Simplify to a single trig function with no fractions.

Answer:

P(x)=[tex](40000)( 0.98)^x[/tex]

A=40000    B=0.98
P(7)= 34700

Step-by-step explanation:

Notice the population can be modeled as:

P(x)=[tex]A B^x[/tex]

For X=0 P(0)=40000=A (Initial population)
So A=40000
For x=1 (one year after) P=39200

[tex]39200=40000 B^1[/tex]
solving for B
[tex]B=\frac{39200}{40000} =0.98[/tex]
So B=0.98

So the population can be modeled as

P(x)=[tex](40000)( 0.98)^x[/tex]

Now at 7 years:

P(7)=[tex](40000)( 0.98)^7[/tex]= 34725.02133
Needs to be rounded to the nearest hundred
This is 34700 bonobos (7 years after the study began)

Answer:

Step-by-step explanation:

[tex]V=\pi r^2 h[/tex]

Substitute [tex]h=l,r=2,V=150,[/tex] we get

[tex]150=\pi \times 2^2 \times l[/tex]

[tex]150=4\pi l[/tex]

[tex]l=\frac{150}{4\pi}=11.94cm[/tex]

Answer:

??

Step-by-step explanation:

is there a picture or something?

Answer:

4/3 from x = 2 to x = 5.

Step-by-step explanation:

What is Lagrange mean value theorem?

Lagrange mean value theorem states that, if a function f is continuous over the closed interval [a, b] and differentiable over the open interval (a, b), then there must be at least one point c in the interval (a, b) where the slope of the tangent at the point c is equal to the slope of the tangent through the curve's endpoints, resulting in the expression f'(c) = {F(b) -F(a)}/(b-a)

According to the given graph,

At point x = 2,

F(2) = 3

At point x = 5,

F(5) = 7

Since the formula for the average rate of change of the function between x = a and x = b is,

The average rate of change = {F(b) -F(a)}/(b-a)

Here a = 2, b = 5 and F(2) = 3, F(5) = 7

Substitute the values in the formula,

So the average rate of change = (7 - 3)/(5 - 2) = 4/3.

Hence, the average rate of change of the function is 4/3.

Check the picture below.

[tex]\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x_1=2\\ x_2=5 \end{cases}\implies \cfrac{f(5)-f(2)}{5 - 2}\implies \cfrac{7-3}{5-2}\implies \cfrac{4}{3}[/tex]

Answer: x=7 y=5

Step-by-step explanation:

I believe this is the answer to the question  

The time between the injections that is injected again when the dosage reaches 9 is solved to be 1.8 to the nearest tenth

The information given by the problem include the expression

D(h) = 20e^(-0.45h)

The expression is an exponential function that shows the relationship between time and the dosage in milligrams  

given that D = 9 we find h

D(h) = 20e^(-0.45h)

9 = 20e^(-0.45h)

9/20 = e^(-0.45h)

take ㏑ of both sides

㏑(9/20) = -0.45h

isolating h by dividing by -0.45

h = ㏑(9/20) / -0.45

h = 1.774

h = 1.8 to the nearest tenth

This shows that the time is 1.8 hours

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Answer:

22

Step-by-step explanation:

The time required for a radioactive isotope to decay to a certain percentage of its initial amount can be found using the following formula:

t = (t1/2 / ln(2)) * ln(N0/N1)

where:

t is the time elapsed since the release of the radioactive material

t1/2 is the half-life of the radioactive isotope (6 days in this case)

N0 is the initial amount of the radioactive isotope

N1 is the remaining amount of the radioactive isotope (14% of N0 in this case)

ln is the natural logarithm

We can solve for t by plugging in the given values:

t = (6 / ln(2)) * ln(1 / 0.14)

t ≈ 22.4 days

Therefore, the farmers needed to wait about 22.4 days to use the hay safely.

Step-by-step explanation:

3)

a. To find the height of the kite, we can use trigonometry. The sine function relates the opposite side (the height of the kite) to the hypotenuse (the length of string used) and the angle of elevation. Therefore, we can write:

sin(65°) = height/75

Solving for the height, we get:

height = 75 sin(65°) = 67.8 ft

Therefore, the kite is 67.8 feet high.

b. To find the distance between Joe and Jill, we can use trigonometry again. The tangent function relates the opposite side (the distance between Joe and Jill) to the adjacent side (the height of the bell tower) and the angle of depression. Therefore, we can write:

tan(angle of depression) = opposite/adjacent

tan(angle of depression) = Jill's height/210

Solving for the distance between Joe and Jill, we get:

distance = adjacent * tan(angle of depression)

distance = 210 * tan(angle of depression)

We need to know the angle of depression to solve for the distance, which is not given in the problem.

