If f(x) and f^1(x)
are inverse functions of each other and f(x) - 2x+5, what is f^-1(8)?

-1
3/2
41/8
23

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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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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Area of sector = 161.1 square inches

Activity 1:

The radius of the pizza is half of its diameter, which is 16/2 = 8 inches.

The central angle of the sector is 80%, which is 0.8 times 360 degrees = 288 degrees.

To find the area of the sector, we use the formula:

Area of sector = (central angle / 360) x πr^2

Area of sector = (288 / 360) x π x 8^2

Area of sector = (0.8) x π x 64

Area of sector = 161.1 square inches (rounded to the nearest tenth)

Activity 2:

The unit of measurement for the area of the sector is square inches (in²).

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Answer:

A≈166.28

Step-by-step explanation:

Answer:

A≈166.28

Step-by-step explanation:

The probability that Jade gets the specific outcome you're interested in when rolling two dice is 1/6.

To find the probability that Jade gets a specific outcome when rolling two dice, we will use the sample space for rolling two dice, which consists of 36 possible outcomes (since there are 6 sides on each die, and we have 2 dice: 6 x 6 = 36).

Step 1: Determine the specific outcome you are interested in (for example, the sum of the numbers on the dice being 7).

Step 2: Count the number of ways this outcome can occur. For example, if we want a sum of 7, there are 6 possible outcomes: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).

Step 3: Calculate the probability by dividing the number of successful outcomes by the total number of possible outcomes in the sample space.

In our example, there are 6 successful outcomes, and there are 36 possible outcomes in the sample space:

Probability = (Number of successful outcomes) / (Total number of possible outcomes) = 6/36

Step 4: Express the probability in its simplest form by reducing the fraction. In our example, 6/36 can be reduced to 1/6.

So, the probability that Jade gets the specific outcome you're interested in when rolling two dice is 1/6.

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Jose drove the truck for approximately 213 miles.

Let's assume that Jose drove the truck for m miles.

We know that there was a base fee of $17.99, so the remaining amount after that base fee went towards the additional charge of 83 cents per mile.

So, the additional charge for the miles driven can be represented as 0.83m.

The total cost that Jose had to pay was $194.78. Therefore, we can write the equation:

17.99 + 0.83m = 194.78

Solving for m:

0.83m = 194.78 - 17.99

0.83m = 176.79

m = 213.072

So, Jose drove the truck for approximately 213 miles.

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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 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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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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This evaluated expression is a scalar multiple of -4, which projects that vectors CD and CE are collinear. Then, points C, D, and E lie on the same straight line.

Let us proceed by  evaluating the vector CD. Then C is the midpoint of OA, we can evaluate the vector CD by subtracting vector CO from vector OD.

Vector CO = 1/2 × Vector OA
= 1/2 × (a + b)
= 1/2a + 1/2b

Vector OD = 3/4 × Vector AD
= 3/4 × (3/4a - 1/4b)
= 9/16a - 3/16b

Vector CD = Vector OD - Vector CO
= (9/16a - 3/16b) - (1/2a + 1/2b) = 5/16a - 5/16b

Then the value of the vector CE is
OB = 2BE,
we can evaluate the vector BE by dividing vector OB by 2.

Vector BE = 1/2 × Vector OB
= 1/2 × b
= 1/2b

Vector CE = Vector CO + Vector OE

Vector OE = Vector OB - Vector OE
= b - Vector BE
= b - 1/2b
= 1/2b

Vector CE = Vector CO + Vector OE
= (1/2a + 1/2b) + (1/2b)
= 1/2a + b

Then we have to show that vectors CD and CE are collinear. Two vectors are collinear if one is a scalar multiple.

CE can be expressed in terms of CD
CE / CD
= ((1/2a + b) / (5/16a - 5/16b))

Applying simplification for this expression

CE / CD
= (-8a - 8b) / (5a - 5b)

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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:

f(x) = x² - 10x + 22

Step-by-step explanation:

Let's assume the quadratic polynomial as:

f(x) = ax² + bx + c

Now we know that if one of the zeroes is 5 + √3, then the other zero must be 5 - √3 (because complex roots always come in conjugate pairs).

