It's important to know that a rational number can be expressed as fractions, but also when they are expressed as decimals, the decimal part repeats infinitely, that is, it has a pattern or finite decimal digits.
Having said that, we can deduct that all the answer choices are rational numbers.PLEASE I NEED THIS ANSWER ASAP!!!!!!
46% of employees judge their peers by the cleanliness of their workspaces. You randomly select 8 employees and ask them whether they judge their peers by the cleanliness of their workspaces. The random variable represents the number of employees who judge their peers by the cleanliness of their workspaces. Complete parts (a) through (c) below.
Using the binomial distribution, the probabilities are given by the image at the end of the answer.
Binomial distributionThe probability mass function is given as follows:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters of the function are described as follows:
n is the number of trials of the experiment.p is the probability of a success on a single trial of the experiment.x is the number of successes that we want to find the probability of.In the context of this problem, the values of these parameters are given as follows:
p = 0.46, as 46% of employees judge their peers by the cleanliness of their workspaces.n = 8, as you randomly select 8 employees and ask them whether they judge their peers by the cleanliness of their workspaces.To complete the table, we find each probability, as follows:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{8,0}.(0.46)^{0}.(0.54)^{8} = 0.0072[/tex]
[tex]P(X = 1) = C_{8,1}.(0.46)^{1}.(0.54)^{7} = 0.0493[/tex]
[tex]P(X = 2) = C_{8,2}.(0.46)^{2}.(0.54)^{6} = 0.1469[/tex]
[tex]P(X = 3) = C_{8,3}.(0.46)^{3}.(0.54)^{5} = 0.2503[/tex]
[tex]P(X = 4) = C_{8,4}.(0.46)^{4}.(0.54)^{4} = 0.2665[/tex]
[tex]P(X = 5) = C_{8,5}.(0.46)^{5}.(0.54)^{3} = 0.1816[/tex]
[tex]P(X = 6) = C_{8,6}.(0.46)^{6}.(0.54)^{2} = 0.0774[/tex]
[tex]P(X = 7) = C_{8,7}.(0.46)^{7}.(0.54)^{1} = 0.0188[/tex]
[tex]P(X = 8) = C_{8,8}.(0.46)^{8}.(0.54)^{0} = 0.0020[/tex]
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N8) solve the system using substitution method and then graph the equations.2x - 4y = -23x + 2y = 3-
Solution
Given:
2x - 4y = -2
3x + 2y = 3
Substitution method
[tex]\begin{gathered} From\text{ 3x+2y=3} \\ 3x=3-2y \\ x=\frac{3-2y}{3} \end{gathered}[/tex][tex]\begin{gathered} Substitute\text{ }x=\frac{3-2y}{3}\text{ into the first equation} \\ 2x-4y=-2 \\ 2(\frac{3-2y}{3})-4y=-2 \\ \frac{6-4y}{3}-4y=-2 \\ Multiply\text{ }trough\text{ by 3} \\ 6-4y-12y=-6 \\ 6-16y=-6 \\ -16y=-6-6 \\ -16y=-12 \\ y=\frac{-12}{-16} \\ y=\frac{3}{4} \end{gathered}[/tex][tex]\begin{gathered} Substitute\text{ y=}\frac{3}{4}\text{ into }x=\frac{3-2y}{3} \\ x=\frac{3-2(\frac{3}{4})}{3}=\frac{3-\frac{3}{2}}{3}=\frac{\frac{6-3}{2}}{3}=\frac{\frac{3}{2}}{3} \\ x=\frac{3}{6} \\ x=\frac{1}{2} \end{gathered}[/tex][tex]Thus,\text{ x=}\frac{1}{2},y=\frac{3}{4}[/tex]Graphical method:
Plot the graph of the two equations on the same graph
The point of intersection of the two graphs gives the solution to the system of equations
The point of intersection is (0.5, 0.75)
Which in fraction gives (1/2, 3/4)
Thus. x = 1/2, y= 3/4
which of the following are the coordinates of point B on the directed line segment AC, such that AB is 1/5 of AC?
