If there are 4 boys and 4 girls , then the number of ways in which
(i) there is no restriction is 40320 ways ,
(ii) not all girls sit together is 37440 ways ,
(iii) no two girls sit together is 2880 ways .
the number of boys in the class is = 4 boys ;
the number of girls in the class is = 4 girls ;
Part(i) ,
we have to find number of ways in which they can sit in a row where there is no restriction .
there are total of 8 persons ,
So ,the number of ways in which 8 persons can be arranged without any restrictions is = 8! ways.
= 40320 ways
Part(ii) ,
we have to find number of ways in which they can sit in a row where not all girls sit together.
the cases where all gets sit together is written as :
= B₁ , B₂ , B₃ , B₄ , G ⇒ 5! ways ;
= B: G₁ , G₂ , G₃ , G₄ ⇒ 4! ways ;
the number of ways all girls are together is = 5! × 4! ;
So , the number of ways not all girls are together is = 8! - 5!×4! ;
= 37440 ways
Part(iii) ,
we have to find number of ways in which they can sit in a row where no two girls sit together.
this case can be denoted as : G , B₁ , G , B₂ , G , B₃ , G , B₄ .
So , the boys can be arranged in 4! ways ;
and there will be 5 places for 4 girls , which can be arranged in ⁵P₄ ways ;
total number of ways is = 5!×4! = 2880 ways .
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Suppose H(x)=7 root x-3 . Find two functions f and g such that (fog)(x)=H(x) .
The two functions such that composition (fog) (x) = H(x) are f(x) = 7x and g(x) = [tex]\sqrt{x-3}[/tex].
What is Function?A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.
We have a function H(x) = 7 [tex]\sqrt{x-3}[/tex]
We have to find two functions f and g such that (fog) (x) = H(x).
Let f(x) = 7x
g(x) = [tex]\sqrt{x-3}[/tex]
The composition of these functions is,
(fog)(x) = f (g(x))
= 7 . g(x)
= 7 [tex]\sqrt{x-3}[/tex]
= H(x)
Hence the functions are f(x) = 7x and g(x) = [tex]\sqrt{x-3}[/tex].
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the heptagon ($7$-sided polygon) and hexagon are drawn in plane so that they intersect at a finite number of points. let $n$ be the maximum possible number of points where two polygons can intersect. which one is correct?
The maximum possible number of points where two polygons can intersect is [tex]$n = 4$[/tex]. This is achieved when the two polygons are tangent to each other at four points.
Maximum Intersection Point CountThere are different ways to approach the problem of determining the maximum possible number of points where two polygons can intersect, but one possible method is to use geometric principles and constructive examples. Here is one possible way to do it:
Start with a heptagon and a hexagon that are both regular (i.e., all sides and angles are equal).Place the heptagon in the plane so that one of its vertices coincides with the center of the hexagon.Draw an equilateral triangle with sides of the same length as the heptagon's sides, and one of its vertices at the center of the hexagon.Connect the center of the hexagon to each vertex of the triangle, forming six congruent isosceles triangles.Draw a line segment from the center of the hexagon to the middle point of each side of the heptagon. This divides the heptagon into 14 congruent isosceles triangles.Now, it is clear that the heptagon and hexagon intersect at 4 points.This is a constructive example for showing that the maximum possible number of intersection points between heptagon and hexagon is 4.
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An anesthetic is given at the dosage of 15 mg per 10 lb body weight. It is packaged in vials at a concentration of 100 mg per mL.
How many mL should be given to a patient who weighs 58 kg?
0.1920 mL of the anesthetic should be given to the patient.
Calculation of Dosage
First, we need to convert the patient's weight from kilograms to pounds:
58 kg x 2.20462 lb/kg = 128.01 lbs
Next, we can calculate the total amount of anesthetic to be given to the patient:
15 mg/10 lbs x 128.01 lbs = 19.20 mg
Finally, we can divide the total amount of anesthetic by the concentration of the anesthetic in each mL:
19.20 mg / 100 mg/mL = 0.1920 mL
So, 0.1920 mL of the anesthetic should be given to the patient.