So the sum of the zeroes will be:

(5 + √3) + (5 - √3) = 10

10 = 2 * 5

The product of the zeroes will be:

(5 + √3) * (5 - √3) = 25 - 3 = 22

Now, using the sum and product of zeroes, we can write:

b/a = 10

c/a = 22

Solving for b and c, we get:

b = -10a

c = 22a

Substituting these values in f(x), we get:

f(x) = a(x - 5 - √3)(x - 5 + √3)

Expanding the right-hand side:

f(x) = a[(x - 5)² - (√3)²]

f(x) = a(x² - 10x + 22)

Comparing the coefficients of f(x) with ax² + bx + c, we get:

a = 1, b = -10, c = 22

Therefore, the quadratic polynomial is:

f(x) = x² - 10x + 22

The total sum of all edges of the regular quadrilateral prism is 108 cm.

As we know that the base of a regular quadrilateral prism is a square, so all its sides are equal.

The edges of the base can be calculated as:

P = 4 × L

Each side of the base has a length of 8 cm. Then, put L=4,

P = 4 (8)

P = 32 cm

The edges of the top of the prism can be calculated as:

P = 4L

P = 4 (8)

P = 32 cm

The edges of the prism height can be calculated as:

P = 4h

P = 4 (11)

P = 44 cm

The total sum of all edges can be calculated as:

= 32 + 32 + 44

= 108 cm

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Answer:

The data in the table represent an exponential function.

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

The answer is reasonable because it is positive and also the equation is true . it would take Lara about 5.3 hours to clean the tank by herself.

Let's denote the time it takes for Lara to clean the tank alone as "L". We can use the formula for the combined work rate of two people, which is:

(1/7) + (1/L) = (1/3)

Multiplying both sides by the least common denominator, 21L, gives:

3L + 21 = 7L

Subtracting 3L from both sides, we get:

21 = 4L

Dividing both sides by 4, we get:

L = 5.25 hours (to the nearest tenth)

The answer is reasonable because it is positive, and it is also less than 7 hours, which is Kevin's time. When substituted back into the original equation, we get:

(1/7) + (1/5.25) = (1/3)

0.1429 + 0.1905 = 0.3333

0.3334 ≈ 0.3333

The equation is true, so the answer is reasonable. Therefore, it would take Lara about 5.3 hours to clean.

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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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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 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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The parametric equations to plot points. The sixth point is (0, 1).

The parametric equations given are:

x = sin t

y = 1 − cos t

To plot points for these equations, we can choose some values of t and substitute them in the equations to get the corresponding values of x and y. Here are some points we can plot:

When t = 0, x = sin 0 = 0 and y = 1 − cos 0 = 1.

So the first point is (0, 1).

When t = π/4, x = sin (π/4) = √2/2 and y = 1 − cos (π/4) = 1 − √2/2.

So the second point is (√2/2, 1 − √2/2).

When t = π/2, x = sin (π/2) = 1 and y = 1 − cos (π/2) = 0.

So the third point is (1, 0).

When t = π, x = sin π = 0 and y = 1 − cos π = 2.

So the fourth point is (0, 2).

When t = 3π/2, x = sin (3π/2) = −1 and y = 1 − cos (3π/2) = 0.

So the fifth point is (−1, 0).

When t = 2π, x = sin 2π = 0 and y = 1 − cos 2π = 1.

So the sixth point is (0, 1).

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The conclusion is YEN ≈ 63.43°.and  YN = 4√5 cm.

We can begin by drawing a diagram of the triangle LNM with the given measurements:

     N

     |\

     | \

    y|  \ x

     |   \

     |____\

     L  8cm M

Since X is the midpoint of MN, we know that MX = NX = 6/2 = 3cm. Similarly, Y is the midpoint of LN, so LY = NY = 8/2 = 4cm.

To find YN, we can use the Pythagorean theorem:

 Y________N

  |\     |

  | \    |

  |  \   | 6cm

  |   \  |

  |    \ |

 L|_____Y\|

     4cm

YN² = YL² + LN²

YN² = 4² + 8²

YN² = 80

YN = √80 = 4√5 cm

Therefore, YN = 4√5 cm.

To find YẼN, we need to find the angle YLN. Since Y is the midpoint of LN, YL is half the length of LN, which is 8cm. So YL = 4cm. We can use trigonometry to find the angle YLN:

tan(YLN) = opposite/adjacent

tan(YLN) = YL/LN

tan(YLN) = 4/8

tan(YLN) = 0.5

YLN ≈ 26.57°

Since LÑN = 90°, we know that YEN is the complement of YLN:

YEN = 90° - YLN

YEN ≈ 63.43°

Therefore, YEN ≈ 63.43°.

