find the inverse of each function

The exponential equation a¹⁰ = 200 is equivalent to the logarithmic equation Log = 200 a.

The rule of logarithms that we should use here is given below:

[tex]\log \text{x} = \text{a} \iff 10^{\text{a}} = \text{x}[/tex]

The given expression is Log = 200 a.

We can follow the rule log x = a ⇔ 10^a = x to convert this logarithmic equation to an exponential one.

Log = 200 a can be rewritten as a¹⁰ = 200

Therefore, we have found that the exponential equation a¹⁰ = 200 is equivalent to the logarithmic equation Log = 200 a.

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The solution for "r" is 21/24, which can be simplified to 7/8. The answer can be written as a fraction of 7/8.

To solve for "r" in the given equation, we can use algebraic manipulation.

First, we can multiply both sides of the equation by the reciprocal of 6/7, which is:

7/6: 6/7 · r = 3/4  

7/6 · 6/7 · r = 7/6 · 3/4

r = 21/24.

Thus, the solution for "r" is 21/24, which can be simplified to 7/8.

Therefore, the answer can be written as a fraction of 7/8.

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Answer:

perimeter = 75 cm , area ≈ 28.3 in²

Step-by-step explanation:

23 (a)

since the figure is being enlarged by a scale factor of 5

then the perimeter is increased by a factor of 5.

perimeter of larger shape = 5 × 15 = 75 cm

24

the area (A) of the sector is calculated as

A = area of circle × fraction of circle

  = πr² × [tex]\frac{90}{360}[/tex]

  = π × 6² × [tex]\frac{1}{4}[/tex]

  = 36π × [tex]\frac{1}{4}[/tex]

  = 9π

  ≈ 28.3 in² ( to 1 decimal place )

Answer:

The answer is number 3

Y =-3/7X + 3

Step-by-step explanation:

Substitue with the value of the two points in all answers

The value of the left side must equal the value or the right side

For instance,

The two points are (0,3) & (7,0)

Substitue in the first answer with the point (0,3)

3 = - 3 (rejected)

Second answer

3 = 3 works for the point (0,3) then also Substitue with the other point (7,0)

0 = 6 (rejected)

sub. With point (0,3).

Sub. With point (7,0)

Then that's the right one

Answer:

x = 110°

Step-by-step explanation:

The opposite angles are equal in a parallelogram

3x - 60 = 2x + 50

⇒ 3x - 2x = 60 + 50

⇒ x = 110°

Answer:

x = 110°

Step-by-step explanation:

As the top and bottom line segments of the given shape are the same length and parallel (indicated by the tick marks and arrows), the shape is a parallelogram.

As the opposite angles of a parallelogram are equal, to find the value of the variable x, equate the two angle expressions and solve for x:

[tex]\begin{aligned}3x-60^{\circ}&=2x+50^{\circ}\\3x-60^{\circ}-2x&=2x+50^{\circ}-2x\\x-60^{\circ}&=50^{\circ}\\x-60^{\circ}+60^{\circ}&=50^{\circ}+60^{\circ}\\x&=110^{\circ}\end{aligned}[/tex]

Therefore, the value of x is 110°.

Note: There must be an error in the question. If x = 110°, each angle measures 270°, which is impossible since the sum of the interior angles of a quadrilateral is 360°.

The image coordinates of point U, after translating the RSTU rectangle by T(2,-3), would be U(6, 6).

To find the image coordinates of U, we need to apply the translation vector T(2,-3) to each of the original coordinates.

The translation vector represents the horizontal and vertical distances by which each point is moved.

Starting with the original coordinates of point U, which are (4, 9), we add the horizontal distance of 2 to the x-coordinate and subtract the vertical distance of 3 from the y-coordinate.

Therefore, the new x-coordinate of U is 4 + 2 = 6, and the new y-coordinate is 9 - 3 = 6.

Thus, the image coordinates of point U after the translation are (6, 6). This means that U has been moved 2 units to the right and 3 units downward from its original position.