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A cell phone carrier charges a fixed monthly fee plus a constant rate for each minute used.
Part 1. In January, the total cost for 100 minutes was $47 while in February, the total cost for 350 minutes
was $52. The constant charge for each minute used is:
O 0.02
O 0.04
O 0.03
The required constant charge for each minute used is $0.02.
What is linear equation of two variable?Algebraically speaking, an equation is a statement that shows the equality of two mathematical expressions. For instance, the two equations 3x + 5 and 14, which are separated by the 'equal' sign, make up the equation 3x + 5 = 14.
According to question:Let fixed charge of monthly be x and constant rate of each minute used is y.
So,
In january
x + 100y = 47--------(1)
and in February
x + 350y = 52-----------(2)
equation(2) - equation(1)
(x + 350y) - (x + 100y) = 52 -47
250y = 5
y = 0.02
Thus, $0.02 used per minutes.
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An exit poll shows how certain parties voted on Proposition #1. Explain how the given graph is deceptive.
Complete the statements based on the pie chart.
Pie charts are used to compare
, not
. This misleads the viewer to think
half the people in all three parties voted Yes on Issue #1, when
half of Republicans and Independents actually voted Yes on Issue #1.
1) ✔ parts of a whole
2)✔ the difference between groups
3)✔ less than
4)✔ more than
Pie charts are used to compare parts of a whole, not the difference between groups. This misleads the viewer to think more than half the people in all three parties voted Yes on Issue #1, when less than half of Republicans and Independents actually voted Yes on Issue #1.
What is a pie chart?This is a type of graph that resembles a pie because it is a circle that shows the proportion of different groups.
How to interpret the pie chart?Pie charts are used to determine percentages out of 100%, however, you cannot use them to compare groups.Due to this, you should be careful when interpreting the information. For example, in this case, you might think more than half the people in all three parties voted Yes but this might not be true.Note: Here is the missing information
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A student looks out of a second story school window and sees the top of the school flagpole at an angle of evelation of 22 degrees. The student is currently 18 feet above the ground. The building is 50 ft from the base of the flagpole. a. Draw a diagram of the above scenario. b. Find the height of the flagpole
The flagpole is visible to the student from 18 feet away and at a 22-degree inclination to the ground. The flagpole is 40.3 feet tall and 50 feet away from the structure.
At an elevation of 22 degrees, the student is looking out of a second-story school window. The building is 50 feet from the flagpole's base, and the window is 18 feet above the ground. We may compute the angle of the triangle formed by the window, the ground, and the top of the flagpole using the tangent ratio to determine its height. The opposing side (the flagpole) divided by the neighboring side equals the tangent of 22 degrees (the window to the ground). Consequently, the flagpole is 40.3 feet tall (50 divided by 1.224). This indicates that the student is 18 feet above the ground and is looking down at the flagpole.
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Caitlin currently owes $13,000 to a friend who is charging him interest of 2.90% p.m. He is required to settle the loan with two equal payments, one today and the other in six months. Calculate the size of the payments using six months as the focal date.
The size of the payments that need to be made today and in six months are both $ 6, 961.69.
How to find the payment ?To find the payment that Caitlin needs to make today and in six months, assume this amount of x.
The payment in 6 months would be:
= (13, 000 - x) ( 1 + 2.4 / 100)⁶
The payment today would be x.
Equate them to find the payment:
x = (13, 000 - x) ( 1 + 2.4 / 100)⁶
x = 14, 988 - 1.1529x
x = 14, 987.979 / 2.1529
= $ 6, 961.69
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Indicate the equation of the given line in standard form. The line containing the median of the trapezoid whose vertices are R(-1, 5) , S(1, 8), T(7, -2), and U(2, 0). Show calculations.
