Answer:
Mr. Smith gave $1110.7 to the agency
Explanation:
Given that the job Mr. Smith took pays $76k per year.
In his first 4 weeks, taking the number of weeks per year to be 52, he is paid:
4 × ($76000)/52
= 4 × $1461.5 (Approximately)
= $5846
He is charged 19% of this earnings:
19% of $5846 is:
0.19 × $5846
= $1110.7
What he has in the end is:
$17000 - $1110.7
= $15889.3
A sample has a mean of M = 90 and a standard devia-tion of s 20.=a. Find the z-score for each of the following X values.X = 95X = 80X = 98X = 88X = 105X = 76
Answer:
[tex]\begin{gathered} X=95,z=0.25 \\ X=80,z=-0.5 \\ X=98,z=0.4 \\ X=88,z=-0.1 \\ X=105,z=0.75 \\ X=76,z=-0.7 \end{gathered}[/tex]Explanation:
Given a sample with the following:
• Mean,M = 90
,• Standard deviation, s = 20
To find the z-score for each of the given X values, we use the formula below:
[tex]\begin{equation*} z-score=\frac{X-\mu}{\sigma}\text{ where }\begin{cases}{X=Raw\;Score} \\ {\mu=mean} \\ {\sigma=Standard\;Deviation}\end{cases} \end{equation*}[/tex]The z-scores are calculated below:
[tex]\begin{gathered} \text{When X=95, }z=\frac{95-90}{20}=\frac{5}{20}=0.25 \\ \text{When X=80, }z=\frac{80-90}{20}=\frac{-10}{20}=-0.5 \\ \text{When X=98, }z=\frac{98-90}{20}=\frac{8}{20}=0.4 \end{gathered}[/tex][tex]\begin{gathered} \text{When X=88,}z=\frac{88-90}{20}=\frac{-2}{20}=-0.1 \\ \text{When X=105, }z=\frac{105-90}{20}=\frac{15}{20}=0.75 \\ \text{When X=76, }z=\frac{76-90}{20}=\frac{-14}{20}=-0.7 \end{gathered}[/tex]solve2cos²x+3sin x=0
One of the most important trigonometric identities is the one below
[tex]\cos ^2x+\sin ^2x=1[/tex]Then,
[tex]\Rightarrow\cos ^2x=1-\sin ^2x[/tex]Use this result in the equation given by the problem
[tex]\begin{gathered} 2\cos ^2x+3\sin x=0 \\ \Rightarrow2(1-\sin ^2x)+3\sin x=0 \\ \Rightarrow2-2\sin ^2x+3\sin x=0 \end{gathered}[/tex]Furthermore,
[tex]\begin{gathered} \Rightarrow2\sin ^2x-3\sin x-2=0 \\ \Rightarrow2\sin ^2x-3\sin x=2 \\ \Rightarrow\sin x(2\sin x-3)=2 \end{gathered}[/tex]Set y=sinx
[tex]\begin{gathered} \Rightarrow y(2y-3)=2 \\ \Rightarrow y=-\frac{1}{2},y=2 \end{gathered}[/tex]Thus,
[tex]\begin{gathered} \sin x=-\frac{1}{2},\sin x=2 \\ \Rightarrow\sin x=-\frac{1}{2} \\ \text{if sinx=2, then x is not a real number} \end{gathered}[/tex]Finally,
[tex]\begin{gathered} \sin x=-\frac{1}{2} \\ \Rightarrow x=\sin ^{-1}(-\frac{1}{2})=-\frac{\pi}{6} \end{gathered}[/tex]Then, the answer is
[tex]x=-\frac{\pi}{6}+2n\pi,\text{ n is an integer}[/tex]The answer is x=-pi/6+2n*pi
You roll two 4-sided number cubes. Let X be the sum of the numbers on the two cubes P(X=3)
Notice that, since we are rolling two dices, we are dealing with 2 independent events
Let's suppose that the numbers on each dice are 1,2,3,4.
Then, if we roll two dices, there are 16 possible pairs (in principle)
(1,1)
(1,2)
...
(1,4)
(2,1)
...
(4,4)
Each of those pairs is equally probably than the others
Know, we need to identify the pairs that add up 3:
(1,2) and (2,1) are the only possibilities
Finally, we obtain the probability:
[tex]P(x=3)=\frac{2}{16}=\frac{1}{8}=0.125[/tex]Which is none of the options.