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Answer:

A) The y-intercept(s) is/are 2

Step-by-step explanation:

Y-intercepts are where the graph of a function cross over the y-axis. In this case, the line passes through y=2, which is the y-intercept.

Answer:

d ≈ 8.1

Step-by-step explanation:

calculate the distance d using the distance formula

d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]

with (x₁, y₁ ) = R (5, 7 ) and (x₂, y₂ ) = S (- 2, 3 )

d = [tex]\sqrt{(-2-5)^2+(3-7)^2}[/tex]

  = [tex]\sqrt{(-7)^2+(-4)^2}[/tex]

  = [tex]\sqrt{49+16}[/tex]

  = [tex]\sqrt{65}[/tex]

  ≈ 8.1 ( to 1 decimal place )

It should be a positive graph .

Angle C can be found by subtracting the sum of angles A and B from 180 degrees:

b. A = 34.1°; B = 44.4°; C = 101.5°

To solve a triangle with side lengths a = 4, b = 5, and c = 7, we can use the law of cosines and the law of sines.

First, let's find angle A using the law of cosines:

[tex]cos(A) = (b^2 + c^2 - a^2) / (2\times b \times c)[/tex]

[tex]cos(A) = (5^2 + 7^2 - 4^2) / (2 \times 5 \times 7)[/tex]

cos(A) = (25 + 49 - 16) / 70

cos(A) = 58 / 70

cos(A) ≈ 0.829

A ≈ arccos(0.829)

A ≈ 34.1°

Next, let's find angle B using the law of sines:

sin(B) / b = sin(A) / a

sin(B) = (sin(A) [tex]\times[/tex] b) / a

sin(B) = (sin(34.1°) [tex]\times[/tex] 5) / 4

sin(B) ≈ 0.822

B ≈ arcsin(0.822)

B ≈ 53.4°

Finally, angle C can be found by subtracting the sum of angles A and B from 180 degrees:

C = 180° - A - B

C = 180° - 34.1° - 53.4°

C ≈ 92.5°.

b. A = 34.1°; B = 44.4°; C = 101.5°

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4/21 of the books in the library are both fiction and paperbacks.

To determine the fraction of books in the library that are both fiction and paperback, we need to multiply the fractions representing each condition.

Let's start with the fraction of books in the library that are fiction. If 4/7 of the books are fiction, then this fraction represents the number of fiction books.

Next, we want to find the fraction of fiction books that are also paperbacks. Since 1/3 of the fiction books are paperbacks, we multiply 4/7 (fiction books) by 1/3 (paperback fraction).

Multiplying fractions is done by multiplying the numerators together to get the new numerator and multiplying the denominators together to get the new denominator.

Thus, the fraction of books in the library that are both fiction and paperbacks is:

(4/7) * (1/3) = (4 * 1) / (7 * 3) = 4/21

Therefore, 4/21 of the books in the library are both fiction and paperbacks.

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This flexibility of a money market account makes it an ideal option for people who need to save money without risking their inheritance.

After inheriting $25,000 from a distant uncle, you would like to save it for ten years before doing anything with it. Since you want to save the money, there are several options for keeping it safe and earning interest, including a savings account, a certificate of deposit (CD), and a money market account.

Money market accounts, in my opinion, would be the best place to keep the inheritance. The money market account is a low-risk account with high-interest rates, making it an attractive option for someone who wants to save their money. As a result, it would be reasonable to place the inheritance into a money market account that pays a daily compounded rate of 3%.There are several reasons for choosing this option.

Firstly, the daily compounded interest will generate a higher return over the ten-year period than the simple interest or monthly compounded interest offered by other accounts. Second, the account is FDIC-insured, which means that the account holder is guaranteed to receive their money in the event of a bank failure.

Furthermore, the money market account provides easy access to the account holder's money while still earning interest. Most money market accounts have a limit on the number of withdrawals a person can make per month. Still, the account holder can easily transfer funds into a checking account or withdraw money from an ATM if needed.

