What is the average rate of change of the function g(x)=6x from x=-1 to x=3? show your work or explain how you obtained your response

The equation of the line is P = 2(1.0048)ᵗ and the population in 30 years is 2.31

The equation is represented as

y = abᵗ

Where

a = y when t = 0

The points on the line are

(0, 2) and (20, 2.2)

This means that

a = 2

So, we have

y = 2bᵗ

Using the points, we have

2b²⁰ = 2.2

b²⁰ = 1.1

So, we have

b = 1.0048

This means that the equation is

P = 2(1.0048)ᵗ

Above, we have

a = 2

So, the meaning of the interpretation is that the initial population of the endangered colony is 2

Also, we have

b = 1.0048

So, the meaning of the interpretation is that the endangered colony increases by a factor of 1.0048 every year

Recall that

P = 2(1.0048)ᵗ

Here, we have

t = 30

So, the equation becomes

P = 2(1.0048)³⁰

Evaluate

P = 2.31

Hence, the population in 30 years is 2.31

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Amy received 10 slices of pizza and Mr. Au received 45 slices of pizza.

Let's start by using variables to represent the unknown quantities in the problem. Let m be the number of slices of pizza Michael received, a be the number of slices Amy received, and au be the number of slices Mr. Au received.

We know that Miss Chang gave Michael 60 slices of pizza in total, so we can write:

m = 60

We also know that Amy received twice as many slices of pizza as Michael, so we can write:

a = 2m

Finally, we know that Mr. Au received three times the sum of what Miss Chang gave to Michael and Amy, so we can write:

au = 3(m + a)

We also know that Miss Chang gave 60 slices of pizza in total, so we can write:

m + a + au = 60

Now we can substitute the expressions we found for a and au into the last equation to get:

m + 2m + 3(m + 2m) = 60

Simplifying this equation, we get:

m + 2m + 3m + 6m = 60

12m = 60

m = 5

So Michael received 5 slices of pizza, and we can use the equations we found for a and au to determine how many slices Amy and Mr. Au received:

a = 2m = 2(5) = 10

au = 3(m + a) = 3(5 + 10) = 45

Therefore, Amy received 10 slices of pizza and Mr. Au received 45 slices of pizza.

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You can drive at most 700 miles within the $180 budget for weekly transportation.

To determine how many miles can be driven with a budget of $180 for weekly transportation, we'll use the given information: $75 per week for car rental and $0.15 per mile driven. First, subtract the weekly rental cost from the total budget:

$180 - $75 = $105

Now, divide the remaining budget by the cost per mile to find the maximum number of miles that can be driven:

$105 ÷ $0.15 ≈ 700 miles

So, you can drive at most 700 miles within the $180 budget for weekly transportation.

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Answer:

y = -2x + 14

Step-by-step explanation:

OK so first u have to do y = -2x + b cuz its parallel

then u gotta just plug in the ordered pair so

6 = -2(4)+b

6 = -8 +b

14 = b

So now u have to do

y = -2x + 14

If a man bought a diamond for $32 in 1682 and the man had instead put the $32 in the bank at 3% interest compounded continuously, then the value of the diamond in 2003 would be $554,311.

The given problem is related to exponential growth. In this problem, the continuous compounding formula will be used to find the value of $32 in 2003.

The formula for continuous compounding is given by:

A = Pert   Where,

P is the principal amount,

r is the annual interest rate,

e is the Euler's number which is approximately 2.71828, and

t is the time in years.

Using the formula, we get:

A = 32e^(0.03 x 321)

A = 32e^9.63

A = 32 x 17322.23

A = $ 554311.36

Thus, $32 invested at 3% compounded continuously from 1682 to 2003 would be worth $554,311.

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To show that the series Σα) f(n)/2n^3 - 1 n converges regardless of the rule for f, we can use the Comparison Test. Let's choose a series that is easier to compare to, such as Σ(1/2n^3).

First, note that 0 ≤ f(n) ≤ 1 for all n, since f is a function that is not negative and bounded by 1. Thus, we have: 0 ≤ f(n)/2n^3 - 1 n ≤ 1/2n^3, Now, we can compare the given series to the series Σ(1/2n^3) using the Comparison Test. Since 0 ≤ f(n)/2n^3 - 1 n ≤ 1/2n^3 for all n, we know that: 0 ≤ Σα) f(n)/2n^3 - 1 n ≤ Σ(1/2n^3), The series Σ(1/2n^3) is a convergent p-series with p = 3 > 1, so by the Comparison Test, the given series Σα) f(n)/2n^3 - 1 n also converges. Therefore, we have shown that the series Σα) f(n)/2n^3 - 1 n converges regardless of the rule for f.