The linear function containing the median of the trapezoid whose vertices are R(-1, 5) , S(1, 8), T(7, -2), and U(2, 0). is given as follows:
y = -1.67x + 6.5.
How to define the linear function?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
In which:
m is the slope, representing the rate of change.b is the intercept, representing the value of y when x = 0.The median of the trapezoid is composed by these two points:
Midpoint of RS.Midpoint of TU.The coordinates of the midpoint of RS are given as follows:
x = (-1 + 1)/2 = 0.y = (5 + 8)/2 = 6.5.The coordinates of the midpoint of TU are given as follows:
x = (7 + 2)/2 = 4.5.y = (-2 + 0)/2 = -1.Hence the points of the function are:
(0, 6.5) and (4.5, -1).
When x = 0, y = 6.5, hence the intercept b is given as follows:
b = 6.5.
When x increases by 4.5, y decays by 7.5, hence the slope m is given as follows:
m = -7.5/4.5
m = -1.67.
Hence the function is:
y = -1.67x + 6.5.
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NO LINKS!! NOT MULTIPLE CHOICE!!!
56. Kotkit is the hottest new social media app. It is lowkey fire. It slays! There were no crumbs left. Users post their Tiktoks to Kotkit, but play them backwards. At the start of the year, there are 1500 users, but the number of users grows exponentially at a rate of 12% per day.
a. Write a function K(t) that expresses the number of users t days after the start of the year.
b. How many users are of the Kotkit after 30 days?
.c After how many days will the numbers of users reach one million users?
Answer:
[tex]\textsf{a)} \quad K(t)=1500(1.12)^t[/tex]
b) 44940
c) 57.4 days
Step-by-step explanation:
Exponential Growth Function
[tex]f(t)=a(1+r)^t[/tex]
where:
a is the initial value (y-intercept)r is the growth rate (in decimal form)t is the timeGiven values:
a = 1500r = 12% per day = 0.12t = time in daysPart aTo create a function K(t) that expresses the number of users t days after the start of the year, substitute the given values into the exponential growth formula:
[tex]\implies K(t)=1500(1+0.12)^t[/tex]
[tex]\implies K(t)=1500(1.12)^t[/tex]
Part bTo calculate how many users there are after 30 days, substitute t = 30 into the function K(t) from part a:
[tex]\implies K(30)=1500(1.12)^{30}[/tex]
[tex]\implies K(30)=44939.8831...[/tex]
[tex]\implies K(30)=44940\; \rm users[/tex]
Part cTo calculate after how many days it takes to reach one million users, substitute K(t) = 1,000,000 into the function and solve for t:
[tex]\implies 1000000=1500(1.12)^t[/tex]
[tex]\implies \dfrac{2000}{3}=1.12^t[/tex]
[tex]\implies \ln \left(\dfrac{2000}{3}\right)=\ln 1.12^t[/tex]
[tex]\implies \ln \left(\dfrac{2000}{3}\right)=t \ln 1.12[/tex]
[tex]\implies t=\dfrac{\ln \left(\frac{2000}{3}\right)}{\ln 1.12}[/tex]
[tex]\implies t=57.3755016...[/tex]
[tex]\implies t=57.4\; \rm days[/tex]
A researcher estimates that a fossil is 3200 years old. Using carbon-14 dating, a procedure used to determine the age of an object, the researcher discovers that the fossil is 4000 years old. A. Find the percent error. B. What other estimate gives the same percent error?
