distance from the center of the polygon to a vertex - Radius of a regular polygonThe radius of a regular polygon is defined as the distance from the center of the polygon to any vertex. It is also the distance from the center of the polygon to the midpoint of a side
.4. The perpendicular distance from the center of the polygon to a side - Apothem of a regular polygonThe apothem of a regular polygon is defined as the perpendicular distance from the center of the polygon to a side. It is also the radius of the inscribed circle.
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Find the length of the missing side
A. 21
B. 22
C. 23
D. 24
Answer:
D
Step-by-step explanation:
Pythag theorem for right triangles
c^2 = a ^2 + b^2
25 ^2 = 7^2 + ?^2
25^2 - 7^2 = ?^2
?^2 = 576
? = 24 units
A spherical balloon is inflated so that it’s radius (r) increases at a rate of 2/r cm/sec. How fast is the volume of the balloon increasing when the radius is 4 cm?
Answer:
32π cm³/sec
Step-by-step explanation:
List given information
[tex]\displaystyle V=\frac{4}{3}\pi r^3\\\\r=4\\\\\frac{dr}{dt}=\frac{2}{r}\\ \\\frac{dV}{dt}=\,\,?[/tex]
Solve for dV/dt
[tex]\displaystyle V=\frac{4}{3}\pi r^3\\\\\frac{dV}{dt}=4\pi r^2\biggr(\frac{dr}{dt}\biggr)\\\\\frac{dV}{dt}=4\pi r^2\biggr(\frac{2}{r} \biggr)\\\\\frac{dV}{dt}=4\pi (4)^2\biggr(\frac{2}{4} \biggr)\\\\\frac{dV}{dt}=4\pi(16)\biggr(\frac{1}{2} \biggr)\\\\\frac{dV}{dt}=4\pi(8)\\\\\frac{dV}{dt}=32\pi[/tex]
Hence, the volume of the balloon is increasing at a rate of 32π cm³/sec when the radius is 4 cm.
answer quick please am i correct
Answer:
6/11 = 0.54
Step-by-step explanation:
Answer: Yes you are correct
Step-by-step explanation:
a system of equations is shown below. y=2x+1 and y=x+2 what is the solution to the system? A. (0,1) B. (1,2) C. (1,3) D. (2,4)
Answer:
(1,3)
Step-by-step explanation:
Given system of equations is :-
y = 2x + 1y = x + 2We can solve this by using substitution method by substituting the value of y from equation (1) into equation (2) as ,
2x + 1 = x + 2
Subtract x on both sides,
2x - x + 1 = 2
Simplify,
x = 2 - 1
x = 1
Substitute this value of x into equation (2) as ,
y = 1 + 2
y = 1 + 2
y = 3
Hence the required answer is (1,3) .
and we are done!
Find the circumference of a circle with diameter, d = 1.26m.
Give your answer rounded to 2 DP
Answer: b
Step-by-step explanation: just took it on edge.
i have a small area that i want to place 2 bench press machines. how much room will i need to reserve for those?
To place two bench press machines in a small area, you will need a space of approximately 10 feet by 10 feet. Let's discuss it in detail below. Here are the dimensions of the bench press machine, which can help determine the amount of space required to fit two bench press machines in a small area:
The length of the bench press machine is between 48 inches and 54 inches.
The width of the bench press machine is between 28 inches and 32 inches.
The height of the bench press machine is between 48 inches and 56 inches.
Based on the above dimensions of the bench press machine, two machines can be placed in a small area of 10 feet by 10 feet. However, for safe use, the following guidelines should be followed:
There should be at least 6 feet of distance between the two bench press machines. There should be at least 3 feet of clearance in the front of the bench press machine to allow for safe movement during exercises. There should be at least 2 feet of clearance behind the bench press machine to allow for a safe exit in case of an emergency.
Thus, to place two bench press machines in a small area, you will need a space of approximately 10 feet by 10 feet.
