How to find the missing side of a triangle using the Law of Sines?
Step-by-step explanation:
the law of sine is
a/sin(A) = b/sin(B) = c/sin(C)
a, b, c are the sides. A, B, C are the corresponding opposing angles.
you fill in what you know and then solve for what you don't know. these are just regular equations. you multiply or divide or add or subtract the same things on both sides and try to get the missing side isolated on one side of an equation.
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
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.
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".
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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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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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
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 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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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
Find an explicit formula for the arithmetic sequence 12, 5, -2, -9,. 12,5,−2,−9,. 12, comma, 5, comma, minus, 2, comma, minus, 9, comma, point, point, point Note: the first term should be \textit{a(1)}a(1)start text, a, left parenthesis, 1, right parenthesis, end text. A(n)=
The explicit formula for the arithmetic sequence 12, 5, -2, -9,..is a(n) = -7n + 19.
To find the explicit formula for this arithmetic sequence, we first need to determine the common difference, which is the amount by which each term differs from the previous term.
d = 5 - 12 = -7
d = -2 - 5 = -7
d = -9 - (-2) = -7
Since the common difference is -7, we can use the formula for an arithmetic sequence:
a(n) = a (1) + (n - 1) d
where a(n) is the nth term of the sequence, a (1) is the first term, n is the term number, and d is the common difference.
Plugging in the given values, we get:
a(n) = 12 + (n - 1) (-7)
Simplifying, we get:
a(n) = -7n + 19
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The explicit formula for the arithmetic sequence will be f(n) = 19 -7n.
f(n) = a + (n - 1) d,
where an is the first term and is the explicit formula for an arithmetic series.
d = Common Difference
n = number of terms
The provided arithmetic series is = 12, 5, -2, -9.
First term: a = 12.
Common variation: d= 5-12= 7
[Difference between any two words that follow one another.]
When a = 12 and d = -7 are entered into equation (1), we obtain
f(n) = 12 + (n - 1) ( - 7 )
Thus, the explicit formula for the above arithmetic sequence is
f(n) = 12 = (n - 1). ( - 7 )
If we continue to solve it, we obtain
f(n) = 12 - 7n + 7
f(n) = 19 - 7n
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Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.
A figure consists of two curves labeled Upper A and Upper B. Curve Upper A is shorter and more spread out than curve Upper B, and curve Upper B is farther to the right than curve Upper A.
Select all that apply:
1. A has the larger mean.
2. B has the larger mean.
3. The means of A and B are equal.
4. A has the larger standard deviation.
5. B has the larger standard deviation.
6. The standard deviations of A and B are equal
Answer:
False
True
False
True
False
False
Step-by-step explanation:
From the plot, we can see that curve B is taller and more narrow than curve A, and it is shifted to the right relative to curve A. This tells us that curve B has a larger mean and smaller standard deviation than curve A. Therefore, statement 2 is true, and statements 1, 3, 5, and 6 are false. Finally, since curve A is more spread out than curve B, it has a larger standard deviation. Therefore, statement 4 is true.
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:
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.
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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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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.
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 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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(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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The annual salaries (in $) within a certain profession are modelled by a random variable with the cumulative distribution function F(x)= {1−kx^−3 for x>44000 {0 otherwise, for some constant k. For these problems, please ensure your answers are accurate to within 3 decimals. a)Find the constant k here and provide its natural logarithm to three decimal places. b)Calculate the mean salary given by the model.
a) The constant k is 5.427 x 10^−12 and its natural logarithm is -26.68.
b) The mean salary of the given model by using the probability density function is approximately $270.86.
a) The cumulative distribution function of the given random variable is provided as follows:
F(x) = {1−kx^−3 if x>44000, and 0 otherwise
The cumulative distribution function is given as
F(x) = 1−kx^−3 if x>44000 and F(x) = 0, if x≤44000i)
We need to check the value of the cumulative distribution function at 44000
We have, F(44000) = 0
0 = 1−k(44000)^−3
⇒ 1 = k(44000)^−3
⇒ k = 1/(44000)^−3
⇒ 5.427 x 10^−12
Taking the natural logarithm of k, we have ln(k) = −28.68 (approx.)
Hence, the constant k is 5.427 x 10^−12 and its natural logarithm to three decimal places is -28.68
b) The probability density function is given as,
f(x) = F'(x) = 3kx^−4, for x>44000 and f(x) = 0, otherwise
The mean or expected value of the random variable is given as
E(X) = ∫[−∞,∞]xf(x)dx
= ∫[44000,∞]x(3kx^−4)dx
= 3k∫[44000,∞]x^−3dx
= 3k[(−1/2)x^−2] [∞,44000]
= (3k/2)(44000)^−2
= 270.86 (approx.)