4)

a. In an isosceles triangle with two equal angles of 70°, the third angle must be:

180° - 70° - 70° = 40°

Since the triangle is isosceles, the height must be the perpendicular bisector of the base. Therefore, we can draw an altitude from the top vertex to the base, splitting the base into two equal segments. Let x be the length of each base segment. Then we can use trigonometry to find the height:

tan(70°) = height/x

height = x * tan(70°)

Since the height is given as 16 cm, we can solve for x:

16 = x * tan(70°)

x = 16/tan(70°)

Therefore, the length of each base segment is:

x = 16/tan(70°) = 6.12 cm

And the length of the base is twice the length of each segment:

base = 2x = 2(16/tan(70°)) = 12.25 cm

Therefore, the measures of the rest of the triangle are:

base = 12.25 cm

each equal angle = 70°

height = 16 cm

b. To determine if the ramp meets ADA standards, we need to find the angle of incline. The angle of incline is the angle between the ramp and the horizontal. We can use trigonometry to find this angle:

sin(angle of incline) = rise/run

sin(angle of incline) = 2/25

angle of incline = sin^(-1)(2/25)

Using a calculator, we get:

angle of incline ≈ 4.79°

Since the angle of incline is greater than the maximum allowable angle of 4.76°, the ramp does not meet ADA standards.

Let x be the number of text messages that your older sister sends per day. Then we can write an equation based on the given information:

x + 12 = 26

We can solve for x by subtracting 12 from both sides of the equation:

x = 26 - 12

x = 14

Therefore, your older sister sends about 14 text messages per day.

Answer: Right Picture

Step-by-step explanation:

Mark me as brainleanst

The equivalent fraction of the given expression is: 2(y - 2x)(-5x + 3y)

The fraction expression is given as;

[(-2/x) + (5/y)]/[(3/y) - (2/x)]

Finding the common denominators and writing the numerators above common denominators gives:

[(2y - 4x)/xy]/[(-5x + 3y)/xy)]

Usually we divide a fraction by multiplying its' reciprocal and we have;

[(2y - 4x)/xy] * [(xy/(-5x + 3y)]

Thus, this gives us the expression:

(2y - 4x)/(-5x + 3y)

= 2(y - 2x)(-5x + 3y)

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Answer:

No solution

Step-by-step explanation:

y = 2x + 1 _____(equ 1)

2x - y = 3 _____(equ 2)

from equ 1

y = 2x + 1

2x + 1 = y

2x = y - 1

x = (y - 1)/2

substitute x = (y - 1)/2 into equ 2

2x - y = 3

2((y-1)/2) - y = 3

(2y - 2)/2 - y = 3

y - 1 - y = 3

y - y = 3 + 1

0 = 4

Hello and best regards sanungapatricio1985

This equation has no solution, since -1 cannot be equal to 3. Therefore, the system has no solution.

We have the following equations:

                ⇒ y = 2x + 1

                ⇒ 2x - y = 3

The substitution method is a common method for solving systems of linear equations. It consists of isolating one of the variables from one of the equations and substituting the expression obtained in the other equation to eliminate that variable and obtain an equation with a single variable, which can be solved to find the value of that variable. Then, the found solution can be substituted into any of the original equations to find the value of the other variable.

From the first equation we have a good substitution candidate:

                ⇒ y = 2x + 1

Now we have to plug y = 2x + 1 found from the first equation, into the second equation 2x - y = 3, to find that:

                ⇒ 2x - y = 3

                ⇒ 2x - (2x + 1) = 3

                ⇒ 2x - 2x - 1 = 3

Based on the previous results, the system has no solution.

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Solving for μ and σ simultaneously gives: μ = 3.2 hours (rounded to 1 decimal place) and σ = 0.426 hours (rounded to 3 decimal places)

What is Standard Deviation ?

Standard deviation is a statistical measure that describes the amount of variation or dispersion in a set of data. It measures how spread out the data is from its mean or average.