Another possibility is that the dices are cubes:
So, the possible combinations that can be obtained by throwing 2 cubes are:
(1,1)...(1,6),(2,1),...(6,6) Which are 36 possible combinations
And there are only 2 combinations that add up to 3: (2,1),(1,2)
So, the probability of adding up 3 is:
[tex]P(x=3)=\frac{2}{36}=\frac{1}{16}=0.0625[/tex]find the measure of the angles in the following triangles.x=______missing angle measures:_______ ______
First, we have to use the interior angles theorem
[tex]x-2+72+x=180[/tex]Then, we solve for x
[tex]\begin{gathered} 2x+70=180 \\ 2x=180-70 \\ 2x=110 \\ x=\frac{110}{2} \\ x=55 \end{gathered}[/tex]Then, we use this value to find each angle
[tex]\begin{gathered} (x-2)=55-2=53 \\ x=53 \end{gathered}[/tex]Hence, the interior angles are 72°, 55°, and 53°Is 128 degrees plus 62 degrees supplementary or complimentary?
Let's begin by identifying key information given to us:
One angle = 128 degrees, Second angle = 62 degrees
Complementary angle are angles that sum up to 90 degrees
Supplementary angles are angles that sum up to 180 degrees
We will proceed to sum these two angles together. We have:
[tex]\begin{gathered} 128^{\circ}+62^{\circ}=180^{\circ} \\ \therefore These\text{ angles are supplementary angles since they sum up to 180 degr}ees \end{gathered}[/tex]Therefore, these angles are supplementary angles since they sum up to 180 degrees
Pauline found the inverse of [9/7 4/3] to be [-3/7 4/-9]. Which calculations will confirm that his (or her) answer is correct? Select all that apply.
In this case, we'll have to carry out several steps to find the solution.
Step 01:
data:
matrix
Step 02:
inverse of a matrix:
[tex]AA^{-1}=\begin{bmatrix}{9} & {4} \\ {7} & {3}\end{bmatrix}\begin{bmatrix}{-3} & {4} \\ {7} & -{9}\end{bmatrix}=\begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix}[/tex][tex]A^{-1}A=\begin{bmatrix}{-3} & {4} \\ {7} & {-9}\end{bmatrix}\begin{bmatrix}{9} & {4} \\ {7} & {3}\end{bmatrix}=\begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix}[/tex]That is the full solution.
55°Angle FGH is a right angle.The measure of angle FGJ is 559.The measure of angle JGH is xº.What is the value of x?toХS
Given angle FGH is a right angle. This means angle FGH is 90 degrees
But
factorize 5m^3-10m^2+8m+2
The expression 5m³ - 10m² + 8m + 2 cannot be factorized
How to factorize the polynomial?From the question, the polynomial is given as
5m^3-10m^2+8m+2
Rewrite the polynomial properly
So, we have
5m³ - 10m² + 8m + 2
Next, we represent the above expression on a graph
From the attached graph, we can see that the polynomial expression only cross the x-axis at x = -0.197
This implies that the expression cannot be factored
So, we cannot further rewrite the given expression
Read more about factorized expressions at
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Find the value of b please I’m having some trouble
Given:
The expression given is,
[tex]3-\frac{2}{9}b=\frac{1}{3}b-7[/tex]Required:
To find the value of b.
Explanation:
We have the given expression as:
[tex]3-\frac{2}{9}b=\frac{1}{3}b-7[/tex]Let us find the value of b.
[tex]\begin{gathered} 3-\frac{2}{9}b=\frac{1}{3}b-7 \\ \Rightarrow\frac{b}{3}+\frac{2b}{9}=3+7 \\ \Rightarrow\frac{3b+2b}{9}=10 \\ \Rightarrow\frac{5b}{9}=10 \\ \Rightarrow5b=9\times10 \\ \Rightarrow5b=90 \\ \Rightarrow b=\frac{90}{5} \\ \Rightarrow b=18 \end{gathered}[/tex]Final Answer:
The value of b is,
[tex]b=18[/tex]14. Which points are on the graph by = 3x - 12? Select all that apply. (A) (B) (C) (D) (-2,0) (0,-2) (6,-3) (8,2) M 23 hp
6y=3x-127
Replace the values of x , y in the equation by the coordinates values given on each option, and check if the equality remains.
A. (x,y) = (-2,0)
6y=3x-127
6(0) =3(-2)-127
0 = -6-127
0= -133
zero is not equal to -133.