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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:

To make the polynomial 9x^10 + ax^b + 100 a perfect square trinomial, we need to add a constant term to it such that it becomes a square of a binomial.

Let's first write the square of a binomial in general form:

(a + b)^2 = a^2 + 2ab + b^2

If we compare this general form with our polynomial, we can see that the first term, 9x^10, is equal to (3x^5)^2, which means that we can write our polynomial as:

(3x^5)^2 + ax^b + 100 = (3x^5 + c)^2

Expanding the right-hand side of this equation, we get:

(3x^5 + c)^2 = 9x^10 + 6cx^15 + c^2

Comparing the coefficient of x^15 on both sides, we get:

6c = 0

Since c cannot be zero (otherwise we would end up with the original polynomial), this means that we must have:

c = 0

Therefore, we can write our polynomial as:

(3x^5)^2 + ax^b + 100 = (3x^5)^2

Expanding the right-hand side, we get:

(3x^5)^2 = 9x^10

Therefore, we must have:

a = 0

b = 10

So the values of a and b that make the polynomial 9x^10 + ax^b + 100 a perfect square trinomial are a = 0 and b = 10.

To find the rate of change of the temperature difference between the two spacecraft, we need to first find the temperature at each spacecraft's position at time t=4.

For the first spacecraft, rt sin(t) = r4sin(4) and t=4, so its position is (4sin(4), 4, 0). Using the temperature function, we have T(4sin(4), 4, 0) = (4sin(4))(4)(5-2) = 48.08.

For the second spacecraft, rz cos(t) = r3cos(4) and t=-4/3, so its position is (3cos(4), -4/3, 7). Using the temperature function, we have T(3cos(4), -4/3, 7) = (3cos(4))(-4/3)(5-2) = -9.09.

Therefore, the temperature difference D between the two spacecraft at time t=4 is D = 48.08 - (-9.09) = 57.17.

To find the rate of change of D with respect to time, we use the Chain Rule. Let x = 4sin(t) and y = 4, so D = T(x, y, 0) - T(3cos(t), -4/3, 7). Then,

dD/dt = dD/dx * dx/dt + dD/dy * dy/dt

We already know that D = 48.08 - 9.09 = 57.17, so dD/dx = dT/dx = y(5-2x) = 4(5-2(4sin(4))) = -31.64.

We also have dx/dt = 4cos(4) and dy/dt = 0, since y is constant.

To find dD/dy, we take the partial derivative of T with respect to y, holding x and z constant: dT/dy = x(5-2y) = (4sin(4))(5-2(4)) = -28.16.

Putting it all together, we get:

dD/dt = dD/dx * dx/dt + dD/dy * dy/dt
= (-31.64)(4cos(4)) + (-28.16)(0)
= -126.56

Therefore, the rate of change of the temperature difference between the two spacecraft at time t=4 is -126.56.
Given the paths of the two spacecraft: r1(t) = (t sin(t), t, 0) and r2(t) = (t cos(t), -t, 7), and the temperature function T(x, y, z) = x * y * z^2, we want to determine the rate of change of the temperature difference D at time t=4 using the Chain Rule.

First, let's find the temperature for each spacecraft at time t:

T1(t) = T(r1(t)) = (t sin(t)) * t * 0^2
T1(t) = 0

T2(t) = T(r2(t)) = (t cos(t)) * (-t) * 7^2
T2(t) = -49t^2 cos(t)

Now, find the temperature difference D(t) = T2(t) - T1(t) = -49t^2 cos(t)

Next, find the derivative of D(t) with respect to t:

dD/dt = -98t cos(t) + 49t^2 sin(t)

Now, we need to evaluate dD/dt at t=4:

dD/dt(4) = -98(4) cos(4) + 49(4)^2 sin(4) ≈ -104.32

Thus, the rate of change of the temperature difference D at time t=4 is approximately -104.32 (in decimal notation, rounded to two decimal places).

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Answer:

43.96 cm

Step-by-step explanation:

Given

Radius ( r ) = 7 cm

To find : Circumference of Circle

Formula

Circumference of Circle = 2πr

Note

The Value of π = 3.14

Circumference of Circle

= 2πr

= 2 × 3.14 × 7

= 43.96 cm

Answer:

Not Rounded: 43.9822971503

Rounded: 44

Step-by-step explanation:

r= radius

2[tex]\pi[/tex]r

2[tex]\pi[/tex](7)

= 43.9822971503

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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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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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.

Answer: Yes those are correct good job

Step-by-step explanation:

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

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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