Answer:
The coordinates of point B is;
[tex](5,-7)[/tex]Explanation:
Given the attached image;
The coordinate of point A is;
[tex](8,-8)[/tex]The coordinate of point C is;
[tex](-7,-3)[/tex]If AB is 1/5 of AC;
[tex]\Delta x_{AB}=\frac{1}{5}(\Delta x_{AC})_{}_{}_{}_{}_{}_{}[/tex]So; let (x,y) represent the coordinates of B;
[tex]\begin{gathered} (8-x)=\frac{1}{5}(8-(-7)) \\ 8-x=\frac{1}{5}(15) \\ 8-x=3 \\ x=8-3 \\ x=5 \end{gathered}[/tex]The same applies to y coordinate;
[tex]\Delta y_{AB}=\frac{1}{5}(\Delta y_{AC})_{}[/tex]So;
[tex]\begin{gathered} (-8-y)=\frac{1}{5}(-8-(-3)) \\ -8-y=\frac{1}{5}(-8+3) \\ -8-y=\frac{1}{5}(-5) \\ -8-y=-1 \\ y=-8+1 \\ y=-7 \end{gathered}[/tex]Therefore, the coordinates of point B is;
[tex](5,-7)[/tex]a is less than or equal to 10
The expression of the mathematical statement is a ≤ 10
How to represent the mathematical statement as an expression?From the question, we have the following mathematical statement that can be used in our computation:
a is less than or equal to 10
The key statement less than or equal to in mathematics and algebra can be represented using the following symbol
less than or equal to ⇒ ≤
So, we have the following representation
a is less than or equal to 10 ⇒ a is ≤ 10
This implies that we rewrite the above expression as follows
So, we have
a is less than or equal to 10 ⇒ a ≤ 10
The above expression cannot be further simplified
So, we leave it like that
Hence, the mathematical statement when expressed as an expression is a ≤ 10
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A Census Burcau report on the income of Americans says that with 95% confidence themedian income of all U.S. households is $49,841 to $50,625. The point estimate and margin oferror for this interval are: *Point estimate = $49,841; Margin of error = $784Point estimate = $50,233; Margin of error = $784oPoint estimate = $50,233; Margin of error = $392Point estimate = $50,625; Margin of error = $392
Let the point estimate be x and the margin of error be e.
Then, we must have
[tex]\begin{gathered} x+e=50625----------------------(1) \\ x-e=49841----------------------(2_{}) \end{gathered}[/tex]Add the equation (1) and equation(2) to eliminate the variable e, we have
[tex]\begin{gathered} 2x=100466 \\ \text{ thus} \\ x=\frac{100466}{2}=\text{ \$}50233 \end{gathered}[/tex]Subtracting equation (2) from equation(1) to eliminate the variable x, we have
[tex]\begin{gathered} 2e=784 \\ \text{ thus} \\ e=\frac{784}{2}=392 \end{gathered}[/tex]Hence, the point estimate is $50233 and the margin of error is $392, The Third option
Identify the postulate illustrated by the statement: Line ST connects pointS and point T
We have two points known to be ( S ) and ( T ). A line connects two points.
The minimum number of points that are required to form a straight line in a cartesian coordinate system are ( two ).
The minimum number of points that are required to form a plane in a cartesian coordinate system are ( three ) which will form two vectors i.e it requires two lines formed with a common point.
Two planes always intersect at exactly one point with direction normal to the two plane normal vectors.
Hence, the only possible postulate that relates two points is the formation of a line between two points; hence, the correct postulate for the given statement is:
[tex]\text{\textcolor{#FF7968}{Through any two points there is exactly one line}}[/tex]
There are no solutions to the system of inequalities shown below. y< 4X-6 y > 4x + 2 A.true B. false
The graphs of both inequalities is shown below;
Please note that the red region with the broken lines represents y < 4x - 6
The blue blue region with the broken lines represent y > 4x + 2
Observe carefully that both graphs run parallel to each other and there is no point of intersection. This means there is no values of x and y that can satisfy both inequalities.