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The two true statements are:

A "When the time starts, there are 13.5 gallons of water in the pool."

C "The water is draining at a rate of 1.25 gallons per minute."

Here we have the linaer equation y = –1.25x + 13.5 that represents the gallons of water y left in an inflatable pool after x minutes.

And we want to see which of the given statements are true.

We can see that the slope is -1.25, this means that the volume of water is reducing (due to the negative sign) then the statement C is true.

We also can see that the y-intercept is 13.5, that would be the initial volume of water in the pool, then the statement A "When the time starts, there are 13.5 gallons of water in the pool." is also true.

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Answer: 2

Step-by-step explanation:

Answer:

[tex] \frac{dy}{dx} = \frac{4}{9} [/tex]

Step-by-step explanation:

sol,

2x+3y-6x+6y=0

f(x,y)=2x+3y-6x+6y

f(X,-y)=2x+3y-6y,fx=?

f(x-y)=2x+3y-6y,fy=?

fx=-4

fy=9

[tex] \frac{d}{y} = \frac{ - 4}{9} [/tex]

now,

[tex] \frac{dy}{dx} = \frac{4}{9} [/tex]

Considering the route as a curve of a quadratic function, the information should be completed as follows;

In Mathematics and Geometry, the vertex form of a quadratic function is represented by the following mathematical equation (formula):

y = a(x - h)² + k

Where:

Based on the information provided about this quadratic function, we can logically deduce that a mathematical equation which quickly reveals the vertex of the quadratic function is given by:

y = a(x - h)² + k

0 = a(2.9 - (-1))² + 4

3.9a = -4

a = -4/3.9

a = -1.03

Therefore, the required quadratic function is given by:

y = a(x - h)² + k

y = -1.03(x - (-1))² + 4

y = -1.03(x + 1)² + 4

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Answer:

The data in the table represent an exponential function.

[tex]y = 2( {3}^{x} )[/tex]

The distance along the inside edge of each lane is 170 yards, 180 yards,  190 yards,  200 yards,  210 yards, 220 yards, 230 yards, and 240 yards.

Length of the Beld = 100 yd

Width of the Beld = 50 yd

The radius of the lane = 50/2 = 25 yards

To calculate the length of the overall length of the lane including semicircles is:

100 yards + 2 × 25 yards = 150 yards

The length of the innermost lane is:

150 yards + 2 × 10 yards = 170 yards

To calculate the length of the other lanes is:

170 yards + 10 yards = 180 yards

The length of lane 3 is:

180 yards + 10 yards = 190 yards

The distance between the two lanes is 10 yards. Then the remaining lengths of the lanes will be 200 yards,  210 yards, 220 yards, 230 yards, and 240 yards

Therefore, we can conclude that the distance between the two lanes along the inside edge of each lane is 10 yards.

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The net pay is correct.

To determine if Chloe's net pay is correct, we need to check if all the deductions are calculated correctly. Let's examine each deduction one by one:

1. Social Security tax:
Social Security tax is 6.2% of gross pay.
6.2% of $856 = 0.062 * $856 = $53.07

2. Medicare tax:
Medicare tax is 1.45% of gross pay.
1.45% of $856 = 0.0145 * $856 = $12.41

3. State tax:
State tax is 23% of federal tax.
23% of $80 = 0.23 * $80 = $18.40

Now let's calculate the net pay by subtracting all deductions from the gross pay:
Net pay = Gross pay - (Federal tax + Social Security tax + Medicare tax + State tax)
Net pay = $856 - ($80 + $53.07 + $12.41 + $18.40) = $856 - $163.88 = $692.12

Since all deductions are correct, and the calculated net pay matches the given net pay, the correct answer is:

a. The net pay is correct.

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The primary reason many present-day African nations struggled to unite their people is F. Because many African peoples have strong tribal ties.

The strong tribal identities and loyalties that exist among various ethnic groups inside African states today are one of the main causes of the effort to bring people together. Through colonisation, many African nations were developed, and many of them contained numerous groups with various cultural, linguistic, and ethnic origins.