A. is not
B. (0,-2)
6(-2) = 3(0)-127
-12 = -127
B is not
C, (6,-3)
6(-3) = 3(6)-127
-18 = 18-127
-18 = -109
C is not
D (8,2)
6(2) = 3(8)-127
12 = 24-127
12=-103
D is not
None of the points are in the graph
how many hours, x, do they run?
We know that
• Sean runs 6 miles per hour (rate).
,• Sean's initial condition is 0.25 miles.
,• Darryl runs 0.7 miles per hour faster than Sean (we have to sum).
Sean's expression would be 6x + 0.25.
Darryl's expression would be 6x + 0.7x = 6.7x.
Now, we make them equal
[tex]6x+0.25=6.7x[/tex]Hence, the right answer is C.Solve for w.182- w = 252
Given the equation :
[tex]182-w=252[/tex]solve for w, so:
subtract 182 from both sides:
[tex]\begin{gathered} 182-w-182=252-182 \\ \\ -w=70 \\ \end{gathered}[/tex]multiply both sides by -1
[tex]w=-70[/tex]So, the value of w = -70
Hi thank you so for all your time and help in advance
Question:
Solution:
Remember the following definition :
[tex]\cos (x)=\frac{adjacent\text{ side to the angle x }}{hypotenuse}[/tex]thus, if:
[tex]\cos (x)=\frac{adjacent\text{ side to the angle x }}{hypotenuse}=\frac{\sqrt[]{3}}{2}[/tex]then, according to the given triangles in the problem, the only angle that fulfills this, is:
[tex]x=30[/tex]so that, the correct answer is:
[tex]x=30[/tex]
pls today I have it for tomorrow
Answer:
6/5 x 5/3 is 2
if that's what you need help with
step-by-step explanation
This is a intro to statistics question, I was wondering if i can have the work shown soni can see the process, thanks :)Charges for advertising on a TV show are based on the number of viewers, which is measured by the rating. The rating is a percentage of the population of 110 million TV households. The CBS television show 60 Minutes recently had a rating of 7.8, indicating that 7.8% of the households were tuned to that show. An advertiser conducts an independent survey of 100 households and finds that only 10 were tuned to 60 Minutes. Assuming that the 7.8 rating is correct, find the probability of surveying 100 randomly selected households and getting 10 or fewer tuned to 60 Minutes. What does this suggest? Does the advertiser have grounds for claiming a refund on the basis that the size of the audience was exaggerated?
Step-by-step explanation:
7.8% were viewing the show.
so, when picking a single household the probability that is is watching the show is 0.078.
that also means that the probability that this household is not watching the show is 1 - 0.078 = 0.922.
"only" 10 of 100 were watching the show ? that is 10% and therefore a higher rate than measured by the rating agency.
there is no indication that the audience was exaggerated.
anyway, the probability to get 10 or fewer out of randomly picked 100 households to watch the show is
the probability of exactly 10 households watching +
the probability of exactly 9 households watching +
the probability of exactly 8 households watching +
...
the probability of exactly 1 household watching +
the probability of no (0) household watching.
to get the probabilty of any of these events :
x = number of households watching out of the overall 100.
w = probability of a household to watch = 0.078
a = probability of a household to be absent (= not watching) = 0.922
probability(x) = p(x) = w^x × a^(100-x) × c(100, x)
in other words :
the combined probability of "x" households watching and "a" households not watching (which must be 100-x) multiplied by the number of combinations of picking x households out of 100.
so, for the first one, x = 10, we get
p(10) = 0.078¹⁰ × 0.922⁹⁰ × 100! / (10! × (100 - 10)!) =
= 0.078¹⁰×0.922⁹⁰×100×99×...×91 / 10×9×8×...×2 =
= 8.335775831e-012 ×
0.00066955... ×
1.731030945644e+13 =
= 0.096612596...
the rest is then best via Excel or other spreadsheet applications.
p(9) = 0.125496...
p(8) = 0.145117...
p(7) = 0.147558...
p(6) = 0.129888...
p(5) = 0.096969...
p(4) = 0.059699...
p(3) = 0.0291
p(2) = 0.01053
p(1) = 0.002515...
p(0) = 0.000297...
and the sum of all that is the probability to have 10 or fewer households in randomly picked 100 watching the show :
0.843782...
the vast majority of samples of 100 households is expected to have 10 or fewer households watching the show.
this covers the general rating (that would suggest that 7.8 households of 100 are watching the show), some even better ranges (more than 7.8 households), but also everything that is worse than the original rating (below 7.8 households).
when we look at the individual probabilities, we see that the largest probabilities by far are in the categories of 9, 8, 7 and 6 households out of 100 are watching.
so, these are the main expected results when picking samples of 100 households.
therefore, I don't see any indication that the advertiser was "cheated".