Simply put, there are no solutions to the system of inequalities shown.
The answer is
A: TRUE
PR and SU are parallel lines. Which angles are corresponding angles?
Given
PR and SU are parallel lines.
To find the pair of corressponding angles.
Explanation:
From, the figure,
Since PR and SU are parallel and the corressponding angles lie in the same corner.
Then,
[tex]\begin{gathered} \angle PQO,\angle STQ \\ \text{are corressponding angles.} \end{gathered}[/tex]Hence, the answer is Option c).
if 453 runners out of 620 completed a marathon, what percent of the funners finished the race?
Answer: 73.1%
Step-by-step explanation:
620/453 = 73.1%
Pls check so you can see if correct
A paving company has 24 employees, 15 with gross earnings of $365 per week and 9 with gross earnings of $385 per week. What is the total social security and medicade for the first quarter of the year
The total social security and medicade for the first quarter of the year is $17,781.66.
How to calculate the tax?The computation will be:
Gross earning per week = 15 * 365 + 9 * 385
= $8,940 per week
Here, the number of weeks is 13 in each quarter:
Gross earning =$8,940 * 13
= $116,220
Social security tax = $116,220 * 6.2%
= $7,205.64
Medicare tax = $116,220 * 1.45%
= $1,685.19
Total = $7,205.64 + $1,685.19
= $8,890.83
Now, to involve the employer's share it is required to multiply the total tax by 2
Therefore,
Total tax remitted to IRS = $8,890.83 * 2
= $17,781.66
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If the snow is falling at a rate of 1 inches per hour, how many hours will it take to snow 12 inches?
Imagine the following, we will place a tube under the snow. So we have the following
After one hour, the tube will be filled with 1 inch of snow.
After 2 hours, we will have one inch more
So, one way to calculate the amount of snow after a specific amount of hours, is simply multiplying the hours by the rate at which the height of the snow changes. IN here, the height changes 1 inch per hour. So after x hours the height of the snow would be
[tex]1\cdot\text{ x }[/tex]We want to find x, such that the height of the snow is 12.
So we have the equation
[tex]1\cdot x\text{ = 12}[/tex]which gives us that in 12 hours we will have 12 inches of snow.
If $19,000.00 is invested in an account for 30 years. Find the value of the investment at the end of 30 years if the interestis:(a) 7% simple interest:(b) 7% compounded monthly:
Hello there. To answer this question, we need to remember some properties in simple and coumpound interests investments.
For simple interest, the balance will be equal to P(1 + rt), in which P is the amount invested, r is the interest rate in years and t is the time (can be either years of months).
For compound interest, the balance will be equal to P(1 + r)^t.
So, using the values P = $19,000.00 and the time is equal to 30 years, we have for:
a) 7% simple interest
It means that r = 7% and then we can use the first formula
19000(1 + 0.07*30)
We converted the rate to decimals above
Multiplying the values, we have:
19000(1 + 2.1)
19000*3.1
$58.900
b) 7% compounded monthly
First, we need to convert the time from years to months, multiplying by 12
30*12 = 360 months
Using the second formula, we have:
19000(1 + 0.07)^(360)
Sum the values into parenthesis
19000*1.07^(360)
Given a Cost of $9.00 and a Percent Markup on Cost of 30% find the Selling Price.
Markup (or price spread) is the difference between the selling price of a good or service and cost. It is often expressed as a percentage over the cost.