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The solution for k in the equation 10 - 10|-8k + 4| = 10, expressed in set notation, is {1/2}.

1. Start with the equation: 10 - 10|-8k + 4| = 10.

2. Simplify the expression inside the absolute value brackets: -8k + 4.

3. Remove the absolute value brackets by considering two cases:

  Case 1: -8k + 4 ≥ 0 (positive case):

    -8k + 4 = -(-8k + 4)  [Removing the absolute value]

    -8k + 4 = 8k - 4     [Distributive property]

    -8k - 8k = -4 + 4    [Group like terms]

    -16k = 0             [Combine like terms]

    k = 0               [Divide both sides by -16]

  Case 2: -8k + 4 < 0 (negative case):

    -8k + 4 = -(-8k + 4)  [Removing the absolute value and changing the sign]

    -8k + 4 = -8k + 4     [Simplifying the expression]

    0 = 0                [True statement]

4. Combine the solutions from both cases: {0}.

5. Check if the solution satisfies the original equation:

  For k = 0: 10 - 10|-8(0) + 4| = 10

             10 - 10|4| = 10

             10 - 10(4) = 10

             10 - 40 = 10

             -30 = 10 [False statement]

6. Since k = 0 does not satisfy the equation, it is not a valid solution.

7. Therefore, the final solution expressed in set notation is {1/2}.

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The surface area and volume of the square pyramid is 96 squared centimeter and 48 cubic centimeters respectively.

The surface area of a square pyramid is expressed as:

SA = [tex]a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }[/tex]

The volume of a square pyramid is expressed as:

Volume = [tex]a^2*\frac{h}{3}[/tex]

Where a is the base edge and h is the height.

From the figure a = 6cm

First, we determine the h, using pythagorean theorem:

h² = 5² - (6/2)²

h² = 5² - 3²

h² = 25 - 9

h² = 16

h = √16

h = 4 cm

Solving for surface area:

SA = [tex]a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }[/tex]

[tex]= a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }\\\\= 6^2 + 2*6 \sqrt{\frac{6^2}{4}+4^2 }\\\\= 36 + 12 \sqrt{\frac{36}{4}+16 }\\\\= 36 + 12 (5)\\\\= 36 + 60\\\\= 96 cm^2[/tex]

Solving for the volume:

Volume = [tex]a^2*\frac{h}{3}[/tex]

[tex]= a^2*\frac{h}{3}\\\\= 6^2*\frac{4}{3}\\\\= 36*\frac{4}{3}\\\\=\frac{144}{3}\\\\= 48 cm^3[/tex]

Therefore, the volume is 48 cubic centimeters.

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In this context, the term "zeros" refers to the roots or x-intercepts.

You're looking for where the graph either,

See the diagram below. That example shows two positive roots, because each root is to the right of the vertical y axis.

The output value f(3) in the functions f( x ) = 3x + 5, f( x ) =  [tex]\frac{1}{2}x^2-1.5[/tex] and f( x ) = [tex]\frac{3}{2x}[/tex] is 14, 3 and 1/2 respectively.

Given the functions in the question:

f( x ) = 3x + 5

f( x ) =  [tex]\frac{1}{2}x^2-1.5[/tex]

f( x ) = [tex]\frac{3}{2x}[/tex]

To evaluate each function at f(3), we simply replace the variable x with 3 and simplify.

a)

f( x ) = 3x + 5

Replace x with 3:

f( 3 ) = 3(3) + 5

f( 3 ) = 9 + 5

f( 3 ) = 14

b)

f( x ) =  [tex]\frac{1}{2}x^2-1.5[/tex]

Replace x with 3:

[tex]f(3) = \frac{1}{2}(3)^2 - 1.5\\\\f(3) = \frac{1}{2}(9) - 1.5\\\\f(3) = 4.5 - 1.5\\\\f(3) = 3[/tex]

b)

f( x ) = [tex]\frac{3}{2x}[/tex]

Replace x with 3:

[tex]f(3) = \frac{3}{2(3)} \\\\f(3) = \frac{3}{6} \\\\f(3) = \frac{1}{2}[/tex]

Therefore, the output value of f(3) is 1/2.