The textbook isn't helping with the screenshot problem below.
From the frequency distribution, the measures are given as follows:
a) Total number of observations: 27.
b) The width of each class is of 5.
c) The midpoint of the second class if of 20.5.
d) The modal class is Class 12 - 17.
e) If another class was added, the limits would be 48 - 53.
What is represented by the frequency distribution?The frequency distribution gives the number of observations that is located in each class.
Hence the total number of observations is given by the sum of the frequencies, as follows:
Total = 9 + 1 + 3 + 8 + 6 = 27.
The modal class is class with the highest number of observations, hence it is of:
Class 12-17.
The width of a class is given by the subtraction of the limits, hence:
Width = 47 - 42 = ... = 23 - 18 = 17 - 12 = 5.
For an added class, the lower bound would be one more than the last class, while the upper bound would be five added to the lower bound, hence the limits are:
48 - 53.
The midpoint of each class is given by the mean of the coordinates, hence, for the second class, it is of:
Midpoint = (18 + 23)/2 = 41/2 = 20.5.
More can be learned about frequency distributions at https://brainly.com/question/24623209
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What should be true about the line of best fit? Check all that apply.
A. The line of best fit must have an equal number of points above and below the line.
B. The line must show the relationship between the two variables.
C. The line of best fit should minimize the distance between itself and the data points.
D. The line of best fit has to cross through at least 2 points in the scatter plot.
E. The line of best fit could be used as a prediction tool showing a trend in the data.
F. The line of best fit should be an exact representation of the data points.
The line of best fit could be used as a prediction tool showing a trend in the data and the line of best fit should minimize the distance between itself and the data points thus options (C) and (E) will be correct.
What is a graph?The graph is a geometrical representation of a function and can be plotted by taking x's and y's corresponding values within the interval.
For example plot of y = x³ ,y = sinx.
A scatter point represents the relation or function.
If a line is used to predict the function then the distance between the points to the line must be minimized.
The line is useful for the representation of the data point.
Hence "The line of best fit could be used as a prediction tool showing a trend in the data and the line of best fit should minimize the distance between itself and the data points".
To learn more about graphs,
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The number of fish in a fish tank doubles each week. The function y=equals 3 (2)^x represents the population, where X is the number of one week periodsa. Describe the domains and range of the function. Then graph the functionb. Find and interpret the y intercept c. How many fish are in the tank at the end of first week?d. How many fish are in the tank after 4 weeks?
a. The domain of the function is the set of values of the independent variable (x) on which the function acts, in this case, the independent variable is time, since the time can only take positive values, the domain would be [0,∞) or x≥0.
The range is all the possible values that the dependent variable can take, in this case, the dependent variable is the population of fish, then it can't be a negative number, since you can't have for example -1 fish, then, again, the interval would be [0,∞) or y≥0
A graph of the function looks like this:
b. We can find the y-intercept by making x equals 0 in the formula of the function, like this:
y= 3*(2)^x
y= 3*(2)^0
y= 3*1, since any number raised to zero equals 1
y=3
Since x equals zero represents the initial time, the y-intercept represents the initial population of fish, then at the beginning, there were 3 fish.
c. We just have to replace 1 for x, and then calculate y, like this:
y= 3*(2)^x
y= 3*(2)^1
y= 3*(2)
y=6
At the end of the first week, there will be 6 fish.
d. Now, we just have to replace 4 for x, like this:
y= 3*(2)^4
y= 3*16
y=48
Then, after 4 weeks, there will be 48 fish
What is the slope of the line that goes through the points (-4, 6) and (2, 8)
Hence the slope is 1/3
Linear Models: Glaciersm= 745-1000= -255, 15-0= 15, -255/15= -17The average annual rate of retreat of the glacier is 17 meters per year.2. Fill in the table below using the information about the average annual rate of retreatand the length of each glacier in 2010.YearLength in meters201020202030204020503. Create an equation that represents the length of the Easton Glacier as a function ofthe number of years since 2010.