Given:
cost = $9.00
percent markup = 30%
Let the selling price be x
The formula form percent markup is:
[tex]\text{ \% markup = }\frac{\text{ Selling price - cost}}{\cos t}\text{ }\times\text{ 100 \%}[/tex]Substituting we have;
[tex]30\text{ = }\frac{x\text{ - 9}}{9}\text{ }\times100[/tex]Solving for x:
[tex]\begin{gathered} \text{x - 9 = 2.7} \\ x\text{ = 11.7} \end{gathered}[/tex]Hence, the selling price is $11.7
Answer: $11.7
A cylinder sits on top of the rectangular prism. What is the combined volume? (use the Pi, round to the nearest tenth of an inch) ______ in3
The combined volume is:
[tex]V=V_{rp}+V_c[/tex]The volume of the rectangular prism is:
[tex]V_{rp}=l\cdot w\cdot h[/tex]The volume of a cylinder is:
[tex]V_c=\pi\cdot r^2\cdot h[/tex]Then, the combined volume is:
[tex]\begin{gathered} V=l_{rp}\cdot w_{rp}\cdot h_{rp}+\pi\cdot r^2\cdot h_c \\ \\ V=10m\cdot5m\cdot3m+\pi\cdot(2m)^2\cdot4m \\ V=150m^3+16\pi m^3 \\ V=(150+16\pi)m^3 \\ \\ V\approx200.3\text{ }m^3 \end{gathered}[/tex]Turn into inches:
[tex]200.3m^3\cdot\frac{61023.7in^3}{1m^3}=12223047in^3[/tex]Then, the volume in inches is 12,223,047 cubic inches (200.3 cubic meters)
Simplify (5x + 7) - (x + 2)
You have the following expression:
(5x + 7) - (x + 2)
in order to simplify the previous expression, eliminate parenthesis and take into account that if a parenthesis is preceeded by a minus sign, when you elminate th eparenthesis the sign inside change to the opposite, just as follow:
(5x + 7) - (x + 2) =
5x + 7 - x - 2 =
5x - x + 7 - 2 =
4x + 5
Hence, the simplified expression is 4x + 5
Two methods to solve (X+3)^2=6
The solution of the given equation is [tex]-3+\sqrt{6}[/tex] and [tex]-3-\sqrt{6}[/tex].
Given equation:-
[tex](x+3)^2=6[/tex]
We have to find the value of x by solving the given equation.
We can rewrite the given equation as:-
[tex]x^2+6x+9=6\\x^2+6x+3=0[/tex]
We can solve the the quadratic equation by finding the discriminant.
[tex]x = \frac{-6+-\sqrt{6^2-4*1*3} }{2*1}[/tex]
[tex]x = \frac{-6+-\sqrt{36-12} }{2}[/tex]
[tex]x=\frac{-6+-2\sqrt{6} }{2}=-3+-\sqrt{6}[/tex]
Hence, the values of x are [tex]-3+\sqrt{6}[/tex] and [tex]-3-\sqrt{6}[/tex].
Discriminant
In arithmetic, a polynomial's discriminant is a function of the polynomial's coefficients.
Quadratic equation
The polynomial equation whose highest degree is two is called a quadratic equation or sometimes just quadratics. It is expressed in the form of:
ax² + bx + c = 0
where x is the unknown variable and a, b and c are the constant terms.
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Graph for a 3rd degree polynomial function whose graph crosses the horizontal axis more than one
Given the 3° degree function:
[tex]x^3-4x+2[/tex]Graph:
6. Express the given function h as a composition of two functions f and g
such that H(x) = (fog)(x).
a) H(x) = |3x +2|
b) H(x) = √√√√5x +4
The given function can be represented f(x) and g(x) as below
What are functions?
A function from X to Y is an assign of each constituent of Y to each variable of X. The set X is known as the function's scope, while the set Y is known as the function's image domain. The notation f: XY denotes a function, its domain, and its codomain, and the value of a function f at an element x of X, indicated by f(x), is known as the image of x under f, or the value of f applied to the argument x. When defining a function, the domains and codomain are not often explicitly specified, and without performing some (complicated) calculation, one may only know that perhaps the domain is included in a larger package.