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determine whether or not the given procedure results in a binomial distribution. if not, identify which condition is not met. surveying 33 people to determine which statistics courses they have taken.yes or no
The procedure of surveying 33 people to determine which statistics courses they have taken does not result in a binomial distribution because it does not meet the condition of having only two possible outcomes for each trial.
The outcome of each trial is not a success or failure, but rather a specific course that the individual has taken. Binomial distribution requires a fixed number of independent trials with only two possible outcomes (success or failure) for each trial.
It does not satisfy the four conditions required for a binomial distribution as follows:
The trials must be independent: In this case, the independence assumption is not satisfied as the responses of the 33 people are not necessarily independent. The trials must be identical: In this case, the trials are not identical as the individuals surveyed may have different characteristics and backgrounds. The probability of success must be constant: The probability of success is not constant, as it depends on the number of courses taken by each individual. The number of trials must be fixed: In this case, the number of trials is fixed (33 people were surveyed), but the other three conditions are not met, so the procedure does not result in a binomial distribution.To learn and practice more about 'binomial distribution':
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given the triangle below, what is RS
Answer:
Proofs attached to answer
Step-by-step explanation:
Proofs attached to answer
Why is the sky blue?
Thanks :)
Answer:
Violet and blue light have the shortest wavelengths and red light has the longest. Therefore, blue light is scattered more than red light and the sky appears blue during the day. When the Sun is low in the sky during sunrise and sunset, the light has to travel further through the Earth's atmosphere.
i need a answer please
The car will never have a value of £0 because the formula for depreciation approaches but never reaches zero.
What is percent?Percent is a way of expressing a number as a fraction of 100. The word "percent" comes from the Latin per centum, which means "per hundred". Percentages are often used to represent rates, ratios, or proportions of a quantity. For example, if a student scores 90 out of 100 on a test, their score can be expressed as 90%, or 0.9 as a decimal. Similarly, if a store offers a discount of 20% on a product, it means that the customer can purchase the product for 80% of its original price, or 0.8 as a decimal. Percentages are commonly used in finance, science, and everyday life to represent various types of information, such as interest rates, probabilities, taxes, and discounts.
Here,
1. Amount after 1 year:
The amount after 1 year can be calculated using the formula for compound interest:
A = P(1 + r/n)ⁿˣ
Where A is the amount, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.
Substituting the given values, we get:
A = 1000(1 + 0.05/1)¹*¹= £1050
Therefore, the total amount in the bank account after 1 year is £1050.
2. Amount after 2 years:
Using the same formula, we get:
A = 1000(1 + 0.05/1)¹*² = £1102.50
Therefore, the total amount in the bank account after 2 years is £1102.50.
3. Amount after 3 years:
Using the same formula, we get:
A = 1000(1 + 0.05/1)¹*³ = £1157.63
Therefore, the total amount in the bank account after 3 years is £1157.63.
4. Amount after 6 years:
Using the same formula, we get:
A = 1000(1 + 0.05/1)¹*⁶ = £1345.99
Therefore, the total amount in the bank account after 6 years is £1345.99.
5. The amount of interest earned after 6 years is not equal to double the amount of interest earned after 3 years because the interest earned in each year is added to the principal amount, so the interest earned in subsequent years is based on a larger principal amount. This leads to exponential growth in the amount of interest earned over time.
6. The value of the car after n years can be calculated using the formula:
V = P(1 - r/100)ⁿ
where V is the value of the car after n years, P is the initial value of the car, r is the rate of depreciation per year, and n is the number of years.