Let the mean of the population sleep time be μ and the standard deviation be σ.

From the given information, we know that the distribution of sample means of sleep time follows a normal distribution with mean μ and standard deviation σ/√40 (since each sample size is 40).

We are also given that 5% of the sample means are less than 3 hours and 5% of the sample means are more than 3.4 hours.

Using a standard normal distribution table, we can find the corresponding z-scores for these probabilities:

P(Z < z) = 0.05 when z = -1.645

P(Z > z) = 0.05 when z = 1.645

Now we can use the formula for z-score:

z = (X'  - μ) / (σ / √n)

where X' is the sample mean, n is the sample size (which is 40 in this case).

For the lower bound, we have:

-1.645 = (3 - μ) / (σ / √40)

For the upper bound, we have:

1.645 = (3.4 - μ) / (σ / √40)

Therefore, Solving for μ and σ simultaneously gives: μ = 3.2 hours (rounded to 1 decimal place) and σ = 0.426 hours (rounded to 3 decimal places)

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[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$3000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{\underline{annually}, thus once} \end{array}\dotfill &1\\ t=years\dotfill &6 \end{cases} \\\\\\ A = 3000\left(1+\frac{0.04}{1}\right)^{1\cdot 6} \implies A \approx 3795.96 \\\\[-0.35em] ~\dotfill[/tex]

[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$3000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{\underline{quarterly}, thus four} \end{array}\dotfill &4\\ t=years\dotfill &6 \end{cases} \\\\\\ A = 3000\left(1+\frac{0.04}{4}\right)^{4\cdot 6} \implies A \approx 3809.20 \\\\[-0.35em] ~\dotfill[/tex][tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$3000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{\underline{monthly}, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &6 \end{cases} \\\\\\ A = 3000\left(1+\frac{0.04}{12}\right)^{12\cdot 6} \implies A \approx 3812.23 \\\\[-0.35em] ~\dotfill[/tex]

[tex]~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$3000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ t=years\dotfill &6 \end{cases} \\\\\\ A = 3000e^{0.04\cdot 6} \implies A \approx 3813.75[/tex]

Answer:

a

Step-by-step explanation:

The formula to calculate simple interest is:

I = P * r * t

Where:

I = Interest

P = Principal amount

r = Rate of interest

t = Time period

Given:

Principal amount (P) = $1,700

Rate of interest (r) = 8% per annum

Time period (t) = 4 years

Using the formula of simple interest:

I = P * r * t

I = 1,700 * 0.08 * 4

I = $544

Therefore, the total interest on a 4-year term loan of $1,700 with a simple annual interest rate of 8% is $544.

Answer: A. $544

Answer:

The cost of a specialty pizza is $19.49.

Step-by-step explanation:

To write a subtraction equation that could be used to find the cost c of a specialty pizza, we need to set up an equation that represents the relationship between the cost of a one-topping pizza and the cost of a specialty pizza.

The problem states that a one-topping pizza costs $6.50 less than a specialty pizza. Therefore, we can subtract $6.50 from the cost of a specialty pizza to get the cost of a one-topping pizza:

Cost of specialty pizza - $6.50 = Cost of one-topping pizza

We can rearrange this equation to solve for the cost of the specialty pizza:

Cost of specialty pizza = Cost of one-topping pizza + $6.50

Now, we can substitute the given value for the cost of a one-topping pizza:

Cost of specialty pizza = $12.99 + $6.50

Simplifying the expression, we get:

Cost of specialty pizza = $19.49

Therefore, the cost c of a specialty pizza is $19.49.

Hope this helps! I'm sorry if it doesn't. If you need more help, ask me! :]

Therefore, the value of hydraulic conductivity using the Thiem equation is approximately 10.67 ft/day.

An equation is a mathematical statement that indicates that two expressions are equal. It typically consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. The expressions on both sides of the equation are separated by an equal sign "=" which means that the two expressions have the same value. Equations are used to model relationships between variables, solve problems, and make predictions in various fields of science, engineering, economics, and mathematics. They are an essential tool in algebra, calculus, and other branches of mathematics.