The functions are
(a) f(x) = 3x+2 and g(x) = |x|
so, H(x) = f(g(x)) = |3x+2|
(b) f(x) = 5x+4 and g(x) = √√√√x
so, H(x) = f(g(x)) = √√√√5x+4
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if you halved a recipe that calls for 5 c. chicken broth how much broth would you use
If halved a recipe that calls for 5 c chicken broth, then you would end up using 2.5 c chicken broth (that is two and half c of chicken broth).
help meeeeeeeeee pleaseee !!!!!
For the two given functions, the compositions are:
(f o g)(x) = √(2x + 3)(g o f)(x) = 2*√x + 3How to find the two compositions?
Here we have two functions:
f(x) = √x
g(x) = 2x + 3
Now we want to get the compositions:
(f o g)(x) = f( g(x))
So here we just need to evaluate f(x) in g(x), we will get:
(f o g)(x) = √g(x) = √(2x + 3)
The other composition is:
(g o f)(x) = g(f(x)) = 2*f(x) + 3 = 2*√x + 3
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List all real values of x such that f(x) = 0, if there are no such real x, type DNE in the answer blank. If there is more than one real x, give a comma separated list (i.e: 1, 2) X =
Given the function defined as:
[tex]\begin{gathered} f(x)=-7+\frac{-8}{x-6} \\ \end{gathered}[/tex]The function can further be expressed as:
[tex]f(x)=-7-\frac{8}{x-6}[/tex]Find the LCM of the function;
[tex]\begin{gathered} f(x)=\frac{-7(x-6)-8}{x-6} \\ f(x)=\frac{-7x+42-8}{x-6} \\ f(x)=\frac{-7x+34}{x-6} \\ \end{gathered}[/tex]If f(x) = 0, then the value of x is calculated as:
[tex]\begin{gathered} \frac{-7x+34}{x-6}=0 \\ -7x+34=0 \\ -7x=0-34 \\ -7x=-34 \end{gathered}[/tex]Divide both sides of the equation by -7:
[tex]\begin{gathered} \frac{\cancel{-7}x}{\cancel{-7}}=\frac{\cancel{-}34}{\cancel{\square}7} \\ x=\frac{34}{7} \end{gathered}[/tex]Therefore the value of x if f(x) = 0 is 34/7
Rosa needs to build a wall. She has to start the wall with one postand then every 5.75 feet put another post. The wall will be166.75 feet long. How many posts will she need?
For every 5.75 feet, here is one post.
Determine the number of posts in a wall of 166.75 feet.
[tex]\begin{gathered} p=\frac{166.75}{5.75} \\ =29 \end{gathered}[/tex]So 29 posts needed for the wall.
4. The relationship between temperature expressed in degrees Fahrenheit(F) and degrees Celsius (C) is given by the formula F= (9/5)C + 32. If the temperature is 5 degrees Fahrenheit, what is it in degrees Celsius ?
To calculate which value in Celsius the temperature of 5 Fº equates to, we first need to rewrite the expression isolating the "C" variable on the left side.
[tex]\begin{gathered} F=\frac{9}{5}\cdot C+32 \\ \frac{9}{5}\cdot C=F-32 \\ 9\cdot C=5\cdot F-160 \\ C=\frac{5}{9}\cdot F-\frac{160}{9} \\ \end{gathered}[/tex]We now need to replace F by 5.
[tex]\begin{gathered} C=\frac{5}{9}\cdot5-\frac{160}{9} \\ C=\frac{25}{9}-\frac{160}{9} \\ C=\frac{-135}{9} \\ C=-15 \end{gathered}[/tex]The temperature is -15 degrees in Celsius.
Select the correct answer.Christi is using a display box shaped like a regular pentagonal prism as a gift box. About how much gift wrap does she need to completely coverthe box?A 800 cm²B. 480 cm2C. 1,020 cm²D. 1,600 cm²
Given: A regular pentagonal prism with base edge 8cm and height 20 cm .
Find: wrap need to cover the box.
Explanation: for to find the length of wrap we need to find the area of regular pentagonal prism .