I guess the annual retreat of the glacier is linear, and two points in the line are
[tex](0,1000),(15,745)[/tex]Where x is in years and y in meters.
We can find the equation of a line given two points by using the formula below
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]In our case,
[tex]\begin{gathered} y-1000=\frac{745-1000}{15-0}(x-0) \\ \Rightarrow y-1000=-17x \\ \Rightarrow y=-17x+1000 \end{gathered}[/tex]2) Suppose that x=0 corresponds to the year 2010. Then, the information in the table is:
[tex]\begin{gathered} 2010\to-17(0)+1000=1000 \\ 2020\to-17(10)+1000=830 \\ 2030\to-17(20)+1000=660 \\ 2040\to-17(30)+1000=490 \\ 2050\to-17(40)+1000=320 \end{gathered}[/tex]The answers are 1000,830,660,490,320 to to bottom
c)
We already obtained the equation we are asked for in this part of the problem. Remember that we used the points (0,1000) and (15,745)
The equation is:
[tex]y=-17x+1000[/tex]y=-17x+1000
In the diagram, find the segment length of FD.(Assume FG is tangent.) G X +9 D 18 E 12
ANSWER
FD = 27
EXPLANATION
By the tangent-secant theorem:
[tex]FG^2=EF\cdot FD[/tex]So we have:
[tex]\begin{gathered} 18^2=12\cdot FD \\ FD=\frac{18^2}{12} \\ FD=27 \end{gathered}[/tex]Help me with math and explain it with a Quick solution By explaining the you answer
EXPLANATION
Given the points (x_1,y_1) = (-9,2) and (x_2,y_2) = (-4,2)
We can apply the distance formula in order to get the distance between them as shown as follows:
[tex]\text{distance}=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Replacing terms:
[tex]\text{distance}=\sqrt[]{(-4-(-9))^2+(2-2)^2}[/tex]Adding numbers:
[tex]\text{distance}=\sqrt[]{(5)^2+(0)^2}[/tex]Computing the powers:
[tex]\text{distance}=\sqrt[]{25}=5[/tex]Hence, the distance is equal to 5 units.
you're planning to supply cookies for a staff party. You're thinking of purchasing platters and cookies containing 80 sugar cookies and 15 chocolate chip cookies. what is the ratio of sugar cookies to chocolate chip cookies?
We have the following:
To calculate the proportion, we must calculate the quotient between sugar cookies and chocolate chip cookies, as follows
[tex]\frac{80}{15}=\frac{16\cdot5}{3\cdot5}=\frac{16}{3}[/tex]Which means that for every 3 chocolate chip cookies there are 16 sugar cookies.
write the ratio of the first measurement. make sure to simply if possible
1. Ratio of 10 feet to 5 yards
A yard is equvalent to 3 feet
So we can say that ratio of 10 feet to 5 yards is the same as ratio of 10 feet to 5x3 feet, which is:
ratio of 10 feet to 15 feet = 10/15
10/15 simplified by dividing both numerator and denominator by 5:
2/3
Answer:
2/3
[tex]\frac{2}{3}[/tex]units givenQuestion 2 (3 points)toUsing the Rectangular Prism in the picture, find the Lateral Area, the Area of a Single Base,and the TOTAL Surface Area:.12 cm825 cm10 cmUse the face with dimensions 10cm x 5cm as the base.Lateral Area =cm²Single Base Area =64188 in48cm²
Solution:
Given the rectangular prism;
Where;
[tex]\begin{gathered} length,l=10cm \\ \\ width,w=5cm \\ \\ height,h=2cm \end{gathered}[/tex]Thus, the lateral area, L is;
[tex]\begin{gathered} L=2(l+w)h \\ \\ L=2(10+5)2 \\ \\ L=4(15) \\ \\ L=60cm^2 \end{gathered}[/tex]ANSWER:
[tex]Lateral\text{ }Area=60cm^2[/tex]Also, the single base area, B, is;
[tex]\begin{gathered} B=l\times w \\ \\ B=10\times5 \\ \\ B=50cm^2 \end{gathered}[/tex]ANSWER:
[tex]Single\text{ }Base\text{ }Area=50cm^2[/tex]Then, the surface area, S, is;
[tex]\begin{gathered} S=L+2B \\ \\ S=60+2(50) \\ \\ S=60+100 \\ \\ S=160cm^2 \\ \\ \end{gathered}[/tex]ANSWER:
[tex]Surface\text{ }Area=160cm^2[/tex]What is the surface area of the following solid? *13 ft 12ft 5ft 5ft
In order to calculate the total surface area we just need to sum the area of the different faces.