A. The percent error is 25%
B. Another estimate that gives the same percent error is the Theoretical value.
How to Find the Percent Error?A. To find the percent error, we use the following formula:
percent error = (|experimental value - theoretical value| / theoretical value) x 100
In this case, the experimental value is 4000 years (from carbon-14 dating) and the theoretical value is 3200 years (from the researcher's estimate). Plugging these values into the formula:
percent error = (|4000 - 3200| / 3200) x 100 = (800 / 3200) x 100 = 25%
B. To find an estimate that gives the same percent error, we can use the formula:
Theoretical value = experimental value / (1 + percent error / 100)
If we use the same percent error (25%) and experimental value (4000 years) as before:
Theoretical value = 4000 / (1 + 25/100) = 4000 / 1.25 = 3200
So, an estimate of 3200 years would give the same 25% percent error as the researcher's original estimate.
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(1.1) Convert 500 g to kg. (1.2) convert 1000 km to m. (1.3) Convert 1,65 ton to kg. (1.4) Convert 1,5 litre to millilitre. (1.5) Convert 50 m to cm. (1.6) if 1 inch is 2,54 cm, convert 12,5 cm to inches. (1.7) If 1 cup is 250 ml, how many cups will 1 250 ml be? (1.8) if 3 g of flour is 5 ml, how many ml will 10 g be? (1.9) Round the following number off to two decimal places: 135,2128 (1.10) Round the following number off to the nearest whole number: 250,62 (1.11) Round the following number off to the nearest ten: 123,84.
QUESTION 2 [6 MARKS] If an athlete passed the 5-km mark in a race after 19 min 58 seconds and the 10 km mark after 37 min 25 seconds, how long did it take him to complete the second 5 km?
Answer:
(1.1)
1000g=1kg
500÷1000=0.5kg
(1.2)
1000m=1km
1000×1000= 1000000m
(1.3)
1000kg=1tonne
165×1000= 165,000kg
Karen wants to buy new shoes. There is a promotion for 3
pairs of sneakers for $450.75, how much would one pair of
sneakers cost?
identify the sampling method that was used. three representatives from each employee type (faculty, staff, administrators) are chosen for a salary committee.
The sampling method that was used. three representatives from each employee type (faculty, staff, administrators) are chosen for a salary committee. This uses stratified sampling.
ABOUT STRATIFIED SAMPLEIn a stratified sample, researchers divide a population into homogeneous subpopulations called strata (the plural of stratum) based on specific characteristics (e.g., race, gender identity, location, etc.). Every member of the population studied should be in exactly one stratum.
Each stratum is then sampled using another probability sampling method, such as cluster sampling or simple random sampling, allowing researchers to estimate statistical measures for each sub-population.
Researchers rely on stratified sampling when a population’s characteristics are diverse and they want to ensure that every characteristic is properly represented in the sample. This helps with the generalizability and validity of the study, as well as avoiding research biases like under coverage bias.
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Express the following sum in sigma notation. Use 1 as the lower limit of summation and k for the index of summation. 1+4+9+16+25 +36 Express the sum in sigma notation. Σ" k 1
∑[tex](k^2)[/tex] for k = 1 to 6 is the sum in sigma notation for this series of terms.
The series is 1 + 4 + 9 + 16 + 25 + 36. The lower limit of summation is 1, which means, the sum starts at the initial term in the series.
The index of summation is k, which implies, the ongoing term in the series is represented by the variable k.
In the expression [tex]k^2[/tex] which implies that we are adding up the square of the index of summation.
So, the sigma notation for this series of terms is ∑[tex](k^2)[/tex] for k = 1 to 6, which means, we are adding up the squares of the numbers from 1 to 6 (1, 4, 9, 16, 25, and 36)
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The table shown below shows some values for functions f(x) and g(x). What is/are the solution(s) to f(x)=g(x)? Explain your answer(s).
a. The solutions to the system of equations f(x) = g(x) are given as follows: x = 0 and x = 1.
b. The explanation is that they have the same numeric value at these values of x.
How to interpret the solution of a system of equations?A system of equations is composed by multiple equations.
The solution to the system of equations is the point of intersection of all these equations, which happen when all these equations have the same numeric value.
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what is the constant of proportionality for the relationship shown in this table?