Using this formula, we can calculate the value of the car after:
1 year: V = 1000(1 - 0.05)¹ = £950
2 years: V = 1000(1 - 0.05)² = £902.50
3 years: V = 1000(1 - 0.05)³ = £857.38
6 years: V = 1000(1 - 0.05)⁶ = £735.03
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round 0.956 to one decimal place
Answer: 0.96
Step-by-step explanation:
To round 0.956 to one decimal place, we need to look at the second decimal place (the hundredths place), which is 5. Since 5 is greater than or equal to 5, we need to round up the first decimal place (the tenths place), which is 9. Therefore, the rounded number to one decimal place is:
0.956 rounded to one decimal place = 0.96
Mr hawkins is covering a wall with wallpaper the wall measures 11 feet by 15 feet each square foot costs 3. 10 find the cost of covering the wall with wall paper
The cost of covering the wall with wall paper is 495.
the lengthe of the wall = 11 feet
the width of the wall = 15 feet
so, the area of the wall
A = the length of the wall x the width of the wall
A = 11 x15
A = 165 square foot
then, the total cost is
= 165 x 3
= 495
So, the cost of covering the wall with wall paper is 495.
The surface of a shape's area is measured. You must multiply the length and breadth of a rectangle or square in order to determine its area. A has an area of x time y.
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Select all of the angles that have the same measure as angle 1. Assume the lines are parallel.
028
24
023
27
26
22
315
2
can't be determined
Answer:
Since the lines are parallel, we know that alternate interior angles are congruent. Therefore, angles 1 and 5 are congruent, as they are alternate interior angles. So any angle that is congruent to angle 5 is also congruent to angle 1.
Looking at the answer choices, we see that angle 2 is adjacent to angle 5 and therefore also congruent to angles 1 and 5. So the correct answer is:
2
Therefore, the answer is "2".
The numbers of trading cards owned by 7 middle-school students are given below.
(Note that these are already ordered from least to greatest.)
360, 373, 402, 499, 548, 644, 646
Suppose that the number 646 from this list changes to 639. Answer the following.
a) If the number 646 from the ordered list changes to 639, c) the median stays the same.
b) a) If the number 646 from the ordered list changes to 639, a) the mean decreases by 1.
What are the median and the mean?The median is the middle value in an ordered list (descending or ascending).
The mean refers to the average value.
The mean or average is computed as the quotient of the division of the total data value by the total number of items in the data set.
The ordered numbers of trading cards owned by 7 middle-school students = 360, 373, 402, 499, 548, 644, 646
The total value (sum) = 3,472
The number of cards = 7
The median = 499
The mean = 496 (3,472/7)
If the number 646 changes too 639:
The total value (sum) = 3,465
The median = 499
The mean = 495 (3,465/7)
Difference in mean = 1 (496 - 495)
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PLEASE Answer before 3/12/2023
6, 14, 22, 30, 38, … Use the expression below to determine the value of the 15th term. 6 + 8n, where n is equal to 0, 1, 2, 3, … answers: 112 118 120 126
Answer: 118
Step-by-step explanation: 38 is the highest value, which is the fifth number in the pattern, and we need to find out what the 15th number is. We simply need to add (8x10) to 38.
Answer: 118
Step-by-step explanation:
you just keep going up at the rate of 8
The expression shown in red represents how many hockey pucks and hockey sticks come in one gym set. The expression shown in blue shows how many come in 3
sets.
Use the drop-down menus to complete the statements below to compare the values of the two expressions.
Answer:
three times
Step-by-step explanation:
An electric dipole with its center located at the origin of a Cartesian coordinate system oscillates along the z axis, creating an electromagnetic wave. At a position on the y axis far from the origin, what is the polarization of the wave and which axis are the magnetic (a) The wave is polarized parallel to the a axis and the magnetic field lines are parallel to b The wave is polarized parallel to the z axis and the magnetic field lines are parallel to (c) The wave is polarized parallel to the y axis and the magnetic field lines are parallel to (d) The wave is polarized parallel to the y axis and the magnetic field lines are parallel to (e) The wave is polarized parallel to the z axis and the magnetic field lines are parallel to field lines parallel to? the y axis the axis the r axis the z axis the z axis
The wave is polarized parallel to the y-axis, and the magnetic field lines are parallel to the x-axis. Here option D is the correct answer.