From the picture we see that we have the following faces.
- 2 x A (two triangles)
- S (a rectangle)
- T (a rectangle)
- B (the base rectangle)
Now me calculate the area of each one:
[tex]A\text{ = }\frac{1}{2}\cdot12\cdot5\text{ = }30[/tex][tex]S\text{ = 5}\cdot5=\text{ 25}[/tex][tex]B=12\cdot5=60[/tex][tex]T=13\cdot5=65[/tex]Now that we have the area of each face, we sum all the areas, taking in account the ones that appears twice:
[tex]\text{Total area = 2}\cdot A+S+T+B=2\cdot30+25+60+65=210[/tex]And, because the lenghts are measured in ft, the final answer is:
[tex]\text{Total area = 210 ft}^2[/tex]Note:
[tex]\text{Area of a triangle = }\frac{1}{2}\cdot\text{base}\cdot\text{height}[/tex][tex]\text{Area of a rectangle = base }\cdot\text{ height}[/tex]Complete the general solution to [tex]y = arcsin - \frac{ \sqrt{3} }{2} [/tex]y=___+-2πkSelect all that apply.π/3(2π)/3(4π)/3(5π)/3
To find the general solution to y we need take the sine function in both sides of the equation given:
[tex]\begin{gathered} \sin y=\sin (\sin ^{-1}-\frac{\sqrt[]{3}}{2}) \\ \sin y=-\frac{\sqrt[]{3}}{2} \end{gathered}[/tex]Now, we have to find the value of y for which the sine function is equal to the right side. From the properties of the sine function and its definition we conclude that y has to be:
[tex]y=\frac{5\pi}{3}[/tex]Therefore the general solution is:
[tex]y=\frac{5\pi}{3}\pm2\pi k[/tex]Select the Coordinate point that is not a solution to the given system of linear inequalities. 3x + 5y 2-15 3x + y > 3
3x + 5y ≥ -15 (1)
3x - y > 3 (2)
From (1) let's solve for x:
3x + 5y ≥ -15
Subtract 5y from both sides:
3x ≥ -15 - 5y
Divide both sides by 3:
x ≥ -5 - 5y/3 (3)
Replace (3) into (2)
3(-5 - 5y/3) - y > 3
Using distributive property:
-15 - 5y - y > 3
-15 - 6y > 3
Solving for x:
Add 15 to both sides:
-6y > 18
Divide both sides by -6:
y < -3
Replacing y into (3)
x ≥ 0
3x - y > 3
if x = 4 and y =9
3*4 - 9 >3
12 - 9 > 3
3 > 3
this is false because 3 = 3
What are the coordinates of vertex C'after rotating the figure 180° about theorigin?у6BID42A АEXo024 6A. (-6,4)B. (4.-6)C. (4,6)D. (-4,-6)
Vertex C (4, 6 ) Rotation 180º about the origin CCW c' (-4, -6)
C (x, y ) C ' (-x, -y )
________________
Answer
Option D
Suppose a city's population grows by 5% each year. How long will it take for the population of the city to triple? Answer to the nearest hundredth of a year.
Since it is given that the population grows by 5% each year, it follows that the Exponential Growth Function is the appropriate function that can be used to model the problem.
The Exponential Growth Function is given by:
[tex]y=a(1+r)^t[/tex]Where
• a is the initial amount.
,• r is the percent of increase in decimal.
,• t is the time.
,• y is the amount after time t.
Since we want when the initial population will triple, substitute y=3a into the equation:
[tex]3a=a(1+r)^t[/tex]Substitute r=5%=0.05 into the equation:
[tex]3a=a(1+0.05)^t[/tex]Solve the resulting equation for t:
[tex]\begin{gathered} 3a=a(1+0.05)^t \\ \Rightarrow a(1+0.05)^t=3a \\ \Rightarrow a(1.05)^t=3a \\ Divide\text{ both sides by a:} \\ \Rightarrow(1.05)^t=3 \\ \text{Take logarithm of both sides:} \\ \Rightarrow\ln (1.05)^t=\ln 3 \\ \Rightarrow t\ln (1.05)=\ln 3 \\ \Rightarrow t=\frac{\ln 3}{\ln (1.05)}\approx22.52\text{ years} \end{gathered}[/tex]The population will triple after about 22.52 years.