Here,

To compute the value of hydraulic conductivity using the Thiem equation, we need to substitute the given values into the equation:

K = (Q / π) * ((h2 - h1) / log(r2/r1))

where:

Q = pumping rate = 78,000 ft³/d

h1 = head at the pumping well = 260 ft ASL

h2 = head at the observation well 385 ft away = 291 ft ASL

r1 = distance from the pumping well to the observation well 125 ft away = 125 ft

r2 = distance from the pumping well to the observation well 385 ft away = 385 ft

Substituting the values, we get:

K = (78,000 / π) * ((291 - 260) / log(385/125))

K = 10.67 ft/day

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Answer:

2% of 800, which is 16

Answer:

x = 10 (B)

Step-by-step explanation:

Use trigonometry:

[tex] \sin(30°) = \frac{5}{x} [/tex]

[tex] \frac{1}{2} = \frac{5}{x} [/tex]

Cross-multiply to find x:

[tex]x = 2 \times 5[/tex]

[tex]x = 10[/tex]

Answer: the equation have two solutions: 0.25 and 0.75

[tex]x=\frac{1}{4}[/tex]  or  [tex]x=\frac{3}{4}[/tex]

Step-by-step explanation:

Notice the terms [tex]x+x^2+x^3+x^4+...+x^n[/tex] is just a geometric series (since -1<x<1)

A geometric series is given by the form

[tex]1+x+x^2+x^3+x^4+...+x^n = \frac{1}{1-x}[/tex]
This is
[tex]x+x^2+x^3+x^4+...+x^n = \frac{1}{1-x} -1[/tex]
plugging in in the equation

[tex]-1+\frac{1}{x} +\frac{1}{1-x} -1=\frac{10}{3}[/tex]

grouping terms

[tex]\frac{1}{x} +\frac{1}{1-x} =\frac{16}{3}[/tex]
groupings the fractions

[tex]\frac{1-x+x}{(x)(1-x)}=\frac{1}{(x)(1-x)} =\frac{16}{3}[/tex]

this leads to an simplified quadratic equation since X cannot be zero (notice the initial 1/x term)

[tex]3=16(x^2-x)[/tex]
the solution for this quadratic equation is just

[tex]x=\frac{1}{4}[/tex]  or  [tex]x=\frac{3}{4}[/tex]

Answer:

  1/4 or 3/4

Step-by-step explanation:

You want the value of x in the infinite sum ...

  -1 +1/x +x +x² +x³ +...  = 10/3

for |x| < 1.

After the 2nd term, this looks like a geometric series with a common ratio of x. If we define the series sum as ...

  S = 1/x +1 +x +x² +x³ +...

we see that the equation of interest is ...

  S -2 = 10/3

The series sum is that of a geometric series with first term 1/x and common ratio x:

  S = (1/x)(1/(1 -x)) = 1/(x(1-x))

Substituting for S in the above, we have the equation ...

  1/(x(1-x)) -2 = 10/3

This resolves to a quadratic that will have 2 real roots:

  1/(x(1 -x)) = 16/3 . . . . add 2

  x(1 -x) = 3/16 . . . . . . . invert both sides

  x² -x +3/16 = 0 . . . . . .  subtract the left side expression

  (x -3/4)(x -1/4) = 0 . . . . . factor

Solutions are the values of x that make these factors zero:

  x = 1/4 or x = 3/4

Tt would take Maurice and Olinda approximately 47.37 minutes

What is work done?

Work done in physics is the dot product of force and displacement. When a force applied on an object results in a displacement, it is said to be work is done on the object.

Let's denote the time it takes both Maurice and Olinda working together to weed the garden as t.

We can use the formula: 1 / t = 1 / a + 1 / b

Plugging in the given values, we get: 1 / t = 1 / 45 + 1 / 50

1 / t = (50 + 45) / (50 * 45) = 95 / 2250

Then, we can invert both sides of the equation to get: t / 1 = 2250 / 95

Simplifying the right-hand side by dividing both the numerator and the denominator by 5, we get:

t = 2250 / 95 = 47.37 minutes (rounded to two decimal places)

Therefore, it would take Maurice and Olinda approximately 47.37 minutes to weed the garden if they work together.

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a) The length of x is 5 cm. b)  the total surface area of the frustum is 61[tex]\pi[/tex]√802 + 16[tex]\pi[/tex] square cm (in terms of π).