Therefore, as a result, it has been challenging to forge a strong sense of national identity among many individuals who still identify primarily with their own tribe or ethnic group. This has occasionally resulted in disputes and hostilities amongst various diverse groups, impeding efforts to create a community that is more unified.

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Answer:

A≈166.28

Step-by-step explanation:

Answer:

A≈166.28

Step-by-step explanation:

We can rewrite the integrand as follows:

1 / (2 + e^x + e^(-x))

To evaluate this integral, we use the substitution u = e^x:

du/dx = e^x

dx = du/u

Substituting u and dx in terms of u into the integral, we get:

∫ (1 / (2 + u + 1/u)) du

Multiplying the numerator and denominator by u, we get:

∫ (u / (2u + u^2 + 1)) du

Next, we complete the square in the denominator:

u^2 + 2u + 1 = (u + 1)^2

So, we can write:

∫ (u / ((u + 1)^2 + 1)) du

Now, we use the substitution v = u + 1:

dv/du = 1

du = dv

Substituting v and du in terms of v into the integral, we get:

∫ ((v - 1) / (v^2 + 1)) dv

Using partial fractions, we can write:

(v - 1) / (v^2 + 1) = A(v - i) + B(v + i)

where A and B are constants to be determined, and i = sqrt(-1).

Multiplying both sides by v^2 + 1 and simplifying, we get:

v - 1 = A(v - i)(v^2 + 1) + B(v + i)(v^2 + 1)

Substituting v = i, we get:

-i - 1 = A(0) + B(2i)

B = -(i + 1)/2i = (1 - i)/2

Substituting v = -i, we get:

i - 1 = A(-2i) + B(0)

A = (1 - i)/2i = (i - 1)/2

Therefore, we have:

(v - 1) / (v^2 + 1) = [(i - 1)/(2i)](v + i) - [(1 - i)/(2i)](v - i)

Substituting back for v, we get:

(u / ((u + 1)^2 + 1)) = [(i - 1)/(2i)][(u + 1) + ie^x] - [(1 - i)/(2i)][(u + 1) - ie^x]

Substituting this expression back into the integral and simplifying, we get:

∫ (1 / (2 + e^x + e^(-x))) dx = [(i - 1)/(2i)]*ln(e^x + e^(-x) + 2) - [(1 - i)/(2i)]*ln(e^x - i) + C

where C is the constant of integration.

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a) The distance the ball rebounds on the fifth bounce is approximately 7.59 ft.

b) The total distance the ball has traveled after the fifth bounce is approximately 52.61 ft.

Let's denote the height of the ball after its nth bounce by h_n. Then we can express the relationship between the height of the ball after each bounce in terms of a recursive formula:

h_0 = 16 (initial height)

h_1 = (3/4) * h_0 (rebound distance after the first fall)

h_2 = (3/4) * h_1 (rebound distance after the second fall)

h_3 = (3/4) * h_2 (rebound distance after the third fall)

h_4 = (3/4) * h_3 (rebound distance after the fourth fall)

h_5 = (3/4) * h_4 (rebound distance after the fifth fall)

a) To find the distance the ball rebounds on the fifth bounce, we need to calculate h_5:

h_5 = (3/4) * h_4

= (3/4) * ((3/4) * ((3/4) * ((3/4) * 16)))

= (3/4)^5 * 16

= 7.59375 ft

Therefore, the ball rebounds approximately 7.59 ft on the fifth bounce.

b) To find the total distance the ball has traveled after the fifth bounce, we need to add up all of the distances traveled during the falls and rebounds:

total distance = distance of first fall + rebound distance after first fall + rebound distance after second fall + rebound distance after third fall + rebound distance after fourth fall + rebound distance after fifth fall

total distance = 16 + (3/4) * 16 + (3/4)^2 * 16 + (3/4)^3 * 16 + (3/4)^4 * 16 + (3/4)^5 * 16

total distance = 16 + 12 + 9 + 6.75 + 5.0625 + 3.7969

total distance = 52.6094 ft

Therefore, the ball travels approximately 52.61 ft after the fifth bounce.