The required constant of proportionality for the relationship is 1/2.
What is Constant of proportionality?As an illustration, we may write y = kx, where x is the amount of gas in gallons, y is the price in dollars, and k is the proportionality constant. In other words, the cost is directly related to the number of gallons pumped.
According to question:Let the relation between x and y is
y = kx
Where k is proportionality constant.
Using the given values of x ad y
1 = k(2)
k = 1/2
So, the constant of proportionality for the relationship is 1/2.
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suppose you take a sample of 50 students from your school and measure their height. which one of the following is a random variable? group of answer choices
The mean of the sample is the random variable in the given problem.
The mean of any given data set can be defined as summarizing a given group of data. It is done in order to make data sets and their values easier to understand. It can be obtained by first adding all the given values and then dividing the total sum by the number of data entries.
A random variable is any quantity that depends on random events. Here, we know that the mean of the given sample depends on all of the random events or we can say random values present in the table. Hence, we can conclude that the random variable is the mean of the sample.
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if you are standing 45 feet away from a school and the angle of elevation is 66 degrees how tall is the building?
Answer:
50 m
Step-by-step explanation:
The angle of elevation of the top of the building from the foot of the tower is 30 and the angle of the top of the tower from the foot of the building is 60. the tower is 50 m high
the output column, s, was created from the student id 21444368 using the algorithm: if id digit n (left-to-right) is an even integer, place a 0 in s, row n (top-down) if id digit n (left-to-right) is an odd integer, place a 1 in s, row n (top-down) create a similar table, but with s reflecting the result of the algorithm applied to your student id (do this on scratch paper). on the quiz essay: record your student id, id
My student ID is 21444367.
The algorithm states that if the digit in the student ID (from left to right) is an even number, then a 0 is placed in the s column in the corresponding row (from top to bottom). If the digit in the student ID (from left to right) is an odd number, then a 1 is placed in the s column in the corresponding row (from top to bottom). Thus, the result of the algorithm applied to my student ID is s = 01101000.
This can be calculated by multiplying the even digits of the student ID by 0 and the odd digits of the student ID by 1 and then adding them together.
For my student ID, 21444367, the calculation would be:
0 * 2 + 1 * 1 + 0 * 4 + 1 * 4 + 0 * 4 + 1 * 3 + 0 * 6 + 1 * 7 = 01101000.
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a square fountain in the center of a shopping mall has an area of (4x^2+12x+9)ft^2. The dimensions of the fountain are of the form cx + d, where c and d are whole numbers. Find an expression for the perimeter of the fountain. Find the perimeter when x=2 ft.
A. An expression for the perimeter of the fountain is 8x + 12 ft.
B. The perimeter when x=2 ft. is 28 ft
How do we determine the values?The area of the fountain is given as 4x^2 + 12x + 9 square feet. Since the fountain is a square, we know that the length and the width of the fountain are equal. This means that we can express the area of the fountain in the form of (cx + d)^2.
Expanding (cx + d)^2 gives:
(cx + d)^2 = cx^2 + 2cxd + d^2
We can equate this to the given area of the fountain:
cx^2 + 2cxd + d^2 = 4x^2 + 12x + 9
To find the expression for the perimeter of the fountain, we need to find the dimensions (length and width) of the fountain. This can be done by factoring the area expression:
4x^2 + 12x + 9 = (2x + 3)(2x + 3)
So the dimensions of the fountain are (2x + 3) ft
The perimeter of a square is the sum of all four sides, so we can find the perimeter of the fountain by multiplying the length of one side by 4:
Perimeter = 4(2x + 3) ft = 8x + 12 ft
When x = 2 ft, the perimeter of the fountain is:
8(2) + 12 = 28 ft
So when x = 2 ft, the perimeter of the fountain is 28 ft.
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Simplify the expression. Assume that the denominator does not equal zero.Write any variables in alphabetical order.