A frustum has two bases, which are usually parallel to each other and are either circular or polygonal in shape. The height of the frustum is the perpendicular distance between the two bases, and the slant height is the distance between the apex of the frustum (the point where the original cone or pyramid was cut off) and any point on the perimeter of either base.

a. To find the value of x, we can use the similar triangles formed by the two cones. Let the radius of the small cone be y cm, then we have:

y/x = (35-7)/35 [Using the similarity of triangles]

Simplifying this expression, we get:

y = x(28/35) = 4x/5

Now, we know that the difference in the areas of the two circular bases of the frustum is equal to the area of the missing part. Using this fact, we can find the value of x as:

[tex]\pi[/tex](6²) - [tex]\pi[/tex](y²) = [tex]\pi[/tex](x²) - [tex]\pi[/tex]( (4x/5)² )

Simplifying this expression and solving for x, we get:

x = 5 cm

Therefore, the value of x is 5 cm.

b. The total surface area of the frustum can be calculated as the sum of the curved surface area of the small cone and the curved surface area of the frustum itself.

Curved surface area of the small cone = [tex]\pi[/tex](y²) = [tex]\pi[/tex](4²) = 16[tex]\pi[/tex]

Curved surface area of the frustum = [tex]\pi[/tex](6² + x²) × l

where l is the slant height of the frustum. To find the value of l, we can use the Pythagorean theorem:

l² = (35-7)² + (6-x)²

l² = 784 + (6-x)²

l = √[784 + (6-x)²]

Substituting the value of x, we get:

l = √[784 + (6-5)²] = √802

Therefore, the total surface area of the frustum is:

[tex]\pi[/tex](6² + x²) × √802 + 16[tex]\pi[/tex]

= [tex]\pi[/tex](6² + 5²) × √802 + 16[tex]\pi[/tex]

= 61[tex]\pi[/tex]√802 + 16[tex]\pi[/tex]

Hence, the total surface area of the frustum is 61[tex]\pi[/tex]√802 + 16[tex]\pi[/tex] square cm (in terms of [tex]\pi[/tex]).

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By using technology, the line of best fit include the following: C. y = 6.5x - 279.1.

In this scenario, the average daily temperature would be plotted on the x-axis of the scatter plot while the total ice cream sales would be plotted on the y-axis of the scatter plot.

On the Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an equation for the line of best fit (trend line) on the scatter plot.

From the scatter plot (see attachment) which models the relationship between the average daily temperature and total ice cream sales, a linear equation for the line of best fit is given by:

y = 6.5x - 279.1

In conclusion, the type of correlation between the variables is a strong positive correlation.

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Answer:

ŷ = 6.5x − 279.1

Step-by-step explanation:

Okay so you want to make all of the fractions have a common denominator. Think of the lowest number that 3 and 5 go into, the answer you should get is 15. So 15 is the common denominator.

Now we're going to make 1/3 have a denominator of 15. 1/3 of 15 would be 5. You get that by figuring out what you have to multiply 3 by to get 15 which is 5. Then you multiply 5 and 1 and you get 5.  The answer you will get will be 5/15 which is equal to 1/3.

Next we're going to make 4/5 have a denominator of 15. 4/5 of 15 would be 12/15. You can get this by figuring out what you have to multiply 5 by to get 15 which is 3. Then you take 3 and multiply it by 4 to get that 12 in 12/15. The answer 12/15 is equal to 4/5 of 15.

So now you have 5/15 + a = 12/15.

The denominator of the fraction that "a" will be is 15 because that is what we made the common denominator.

So what we have is 5/15 + x/15 = 12/15.

Think of what you have to add to 5 to get 12. (The answer is 7)

So a is equal to 7/15

Your final answer should be 5/15 + 7/15 = 12/15

The median shows that the average number of students per middle school classroom in both districts is 24, while the mean shows that the average number of students per middle school classroom in one district is 24 and the other is 25.  

Median is the middle value of a set of data. It is the value that divides the data set into two equal halves. It is an important measure of central tendency and is used to compare sets of data.  

The difference between the mean number of students per middle school classroom in each school district is 2 students. The difference between the median number of students per middle school classroom in each school district is 0 students.

Based on the number of students shown for each school district, the median would provide the most accurate picture of the number of students in a middle school classroom, as there are some extreme values that would affect the mean but not the median.

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Answer: 18/14, 27/21, 36/28

Step-by-step explanation:

Answer:

  (f∘g)(1) = 2; (f∘g)'(1) = 2

  (f∘g)(2) = -2; (f∘g)'(2) = -2

Step-by-step explanation:

You want (f∘g)(x) and (f∘g)'(x) for x=1 and x=2 given the function values and derivatives in the table.