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Full Question:

Although part of your question is missing, you might be referring to this full question:

Be sure to show and explain all work using mathematical formulas and terminology. A bouncy ball is dropped from a height of 16ft and always rebounds ¼ of the distance of the previous fall.

a) What distance does it rebound the 5th time?

b) What is the total distance the ball has travelled after this time?

An American in the middle 70% of cheese consumption consumes between 23.1 lbs and 41.5 lbs of cheese per year.

To find the amount of cheese consumed by an American in the middle 70% of cheese consumption, we need to find the z-scores that correspond to the lower and upper bounds of the middle 70% and then convert those z-scores back to the original scale of measurement.

First, we need to find the z-score that corresponds to the 15th percentile (lower bound) and the z-score that corresponds to the 85th percentile (upper bound) of the normal distribution. We can use a standard normal table or a calculator to find these values. Using a calculator, we get:

z_15 = invNorm(0.15) = -1.036

z_85 = invNorm(0.85) = 1.036

Next, we can use the formula:

z = (x - mu) / sigma

where x is the amount of cheese consumed by an American, mu is the mean amount of cheese consumed (32.3 lbs), and sigma is the standard deviation (8.7 lbs), to convert the z-scores back to the original scale of measurement:

For the lower bound:

-1.036 = (x - 32.3) / 8.7

x = -1.036 * 8.7 + 32.3 = 23.1 lbs

For the upper bound:

1.036 = (x - 32.3) / 8.7

x = 1.036 * 8.7 + 32.3 = 41.5 lbs

Therefore, an American in the middle 70% of cheese consumption consumes between 23.1 lbs and 41.5 lbs of cheese per year.

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The relationship between the squares formed by the sides of a right triangle is defined by Pythagorean Theorem, which states that the square on the hypotenuse side is the sum of the squares on the other two sides.

According to Pythagorean Theorem, the square formed by the side length of the hypotenuse side of right triangle is equivalent to the sum of the squares formed by the lengths of the other two sides

Let a, b, and c represent the lengths of the sides of the right triangle, where;

a = The length of the hypotenuse side

b = The length of a leg of the right triangle

c = The length of the other leg of the right triangle

The area of each squares are therefore;

Area of the square formed by the hypotenuse side = a²

Area of the square formed by the legs = b² and c²

Therefore; a² = b² + c²

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Answer:

In this pattern, there are 12 regular pentagons and 20 regular hexagons. Each hexagon shares a vertex with three pentagons and each pentagon shares a vertex with five hexagons.

To find the measure of each gap between the hexagons, we can use the fact that the sum of the angles around any vertex in a regular polygon is always 360 degrees. Let x be the measure of the angle between two adjacent pentagons, and y be the measure of each angle at the center of a hexagon.

At each vertex of the pattern, there are three pentagons and three hexagons meeting. Thus, we have:

3(108) + 3y = 360

Simplifying, we get:

324 + 3y = 360

3y = 36

y = 12

Therefore, each angle at the center of a hexagon measures 12 degrees. Since there are six angles around the center of a hexagon, the total angle around the center of a hexagon is 6(12) = 72 degrees.

To find the measure of each gap between the hexagons, we need to subtract the angle of the hexagon from 180 degrees (since the sum of the angles of a triangle is 180 degrees). Thus, the measure of each gap between the hexagons is:

180 - 72 = 108 degrees

Step-by-step explanation:

Answer: Yes those are correct good job

Step-by-step explanation:

The word sentence for the given number line represents the X is greater than zero. The inequality is the X is not less than or equal to zero.

In the given number line the values are from negative infinity to positive infinity. Though the number line is having all values, there is some part that is shaded only on certain parts of the number line. The part is from number 1 to positive infinity marking zero as a circle.

The circle represents the value starts counting from the next step. In the given number line circle is shaded at number zero and the value started from the next value which is number 1 onwards to positive infinity.

So, the given number line represents X is greater than zero and X is not less than or equal to zero.

In mathematical words,

X < 0 and X≠0

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Given question is not having enough information to solve, for better understanding attaching the number line image below.

The area of the trapezoid in this problem is given as follows:

15 square feet.

The area of a trapezoid is given by half the multiplication of the height by the sum of the bases, hence:

A = 0.5 x h x (b1 + b2).

The dimensions for this problem are given as follows:

h = 3 ft, b1 = 4 ft and b2 = 6 ft.

Hence the area is given as follows:

A = 0.5 x 3 x (4 + 6)

A = 1.5 x 10

A = 15 square feet.