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Answer:

az

Step-by-step explanation:

Due to the presence of spheres X and Y, the electric field strength at point B is [tex]1.01 * 10^6 N/C[/tex] and [tex]-4.05 * 10^6 N/C[/tex], respectively.

Given that two spheres X and Y are carrying charges of 72mC and -72mC respectively, and they are located 4.0 m apart from each other. The electric field strength at points A and B due to the presence of each sphere is to be determined.

Let's begin by calculating the electric field strength at point A due to sphere X. Electric field strength is given by E=kq/r², where k is Coulomb's constant, q is the charge and r is the distance between the two charges. The electric field strength at point A due to sphere X, E₁=kq₁/r₁² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (4.0m)^2 = 4.05 * 10^6 N/C[/tex] (approx.)

Similarly, the electric field strength at point A due to sphere Y can be calculated as follows, E₂=kq₂/r₂² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (4.0m)^2 = 4.05 * 10^6 N/C[/tex] (approx.). Here, the negative sign indicates that the electric field due to sphere Y is in the opposite direction to the electric field due to sphere X. Now, let's calculate the electric field strength at point B. The electric field strength at point B due to sphere X, E₁=kq₁/r₁² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (8.0m)^2 = 1.01 * 10^6 N/C[/tex] (approx.)

Similarly, the electric field strength at point B due to sphere Y can be calculated as follows, E₂=kq₂/r₂² [tex]= (9*10^9Nm^2/C^2) * (-72mC) / (4.0m)^2 = -4.05 * 10^6 N/C[/tex] (approx.). Therefore, the electric field strength at point A due to the presence of sphere X is [tex]4.05 * 10^6 N/C[/tex] and due to the presence of sphere Y is [tex]-4.05 * 10^6 N/C[/tex]. The electric field strength at point B due to the presence of sphere X is [tex]1.01 * 10^6 N/C[/tex] and due to the presence of sphere Y is [tex]-4.05 * 10^6 N/C[/tex].

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The expected return on the portfolio is approximately 13.465%.

To calculate the expected return on the portfolio, we need to consider the weights of each stock in the portfolio.

Let's denote the weight of Stock A as wA and the weight of Stock B as wB. The weight of a stock is the proportion of the total portfolio value that is invested in that stock.

Given that $3,000 is invested in Stock A and $4,100 is invested in Stock B, we can calculate the weights as follows:

wA = $3,000 / ($3,000 + $4,100) = $3,000 / $7,100

wB = $4,100 / ($3,000 + $4,100) = $4,100 / $7,100

Next, we need to calculate the weighted average of the expected returns of the two stocks using their respective weights:

Expected return on the portfolio = (wA * Return on Stock A) + (wB * Return on Stock B)

Expected return on the portfolio = (wA * 10%) + (wB * 16%)

Substituting the calculated weights into the equation:

Expected return on the portfolio = ($3,000 / $7,100 * 10%) + ($4,100 / $7,100 * 16%)

Simplifying the equation:

Expected return on the portfolio = (0.4225 * 10%) + (0.5775 * 16%)

Expected return on the portfolio = 0.04225 + 0.0924

Expected return on the portfolio = 0.13465 or 13.465%

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Let's assume the dimensions of the iPad's screen are represented by length (L) and width (W). We can use the given information to set up two equations.

Perimeter equation:

The perimeter of a rectangle is given by the formula: 2(L + W). In this case, the perimeter is given as 36 inches.

So, 2(L + W) = 36.

Area equation:

The area of a rectangle is given by the formula: L * W. In this case, the area is given as 80 square inches.

So, L * W = 80.