[tex]A=5ah+\frac{1}{2}\sqrt{5(5+2\sqrt{5)}}a^2[/tex]
where a=base edge=8cm and h =height=20 cm
[tex]\begin{gathered} A=5\times8\times20+\frac{1}{2}\sqrt{5(5+2\sqrt{5})}\times8^2 \\ =1020.2211\text{ cm}^2 \end{gathered}[/tex]Final answer: the required answer is 1020 square centimeter.
Answer:
C. 1,020 [tex]cm^{2}[/tex]
Hope this helps!
Step-by-step explanation:
After adding the two equations to eliminate x you are left with 4y=-8
solve for y
[tex]\begin{gathered} \frac{4y}{4}=-\frac{8}{4} \\ y=-2 \end{gathered}[/tex]then, solve for x
[tex]\begin{gathered} 2x-2=4 \\ 2x-2+2=4+2 \\ 2x=6 \\ \frac{2x}{2}=\frac{6}{2} \\ x=3 \end{gathered}[/tex]x = 3
y = -2
Write the equation of a line that passes through the point (-2,-12) is parallel to the equation y= 2x +3
The most appropriate choice for equation of line in slope intercept form will be given by-
y = 2x - 8 is the required equation of line
What is equation of line in slope intercept form?
The most general form of equation of line in slope intercept form is given by y = mx + c
Where m is the slope of the line and c is the y intercept of the line.
Slope of a line is the tangent of the angle that the line makes with the positive direction of x axis.
If [tex]\theta[/tex] is the angle that the line makes with the positive direction of x axis, then slope (m) is given by
m = [tex]tan \theta[/tex]
The distance from the origin to the point where the line cuts the x axis is the x intercept of the line
The distance from the origin to the point where the line cuts the y axis is the y intercept of the line
Here,
The given equation of line is y= 2x +3
Slope of this line = 2
Slope of the line parallel to this line = 2
The line passes through (-2, -12)
Equation of the required line = y - (-12) = 2(x - (-2))
= y + 12 = 2x + 4
= y = 2x + 4 - 12
= y = 2x - 8
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question In photograph
The equation that represents the relationship between x and y in the table is (L.) y = -5x + 3.
What is an Equation in Math?In mathematics, an equation is a relationship between two expressions that are expressed as equality on each side of the equal to sign.
Given in the table is the relationship between x and y respectively.
Substitute the values of x in the respective equations to find the value of y, the resulting value which matches the value of y in the table determines the correct equation.
J. y = -5x -27
⇒ For x = -3, y = -5(-3) - 27 = 15 -27 = -12 ≠ 18
K. y = -5x + 18
⇒ For x = -3, y = -5(-3) + 18 = 15 + 18 = 33 ≠ 18
L. y = -5x + 3
⇒ For x = -3, y = -5(-3) + 3 = 15 + 3 = 18 ≈ 18
For x = -1, y = -5(-1) + 3 = 5 + 3 = 8
For x = 2, y = -5(2) + 3 = -10 + 3 = -7
For x = 6, y = -5(6) + 3 = -30 + 3 = -27
All the values of x and y in the table satisfy the equation y = -5x + 3. Hence this is the required equation that represents the relationship.
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Sales representatives of a new line of computers predict that sales can be approximated by the function (0= 1350 + 6101n(31+ e), where is measured in years.What are the predicted sales in 15 years? Round your answer to the nearest whole number
It is predicted that sales over time can be approximated by the function:
[tex]S(t)=1350+610\ln(3t+e)[/tex]It is required to find the predicted sales in 15 years to the nearest whole number.
To do this, substitute t=15 into the given function:
[tex]S(15)=1350+610\ln(3\cdot15+e)=1350+610\ln(45+e)[/tex]Evaluate and round to the nearest whole number as required:
[tex]S(15)=1350+610\ln(45+e)\approx3708[/tex]The sales in 15 years is about 3708.