12m−3n62m−7n
The simplified expression is 14m -10n
How to simplify the expressionFrom the question, we have the following parameters that can be used in our computation:
12m−3n62m−7n
Express properly
So, we have the following representation
12m - 3n + 2m - 7n
Collect the like terms
This gives
12m + 2m - 3n - 7n
Evaluate the like terms
So, we have the following representation
14m - 10n
Hence, the solution is 14m - 10n
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how to simplify 32/5
Answer: 6 2/5
Step-by-step explanation:
32 divided by 5 equals 6 with a remainder of 2.
How is the work energy theorem represented as an equation? (1 point)
a. ΔW = E
b.W = ΔΕ
c.W = E
d. ΔW = ΔΕ
The work energy theorem represented as an equation is ΔW = ΔΕ is appropriate option.
Option D is correct.
The work-energy theorem is what?A fundamental physics principle known as the work-energy theorem asserts that the amount of work done to an object is proportional to the change in its kinetic energy. To put it another way, the kinetic energy of an object is proportional to the net work done to it.
The work-energy theorem can be expressed as an equation because the change in an object's kinetic energy (W) is equal to the change in its work (W). It states that a change in an object's kinetic energy equals the net work done on it. The conservation of energy is a fundamental physics principle that states that energy cannot be created or destroyed; rather, it can only be transferred or converted from one form to another.
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13. Solve for x. Please help
Answer:
x = 15
Step-by-step explanation:
The sum of the four angles of a quadrilateral add up top 360°
Given:
m∠A = (5x+14)°
m∠b = 109°
m∠C = (3x+8)°
m∠D = 109°°
Here we are given actual values for two angles as 109°. Their sum is: 109 + 109 = 218°
The sum of the remaining two angles which are expressions in x should be:
360 - 218 = 142°
This is the sum of m∠A an m∠C
Substituting the terms for the two angle measures we get
(5x+14) + (3x+8) = 142
5x+14 + 3x+8 = 142
Grouping like terms:
5x + 3x + 14 + 8 = 142
8x + 22 = 142
8x = 142- 22
8x = 120
x = 120/4
x = 15
A wage earner receives a gross pay of $983.15 for 52.5 hours of work. What is his hourly rate of pay if a regular workweek is 44 hours and overtime is paid at timeand-a-half the regular rate of pay?
Answer:
$17.32
Step-by-step explanation:
44 hours
52.5 - 44 = 8.5 hours over time for 1.5x regular pay
assume regular pay is x
so 44x + 8.5 * 1.5x =983.15
44x + 12.75x = 983.15
56.75x=983.15
x=17.324
check to see if it's correct
44*17.324 + 12.75*17.324 = 983.14
On January 1, 2017, the capital balances in Hollings worth Company are Lois Holly $26,000 and Jim Worth $24,000. In 2017 the partnership reports net income of $30,000. The income ratio provides for salary allowances of $12,000 for Holly and $10,000 to Worth and the remainder to be shared equally. Neither partner had any drawings in 2017, Instructions
(1) Prepare a schedule showing the distribution of net income in 2017.
Answer:
see below
Step-by-step explanation:
1-1-2017
H 26K
W 24K
Net income 30K
H salary 12K
W salary 10K
remain: 8K shared EQUALLY so each gets 4K extra
Distribution:
30K - H gets12+4 =16K, and W gets 10+4 = 14K
now view the partitioned circle as a protractor, and think of using it to measure angles that are drawn on a page. give examples and make (rough) math drawings to show how this protractor could measure the following kinds of angles. in each case, determine the size of the angle in degrees and explain your answer.
The instrument used here is a protractor.
ProtractorAn instrument used to measure angles in degrees is a protractor. When measuring angles using a protractor, it's crucial to check that the vertex of the angle is in the protractor's centre and that the angle's base is lined up with the protractor's baseline.