We now have a system of two equations:

2(L + W) = 36

L * W = 80

From equation 1, we can simplify it to L + W = 18 and rewrite it as L = 18 - W.

Now substitute this value of L into equation 2:

(18 - W) * W = 80

Expanding the equation:

18W - W^2 = 80

Rearranging the equation to quadratic form:

W^2 - 18W + 80 = 0

Factoring or using the quadratic formula, we find the two possible values for W.

The solutions are W = 8 and W = 10.

Now we can substitute these values back into the equation L = 18 - W to find the corresponding values of L.

For W = 8:

L = 18 - 8 = 10

For W = 10:

L = 18 - 10 = 8

Therefore, the dimensions of the iPad's screen can be either 8 inches by 10 inches or 10 inches by 8 inches.

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Answer:

C-351

Step-by-step explanation:

Todd should expect approximately 624 people at his gym during the week of July 15.

A: To find the quadratic equation that models the data, we can use the vertex form of a quadratic equation:

[tex]y = a(x - h)^2 + k[/tex] where (h, k) represents the vertex of the parabola.

Let's analyze the data to determine the vertex. We observe that the number of people is highest during the first week and gradually decreases over the following weeks.

This suggests a downward-opening parabola.

From the data, the highest point occurs during the week of 6/3 to 6/9 with 624 people.

Therefore, the vertex is located at (6/3 to 6/9, 624).

Using the vertex form, we have:

[tex]y = a(x - 6/3 to 6/9)^2 + 624[/tex]

Now, we need to find the value of 'a.'

To do this, we can substitute any other point and solve for 'a.' Let's use the data from the week of 5/27 to 6/2:

[tex]618 = a(5/27 to 6/2 - 6/3 to 6/9)^2 + 624[/tex]

Simplifying the equation and solving for 'a,' we find:

[tex]618 - 624 = a(-6/3)^2[/tex]

-6 = 4a

a = -3/2

Therefore, the quadratic equation in vertex form that models the data is:

[tex]y = (-3/2)(x - 6/3 to 6/9)^2 + 624[/tex]

B: To predict the number of people Todd should expect during the week of July 15, we substitute x = 7/15 into the equation and solve for y:

[tex]y = (-3/2)(7/15 - 6/3 to 6/9)^2 + 624[/tex]

Simplifying the equation, we find:

[tex]y = (-3/2)(1/15)^2 + 624[/tex]

y = (-3/2)(1/225) + 624

y = -3/450 + 624

y = -1/150 + 624

y = 623.993

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The solution to the inequality f(x) > -5 is x < -5 or x > -1.

To solve the inequality f(x) > -5 for the quadratic function f(x) = x^2 + 6x, we need to find the values of x that satisfy the inequality.

First, set up the inequality:

x^2 + 6x > -5

Next, move all terms to one side of the inequality to get a quadratic expression:

x^2 + 6x + 5 > 0

To solve this quadratic inequality, we can factor it:

(x + 5)(x + 1) > 0

Now, we need to determine the sign of the expression for different intervals on the x-axis.

a) When x < -5:

If x is less than -5, both (x + 5) and (x + 1) are negative, so their product is positive.

Thus, the inequality is satisfied for x < -5.

b) When -5 < x < -1:

If x is between -5 and -1, (x + 5) is positive, but (x + 1) is negative. The product of a positive and a negative number is negative.

Thus, the inequality is not satisfied for -5 < x < -1.

c) When x > -1:

If x is greater than -1, both (x + 5) and (x + 1) are positive, so their product is positive.

Thus, the inequality is satisfied for x > -1.

Therefore, x -5 or x > -1 is the answer to the inequality f(x) > -5.

In summary:

x < -5 or x > -1.

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The minimum distance between the point (0, 4) on the y-axis and points on the graph is 4.

To find the minimum distance between the point (0, 4) on the y-axis and points on the graph of the function y = x², we can use the concept of perpendicular distance.