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Tricia has a total of $6.35. The answer is D.

Tricia has 10 quarters, which is equivalent to $2.50 (since 1 quarter is $0.25).

She also has 17 dimes, which is equivalent to $1.70 (since 1 dime is $0.10).

Tricia has 40 nickels, which is equivalent to $2.00 (since 1 nickel is $0.05).

Finally, she has 15 pennies, which is equivalent to $0.15 (since 1 penny is $0.01).

To find the total amount of money Tricia has, we can add up the values of each coin:

$2.50 + $1.70 + $2.00 + $0.15 = $6.35

Therefore, Tricia has a total of $6.35. The answer is option D.

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Answer:

Step-by-step explanation:

True

Answer:

Amal's mother was 11.4 years old when Amal was born.

Step-by-step explanation:

Let's start by using variables to represent the ages of each person:

From the problem, we know:

Substituting the first three equations into the fourth, we get:

[tex]\sf:\implies A + 0.5A + 3A + 4(3A) = 94[/tex]

Simplifying:

[tex]\sf:\implies A + 0.5A + 3A + 12A = 94[/tex]

[tex]\sf:\implies 16.5A = 94[/tex]

[tex]\sf:\implies A = 5.7[/tex]

So Amal is 5.7 years old. To find the age of Amal's mother when Amal was born, we need to subtract Amal's age from his mother's age:

[tex]\sf:\implies M - A = 3A - A = 2A[/tex]

So Amal's mother was 2A = 2(5.7) = 11.4 years old when Amal was born.

The data gathering technique used by Dr. Jackson in her study of the daily activities of 4th graders is an example of a quantitative approach. This approach involves the collection of numerical data that can be analyzed using statistical methods to identify patterns and relationships.

By asking the children to indicate the amount of time they spend on different activities, Dr. Jackson is collecting quantitative data that can be used to compare and contrast the daily routines of different children. This approach is particularly useful for identifying trends and patterns in the data, such as the amount of time spent watching television or playing video games.

Quantitative data gathering techniques are commonly used in social sciences research, including education, psychology, and sociology. They are useful for identifying patterns and trends in large data sets, and can be used to make predictions about future behavior or outcomes.

Overall, Dr. Jackson's study of the daily activities of 4th graders is an example of the quantitative approach to data gathering. By collecting numerical data on the amount of time spent on different activities, Dr. Jackson can analyze the data using statistical methods to identify patterns and trends, and make predictions about future behavior or outcomes.

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The total number of zeros in the mass of the sun when written in standard notation is 30, which is option B.

When we write the number in standard notation, we move the decimal point to the left or right to express the number in terms of powers of 10. In this case, we can write the mass of the sun as:

1.9891 x 10^30

To count the number of zeros in this number, we only need to count the number of digits to the right of the decimal point, which is zero in this case. Then we add the exponent, which tells us the number of places we need to move the decimal point to the right to express the number in standard notation. In this case, the exponent is 30, so we need to move the decimal point 30 places to the right, which means adding 30 zeros.

Therefore, the total number of zeros in the mass of the sun when written in standard notation is 30, which is option B.

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Answer:

If the pattern is to continue, the next value would be a decrease of 7 from the previous value because 20-13=7. Therefore, the value for this week would be:

13 - 7 = 6

So, if you were to continue the pattern, you would have bought 6 cupcakes this week.

Answer:

4.28

Step-by-step explanation:

To find what the length of AC is, we can use the tangent of angle A, which is 35°.  Here's the equation:

tan(35)=3/x

multiply both sides by x

x·tan(35)=3

divide both sides by the tangent of 35

x=4.28 (rounded to the nearest hundredth)

This means that AC=4.28

Hope this helps! :)

The largest two digit number that is divisible by both 6 and 8 that is Dexter's number is equals to the ninty-six.

Two digit numbers : 2-digit numbers are the numbers that have two digits and they start from the number 10 and end on the number 99. They cannot start from zero. We have specify that Mr. Phil asks his students to determine the largest 2 -digit number that is divisible by both 6 and 8. Let the dexter's two digit number be 'x'.

From the above list of 11 numbers the largest number that is multiple of 2 and 3 both. That is 96. So, the students answer is 91. The answer of one of his student is less than 5 the correct number. Hence, required value is 96.

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