The oscillating electric dipole along the z-axis creates an electromagnetic wave with electric and magnetic fields perpendicular to each other and to the direction of wave propagation. At a position on the y-axis far from the origin, the electric field will be parallel to the y-axis.
The polarization of the wave refers to the orientation of the electric field vector. Since the electric field is parallel to the y-axis, the wave is polarized parallel to the y-axis.
According to the right-hand rule, the direction of the magnetic field lines will be perpendicular to both the electric field and the direction of wave propagation, which is along the z-axis. Therefore, the magnetic field lines will be parallel to the x-axis.
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(a) In the figure below, m CED=50° and m AB = 140°. Find m CD.
(b) In the figure below, m VYW=36 and m VX = 62. Find m VW
a) Measurement of arcCD is 40°
b) Measurement of arcVW is 134°
Define the term Circle identities?The Circle identities refer to a set of fundamental identities in trigonometry that relate the six trigonometric functions of an angle in a right-angled triangle.
a) Given that, ∠CED=∠AEC =50° arcAB=140° we have to find out arcCD
We know the formula for external angle as per given diagram,
∠AEC = [tex]\frac{1}{2} (arcAB-arcCD)[/tex]
50° = [tex]\frac{1}{2} *(140-arcCD)[/tex]
Simplify, arcCD = 140° - 100° = 40°
Therefore, measurement of arcCD is 40°
b) Given that, ∠VYW =36° arcVX=62° we have to find out arcVW
We know the formula for external angle as per given diagram,
∠VYW = [tex]\frac{1}{2} (arcVW-arcVX)[/tex]
36° = [tex]\frac{1}{2} *(arcVW-62)[/tex]
Simplify, arcVW = 72° + 62° = 134°
Therefore, measurement of arcVW is 134°
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According to the question the a) Measurement of arcCD is 40° and b) Measurement of arcVW is 134°
Define the term Circle identities?The Circle identities refer to a set of fundamental identities in trigonometry that relate the six trigonometric functions of an angle in a right-angled triangle.
a) Given that, ∠CED=∠AEC =50° arcAB=140° we have to find out arcCD
We know the formula for external angle as per given diagram,
∠AEC = 1/2 (arcAB - arcCD)
50° = 1/2*(140 - arccd)
Simplify, arcCD = 140° - 100° = 40°
Therefore, measurement of arcCD is 40°
b) Given that, ∠VYW =36° arcVX=62° we have to find out arcVW
We know the formula for external angle as per given diagram,
∠VYW = 1/2(arcVW - arcVX)
36° = 1/2*(arcVW - 62)
Simplify, arcVW = 72° + 62° = 134°
Therefore, measurement of arcVW is 134°
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A baby weighs 10 pounds at birth, and four years later the child's weight is 40 pounds. Assume that childhood weight W (in pounds) is linearly related to age t (in years).
(a) Express W in terms of t.
W in terms of t can be represented as W = 7.5t + 10.
When a baby weighs 10 pounds at birth and 40 pounds four years later, we must apply the linear equation to represent W in terms of t.
y = mx + b,
where
m indictaes the slope of the line while
b is the y-intercept.
The formula to find the slope of a line is as follows:
Slope (m) = (y2 - y1) over (x2 - x1)
Given that a baby weighs 10 pounds at birth, and four years later, the child's weight is 40 pounds, we can determine the slope of the line as:
Slope (m) = (40 - 10) over (4 - 0) = 30 / 4 = 7.5
Thus, the linear equation relating childhood weight W (in pounds) to age t (in years) is:
W = 7.5t + 10
Therefore, we can express W in terms of t as W = 7.5t + 10.
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The product of two rational number is -5
Answer:
1/5 and 25
5 and -1
Step-by-step explanation:
There are infinitely many pairs of rational numbers that have a product of -5. However, two possible pairs are:
1/5 and 25
5 and -1
In general, the product of two rational numbers can be negative if one of them is positive and the other is negative.
Draw a model to show that 3=6/2 explain your drawing
Equivalent fractions are defined as fractions that are equal to the same value, regardless of their numerators and denominators.