The distance between a point (x, y) on the graph and the point (0, 4) is given by the formula:

distance = √((x - 0)² + (y - 4)²) = √(x² + (y - 4)²)

Substituting the function y = x² into the distance formula, we get:

distance = √(x² + (x² - 4)²) = √(x² + (x⁴ - 8x² + 16))

Simplifying further, we have:

distance = √(x⁴ + x² - 8x² + 16) = √(x⁴ - 7x² + 16)

To find the minimum distance, we need to minimize the expression x⁴ - 7x² + 16. Since this is a quadratic-like expression, we can use calculus to find the minimum.

Taking the derivative of x⁴ - 7x² + 16 with respect to x, we get:

d/dx (x⁴ - 7x² + 16) = 4x³ - 14x

Setting the derivative equal to zero to find critical points:

4x³ - 14x = 0

Factorizing, we have:

2x(2x² - 7) = 0

This gives us two critical points: x = 0 and x = ±√(7/2).

Next, we evaluate the expression x⁴ - 7x² + 16 at these critical points and the endpoints of the interval:

f(0) = 0⁴ - 7(0)² + 16 = 16

f(±√(7/2)) = (√(7/2))⁴ - 7(√(7/2))² + 16 ≈ 4.157

Comparing these values, we find that the minimum distance occurs at x = 0, giving us a minimum distance of √(0⁴ - 7(0)² + 16) = √16 = 4.

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The true statements are:

a. f(-1) = 2

c. f(1) = 0

Let's evaluate each statement using the given piecewise function f(x):

a. f(-1) = -(-1) + 1 = 2

b. f(-2) = -(-2) + 1 = 3 (Not 0, so this statement is false)

c. f(1) = (1)^2 - 1 = 0

d. f(4) = (4)^2 - 1 = 16 - 1 = 15 (Not 7, so this statement is false)

Therefore, the correct statements are:

a. f(-1) = 2

c. f(1) = 0

Statement a is true because when x = -1, we use the first piece of the piecewise function, which gives us -(-1) + 1 = 2.

Statement c is true because when x = 1, we use the third piece of the piecewise function, which gives us (1)^2 - 1 = 0.

Statements b and d are false because they do not match the corresponding values obtained from evaluating the piecewise function at the given inputs.

Therefore, the true statements are:

a. f(-1) = 2

c. f(1) = 0

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The circles are similar because you can translate Circle 1 using the transformation rule (9, 5) and then dilate it using a scale factor of 2.

To prove that Circle 1 and Circle 2 are similar, we need to identify the transformations that can be applied to Circle 1 to obtain Circle 2.

First, let's consider the translation of Circle 1. The translation rule is given by (a, b), where a represents the horizontal shift and b represents the vertical shift.

In this case, to translate Circle 1 to align with Circle 2, we need to shift it 9 units to the right and 5 units up. Therefore, the translation rule for Circle 1 is (9, 5).

Next, let's consider the dilation. A dilation is a transformation that changes the size of the figure but preserves its shape. The scale factor, denoted by k, determines the amount of scaling. In this case, Circle 1 needs to be dilated to match the size of Circle 2.

The scale factor can be determined by comparing the radii of the two circles. The radius of Circle 1 is 3 centimeters, while the radius of Circle 2 is 6 centimeters. The scale factor is obtained by dividing the radius of Circle 2 by the radius of Circle 1: 6/3 = 2.

Therefore, the transformation applied to Circle 1 to prove that the circles are similar is a translation by (9, 5) followed by a dilation with a scale factor of 2.

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The value of the required arc in the figure is solved to be

The inscribed angle is given in the problem as 40 degrees. This is the angle formed at the circumference of the circle

The relationship between inscribed angle and the intercepted arc is  

intercepted arc = 2 * inscribed angle

in the problem, we have that

intercepted arc = ?

inscribed angle = 40

plugging in the values

intercepted arc = 2 * 40

intercepted arc = 80 degrees

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