This composition means f(g(x)). The value is found by first determining the value of z = g(x), then using that to find the value of f(z).

For x=1, the value of g(1) is seen to be -2.

For x=-2, the value of f(-2) is seen to be 2.

This means f(g(1)) = 2.

For x=2, the value of g(2) is 0.

For x= 0, the value of f(0) is -2.

This means f(g(2)) = -2.

This is a little trickier, as you need to find the derivative of the composition:

  f(g(x))' = f'(g(x))·g'(x)

In the attached table, we have made a column for f'(g(x)) to help find this product.

For x=1, f'(g(1)) = f'(-2) = 1; and g'(1) = 2, so f'(g(1))g'(1) = 1·2 = 2 = (f∘g)'(1)

For x=2, f'(g(2)) = f'(0) = 2; and g'(2) = -1, so f'(g(2))g'(2) = 2(-1) = -2 = (f∘g)'(2)

Both f(x) and g(x) include domain values of [12,), and both functions increase over the interval (12, ∞).

A relationship between a set of inputs (the domain) and a set of potential outputs (the range) is known as a function. This relationship has the feature that each input is associated to exactly one output.

The notation f(x) is often used to express a function in mathematics, where f is the name of the function and x is the input value, also known as the argument of the function. When f(x) is calculated, it indicates the function's output value for the given input value x.

First, both functions have a domain that includes all values of x greater than or equal to 12, which is expressed as [12,) in interval notation.

Second, both functions increase over the interval (12, ∞), meaning that as x increases beyond 12, the values of the functions also increase without bound. This is because the exponential function 12^x and the square root function √(x - 12) both become increasingly steep as x increases, leading to a sharp increase in their values.

Therefore, Both f(x) and g(x) include domain values of [12,), and both functions increase over the interval (12, ∞).

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Answer:

4+5/8

or

5/8+4

Step-by-step explanation:

"increased by" is adding, and "quotient" id dividing two numbers.

The mathematical expression '4 increased by the quotient of 5 and 8' is written as 4 + 5/8 in standard mathematical notation, meaning you first divide 5 by 8 and then add the result to 4.

The sequence of operations described can be converted into an expression in the following way: In terms of mathematics, the term 'increased by' refers to addition. The 'quotient of 5 and 8' refers to the result of dividing 5 by 8. Therefore, the expression can be written as '4 + 5/8'.

So, the mathematical expression '4 increased by the quotient of 5 and 8' becomes 4 + 5/8 in standard mathematical notation. This means you first divide 5 by 8 and then add the result to 4.

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Answer:

Step-by-step explanation:

To calculate the payback period of the investment, we need to find out how long it will take for the company to recover the initial investment of 80000000 through the cash flows generated by the investment.

Step 1: Calculate the cumulative cash flow for each year.

Year 1: 25000000

Year 2: 25000000 + 25000000 = 50000000

Year 3: 50000000 + 20000000 = 70000000

Year 4: 70000000 + 20000000 = 90000000

Step 2: Determine the year in which the cumulative cash flow exceeds the initial investment.

Based on the calculations above, the cumulative cash flow exceeds the initial investment of 80000000 in Year 4.

Step 3: Calculate the payback period.

The payback period is the time it takes for the cumulative cash flow to equal the initial investment. In this case, the payback period is the end of Year 3 plus the portion of Year 4 needed to recover the remaining investment, which is calculated as follows:

80000000 - 70000000 = 10000000

10000000 ÷ 20000000 = 0.5

Therefore, the payback period for this investment is 3.5 years.

To confirm this result, we can also calculate the cumulative cash flow at the end of Year 3 and check that it is less than the initial investment, while the cumulative cash flow at the end of Year 4 exceeds the initial investment:

Year 1: 25000000

Year 2: 25000000 + 25000000 = 50000000

Year 3: 50000000 + 20000000 = 70000000 (cumulative cash flow at end of Year 3)

Year 4: 70000000 + 20000000 = 90000000 (cumulative cash flow at end of Year 4)

Since the cumulative cash flow at the end of Year 3 is less than the initial investment and the cumulative cash flow at the end of Year 4 exceeds the initial investment, we can confirm that the payback period is between Year 3 and Year 4, or 3.5 years.