Equivalent fractions can be defined as fractions that may have different numerators and denominators but represent the same value. For example, 9/12 and 6/8 are equivalent fractions because both simplify to 3/4. 1/2, 2/4, 3/6 and 4/8 are equivalent fractions. Let's see how their values are equal. We will represent each of these fractions with a shaded circle.
It can be seen that, seen as a whole, the hatched parts in all the figures represent the same parts.
Equivalent fractions can be written by multiplying or dividing the numerator and denominator by the same number. This is why these fractions reduce to the same number when they are simplified.
Multiply the numerator and denominator by the same number.Numerator and denominator divide by the same number.Learn more about Fraction:
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A shopkeeper bought 26 apples from a fruit vendor for $37.70.How much did each apple cost?
Answer:
The cost per apple is $1.45
Step-by-step explanation:
I will help with the solution, as I do not know the standard for the picture/estimate.
$37.70 / 26 is simply the cost per apple.
The cost per apple is $1.45
Hope this helps!
the mean score on a statistics exam is 82. if your exam score is 2.12 standard deviations below the mean, which of the following scores could not be your exam score? (there may be multiple correct answers, click all that apply) group of answer choices a.70 b.85 c.80 d.90
The score that could not be your exam score is 70.The correct option is (a).To calculate the standard deviation of a set of data, you take the square root of the average of the squared differences of the values from the mean.
The question is asking which of the given scores could not be your exam score if your exam score is 2.12 standard deviations below the mean. The mean score on the exam is 82, so the score that is 2.12 standard deviations below the mean is: 82 - (2.12 x standard deviation) = 82 - (2.12 x 10) = 70.Therefore, the score that could not be your exam score is 70.
To explain further, standard deviation is a measure of how spread out the numbers in a set of data .The other options (85, 80, and 90) are not correct because they are not 2.12 standard deviations below the mean. To summarize, the score that is 2.12 standard deviations below the mean score of 82 is 70.
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some rate functions require algebraic manipulation or simplification to set the stage for undoing the chain rule or other antiderivative techniques. find an equivalent closed form for each function.a. S π / π /4 5t+4 / t² + 1 dtHint : begin by writing as a sum of two functions ____ previewb. S π/t 4tan (t) dt Hint : begin by using a trig identity to change the form of the rate function___ preview
the equivalent closed form for the original rate function is
.[tex]S π/t 4tan (t) dt = -4 ln |cos (t)| + C.[/tex]
a. [tex]S π / π /4 5t+4 / t² + 1 dt[/tex]We can write this rate function as a sum of two functions by expanding the denominator:
[tex]S π / π /4 5t+4 / (t² + 1) dt = S π / π /4 5t+4 / (t² + 1) (t+1/t - 1/t) dt = S π / π /4 5t+4 (t+1 - 1/t²) dt = S π / π /4 5t+4 (t+1) dt - S π / π /4 5t+4 (1/t²) dt[/tex]
Now, integrating the two functions gives us the closed form:
[tex]S π / π /4 5t+4 (t+1) dt = 5/2 (t² + 2t + 2) + CS π / π /4 5t+4 (1/t²) dt = -5/2 (t + 1/t) + C[/tex]
Therefore, the equivalent closed form for the original rate function isS π / π /4 5t+4 / t² + 1 dt = 5/2 (t² + 2t + 2) - 5/2 (t + 1/t) + C.
b. S π/t 4tan (t) dt
We can use a trig identity to change the form of this rate function:
S π/t 4tan (t) dt = S π/t 4 (sin (t) / cos (t)) dt = 4/cos (t) S π/t sin (t) dt
Integrating the rate function gives us the closed form:
S π/t 4tan (t) dt = 4/cos (t) S π/t sin (t) dt = -4 ln |cos (t)| + C.
Therefore, the equivalent closed form for the original rate function is:
S π/t 4tan (t) dt = -4 ln |cos (t)| + C.
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