Acute Angle: - An angle that is less than 90 degrees is considered to be acute. For instance, if the angle is 30 degrees, we would line up the vertex of the angle with the protractor's centre and one of its sides with the baseline. The measurement would be read off the protractor scale, which in this instance would be 30 degrees.
Right Angle: - A right angle is a triangle with a 90 degree angle on each side. Using a protractor, we would line up the vertex of the angle with the protractor's centre and one side of the angle with the baseline to measure a right angle. The measurement would be read off the protractor scale, which in this instance would be 90 degrees.
Obtuse Angle: - An angle that is more acute than 90 degrees is referred to as an obtuse angle. When using a protractor to measure an obtuse angle, we would line up the vertex of the angle with the protractor's centre and one side of the angle with the baseline. The measurement would then be taken from the protractor scale and would be more than 90 degrees in this instance.
Straight Angle: - A straight angle is one that is 180 degrees in length. Using a protractor, we would line up the vertex of the angle with the protractor's centre and one side of the angle with the baseline to measure a straight angle. The measurement would then be taken from the protractor scale, which would be 180 degrees in this instance.
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Circle E has a radius of 5.1 meters. FG is a chord and is a perpendicular bisector of FG. If FG = 6.3 meters, calculate the distance of IH.
The distance of IH is 1.089 meters.
What is Circle?Circle is a two dimensional figure which consist of set of all the points which are at equal distance from a point which is fixed called the center of the circle.
Given that,
Radius = 5.1 meters
EF = EI = 5.1
FG = 6.3 meters
HE is a perpendicular bisector of FG.
FH = HG = 6.3 / 2 = 3.15
Also, FH is the projection of EF in the vertical axis.
FH = EF sin (θ), where θ is the angle FEH.
3.15 = 5.1 sin (θ)
sin (θ) = 0.6176
θ = sin⁻¹ (0.6176)
θ = 38.14°
IH = EI - EH
We have, EH is the projection of EF into the horizontal axis.
EH = EF cos (θ)
Substituting,
IH = 5.1 - EF cos (θ)
IH = 5.1 - 5.1 cos (38.14)
IH = 5.1 ( 1 - cos (38.14))
IH = 1.089 meters
Hence the length of the line segment IH is 1.089 meters.
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57. The population of Asheville, NC can be modeled by P(t) = 90000 · e^(0.027t) as a function of years since the start of 2020.
a. State the growth rate. Round to three places
b. How many residents does the model predict will live in Asheville, NC in 2030.
Answer:
a) Growth rate of 2.737% (3 d.p.) per year
b) 117,897
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function with base $e$}\\\\$f(t)=ae^{rt}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $e$ is Euler's number. \\ \phantom{ww}$\bullet$ $r$ is the continuous growth/decay rate.\\ \phantom{ww}$\bullet$ $t$ is time.\\\end{minipage}}[/tex]
If a > 0 and r > 0, the function is an exponential growth function.
If a > 0 and r < 0, the function is an exponential decay function.
Given exponential function:
[tex]P(t) = 90000\cdot e^{0.027t}[/tex]
As r > 0 the function is an exponential growth function.
The annual growth rate is:
[tex]\begin{aligned}\textsf{Annual growth rate}& = e^{0.027}-1\\&=1.02736780...-1\\&=0.0273678027...\\&=2.73678027...\%\\&=2.737\%\; \sf (3\;d.p.)\end{aligned}[/tex]
To calculate how many residents the model predicts will live in Asheville, NC in 2030, substitute t = 10 into the given function:
[tex]\begin{aligned}\implies P(10) &= 90000\cdot e^{0.027 \cdot 10}\\&= 90000\cdot e^{0.27}\\ &= 90000\cdot 1.30996445...\\&=117896.800...\\&=117897\; \sf (nearest\;whole\;number)\\\end{aligned}[/tex]
Therefore, the model predicts that there will be a population of approximately 117,897 living in Asheville, NC in 2030.