Therefore, the mean salary given by the model is $270.86 (approx.)
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find the number of ways of arranging the numbers ${}1,$ ${}2,$ ${}3,$ ${}4,$ ${}5,$ $6$ in a row so that the product of any two adjacent numbers is even.
Combining these, we have[tex]$6 \times 6 = \boxed{36}$[/tex] different arrangements of the numbers [tex]${}1,$ ${}2,$ ${}3,$ ${}4,$ ${}5,$ $6$[/tex] in a row where the sum of any adjacent numbers is even.
What are the fundamental products?Products intended for exporting after processing into goods or processed products are referred to as "basic products," as are goods planned for export after processing. Samples 1 - 3 Samples 2 - 3.
We may start by noting that at minimum one of the neighboring numbers must be even for the sum of both numbers to be even. This means that a even numbers (2, 4, 6) as well as the odd numbers (1, 3, 5) should be arranged in the appropriate positions.
Let's start by thinking about the even positions. The second, fourth, and sixth places are the only even positions. We can choose any variant of the 3 even numbers to occupy these spots, giving us[tex]$3! = 6$[/tex] ways.
Let's now think about the unusual positions. The first, third, and fifth positions are the only ones that are odd. We have an additional [tex]$3! = 6$[/tex]ways to fill these spots by using any combination of the 3 odd numbers.
Consider the odd locations now. The first, third, and fifth places are the three odd positions. We have an additional[tex]$3! = 6$[/tex]ways by using any permutation of the three odd numbers to fill these positions.
Together, this give us [tex]$6 \times 6[/tex] = [tex]\boxed{36}$[/tex] different ways to arrange the numbers [tex]${}1,$ ${}2,$ ${}3,$ ${}4,$ ${}5,$ $6$[/tex]in a row so that the sum of any two adjacent numbers is even.
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Write this value in order , starting with the smallest 0. 2 1/2 2℅
0, 2%, 2 1/2 The values are ordered from smallest to largest, with 0 being the smallest, followed by 2%, and then 2 1/2.
0 is the smallest value because it represents nothing. It is the absence of a quantity, and therefore, it is always the smallest value.
2% is larger than 0 because it represents a percentage of a whole. Percentages are fractions out of 100, so 2% means 2 out of 100. It is larger than 0, but smaller than 2 1/2.
2 1/2 is the largest value because it represents a whole number and a fraction. It is larger than 2%, because 2 1/2 is equal to 250 out of 10,000 or 25 out of 1,000, which is a larger quantity than 2 out of 100.
Understanding the relative size of different values is an important skill in many areas, including mathematics, science, and finance. Being able to order values from smallest to largest helps us make sense of data and information, and it allows us to make informed decisions based on the relative size of different quantities. It is important to learn how to do this accurately and efficiently in order to be successful in these fields.
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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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AREA/ PERIMETER/ VOLUME
Find the perimeter and area.
12m
10m
9m
8m
Step-by-step explanation:
Perimeter:
12m + 10m + 8m + (10 - 8)m + 9m + (12 - 9)m = 44m
or
12m * 2 + 10m * 2 = 44m
Area:
A = 9m * 8m + 12m * (10 - 8)m
A = 72m² + 24m² = 96m²
given the triangle below, what is RS
Answer:
Proofs attached to answer
Step-by-step explanation:
Proofs attached to answer
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
3
The ratio of desktop computers to laptop computers sold by
a mail-order company last week was 8 to 3. What could be
the numbers of computers sold by the company last week?
A
B
C
D
448 desktops, 168 laptops
448 desktops, 165 laptops
440 desktops, 168 laptops
400 desktops, 165 laptops
using the ratio given, the number of computers could be sold by the company last week is: A. 448 desktops, 168 laptops.
How to Calculate Ratios?To find the actual numbers of desktop and laptop computers sold, we need to choose a common factor for the ratio 8:3.
Let's assume that the total number of computers sold is 33x (where x is a positive integer). Then, the ratio 8:3 corresponds to 8x desktops and 3x laptops. We can check which of the given options satisfies this condition:
A. 8x = 448, 3x = 168 --> This satisfies the condition, as 8:3 = 448:168
B. 8x = 448, 3x = 165 --> This does not satisfy the condition, as 8:3 is not equal to 448:165
C. 8x = 440, 3x = 168 --> This does not satisfy the condition, as 8:3 is not equal to 440:168
D. 8x = 400, 3x = 165 --> This does not satisfy the condition, as 8:3 is not equal to 400:165
Therefore, the answer is option A: 448 desktops and 168 laptops could be the numbers of computers sold by the company last week.
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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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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.
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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