Greg's youth group is collecting blankets to take to the animal shelter. There are 38 people in the group, and they each gave 2 blankets. They got an additional 29 by asking door-to-door. They set up boxes at schools and got another 52. Greg works out that they have collected a total of 121 blankets. Does that sound about right?
We want to know the total of blankets that Greg's collected.
As there are 38 people in the group, and they each gave 2 blankets, they brough a total of 79 blankets.
As they got 29 asking door-to-door, and got another 52, we will sum the values, as shown:
[tex]79+29+52=160[/tex]This means that the Greg group collected a total of 160 blankets, instead of 121, and the Greg statement is false.
10. A city has a population of 125,500 in the year 1989. In the year 2007, its population is 109, 185. A. Find the continuous growth/decay rate for this city. Be sure to show all your work.B. If the growth/decay rate continues, find the population of the city in the year 2021.C. In what year will the population of the city reach 97,890? Be sure to show all your work.
SOLUTION
A.
To solve this question, we will use the compound interest formula.
Which is:
[tex]\begin{gathered} A=P(1-\frac{r}{100})^{nt} \\ Since\text{ we are dealing with a yearly statistics, n = 1} \end{gathered}[/tex][tex]\begin{gathered} \text{From 1989 to 2007, there is a year difference of 18 years} \\ t=18 \\ A=109,185 \\ P=125,500 \\ We\text{ are looking for the continuous growth rate (r)} \\ \text{Now, we will substitute all these given parameters into the formula } \\ \text{above.} \end{gathered}[/tex][tex]\begin{gathered} 109,185=\text{ 125,500(1-}\frac{r}{100})^{18} \\ \frac{195185}{125500}=\frac{125500}{125500}(1-\frac{r}{100})^{18} \\ 0.87=(1-\frac{r}{100})^{18} \\ \text{take the natural logarithm of both sides:} \\ \ln 0.87=18\ln (1-\frac{r}{100}) \\ -0.1393=18\ln (1-\frac{r}{100}) \\ \frac{-0.1393}{18}=\ln (1-\frac{r}{100})_{}_{}_{}_{}_{} \\ -0.007737=\ln (1-\frac{r}{100}) \\ \end{gathered}[/tex][tex]\begin{gathered} e^{-0.007737}=(1-\frac{r}{100}) \\ 0.9922=1-\frac{r}{100} \\ \frac{r}{100}=1-0.9922 \\ \frac{r}{100}=0.007707 \\ r=100\times0.007707 \\ r=0.771\text{ \%} \end{gathered}[/tex]The continuous decay rate is 0.771%
B.
Using the same formula:
[tex]\begin{gathered} A=P(1-\frac{r}{100})^{nt} \\ t=2021-2007=14 \\ P=109,185 \\ n=1 \\ A=\text{?} \\ r=0.771 \\ \text{Substitute all the parameters into the formula above:} \end{gathered}[/tex][tex]\begin{gathered} A=P(1-\frac{r}{100})^{nt} \\ A=109,185(1-\frac{0.771}{100})^{1\times14} \\ A=109,185\times0.89730607 \\ A=97,972.36 \\ A=97,972\text{ (to the nearest person)} \end{gathered}[/tex]The population of the city in the year 2021 is 97,972.
C.
We will use the same formula:
[tex]\begin{gathered} A=P(1-\frac{r}{100})^{nt} \\ A=97,890 \\ P=125,500 \\ r=0.771 \\ t=\text{?} \\ \text{Substitute all these parameters into the formula above:} \\ \end{gathered}[/tex][tex]\begin{gathered} 97890=125,500(1-\frac{0.771}{100})^t^{} \\ \frac{97890}{125500}=\frac{125500}{125500}(0.99229)^t \\ 0.78=0.99229^t \\ \ln 0.78=t\ln 0.99229 \\ -\frac{0.2485}{\ln 0.99229}=t \\ t=32.101 \\ SO\text{ the year that the population will reach 97,890 will be:} \\ 1989+32.101=2021.101 \\ \text{Which is approximately year 2021.} \end{gathered}[/tex]