a circular wheel of diameter 35 cm makes 100 revolutions in 1 min. calculate the distance covered by the wheel in half an hour. Answer in km

Take pie= 22/7

Answer ASAP

]The particle is moving from left to right for t > 2 or 0 < t < 1.5.

The path of a particle defined by s(t) = t⁴/4 − t³ + t², t ≤ 0. The particle is moving from left to right when its velocity is positive.

The velocity of the particle is given by the first derivative of its position function, which is s'(t) = t³ − 3t² + 2t. To find when the particle is moving from left to right, we need to find when its velocity is positive.
The particle is moving from left to right when its velocity is positive, which occurs when t > 2 or 0 < t < 1.5. Therefore, the particle is moving from left to right for t > 2 or 0 < t < 1.5.

The velocity function, s'(t) = t³ − 3t² + 2t, can be factored as s'(t) = t(t − 1)(t − 2).

This means that the velocity of the particle is 0 when t = 0, t = 1, and t = 2.

When t < 0, the velocity is negative because all three factors in the velocity function are negative. When 0 < t < 1, only one factor is negative (t − 2), so the velocity is positive.

When 1 < t < 2, two factors are negative (t − 1) and (t − 2), so the velocity is negative. When t > 2, all three factors are positive, so the velocity is particle .

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[tex] {9}^{ \frac{1}{3} } \\( {3}^{3} )^{ \frac{1}{3} } \\ 3[/tex]

or

[tex] \sqrt[ 3]{9} \\ 3[/tex]

a) The simplified ratio is 2:7.

b) 70 machines would pack approximately 245 crates of apples per day.

a) The ratio of the number of machines to the number of crates packed per day can be calculated by dividing the number of machines by the number of crates packed. In this case, there are 40 machines packing 140 crates per day. Thus, the ratio can be expressed as:

40:140

However, we can simplify this ratio by dividing both numbers by their greatest common divisor, which is 20:

40 ÷ 20 : 140 ÷ 20

2:7

So, the simplified ratio is 2:7.

b) If 40 machines pack 140 crates per day, we can use this information to determine how many crates 70 machines would pack. We can set up a proportion to solve for the unknown value:

40 machines / 140 crates = 70 machines / x crates

To solve for x, we can cross-multiply:

40 * x = 140 * 70

Dividing both sides by 40:

x = (140 * 70) / 40

x ≈ 245 crates

Therefore, 70 machines would pack approximately 245 crates of apples per day.

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The median from the histogram is 5

A histogram is a graph used to represent the frequency distribution of a few data points of one variable.

The median is the value in the middle of a data set.

The total frequency of the distribution is

1+4+5+4+2+1+1+1+1

= 20

The median term will be on

20+1)/2

= 10.5th

Therefore form the graph the 10.5 th term falls in the class of 5

This means that the median class is 5.

Therefore the median from the histogram is 5

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When rounded to the nearest million, 363366 would become 0.

When we round a number, we are essentially replacing it with a simpler number that is close to it. The rule for rounding to the nearest million is as follows:

The number 363366 rounded to the nearest million is 0. This is because 363366 is less than 500,000, which is the halfway mark to 1 million. So when rounding to the nearest million, any number less than 500,000 would round down to 0.

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The sum of the first 9 terms of the series is approximately

11

Given data

3, 15/4, 75/16, ...

To find the sum of the first 9 terms of the series, we'll need to find a pattern and then use the formula for the sum of a geometric series. The series you provided seems to be a geometric series with a common ratio of 5/4.

The first term (a₁) is 3, and the common ratio (r) is 5/4.

The formula for the sum of a geometric series is given by:

Sum = a₁ * (1 - rⁿ) / (1 - r),

where n is the number of terms.

Plugging in the values, we have:

Sum = 3 * (1 - (5/4)⁹) / (1 - 5/4).

Sum ≈ 3 * (1 - 1.953125) / (1 - 1.25)

≈ 3 * (-0.953125) / (-0.25)

≈ 3 * 3.8125

≈ 11.4375.

Rounding to the nearest integer = 11

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The length of one side of the equilateral triangle in terms of x is 6x.

To find the length of one side of the equilateral triangle in terms of x, we need to consider the perimeter of both the rectangle and the equilateral triangle.

The perimeter of a rectangle is given by the formula:

Perimeter of rectangle = 2(length + width)

In this case, the length of the rectangle is 7x cm, and the width is 2x cm. Substituting these values into the formula, we have:

Perimeter of rectangle = 2(7x + 2x) = 2(9x) = 18x

We are told that the equilateral triangle has the same perimeter as the rectangle.

Since an equilateral triangle has all sides equal, the perimeter can be calculated by multiplying the length of one side by 3.

Therefore, we have:

Perimeter of equilateral triangle = 3(side length)

Since the perimeter of the equilateral triangle is equal to the perimeter of the rectangle (18x), we can set up the equation:

3(side length) = 18x

Dividing both sides of the equation by 3, we get:

side length = 6x

Hence, the length of one side of the equilateral triangle in terms of x is 6x.

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The complete question may be like: A rectangle measures 2x cm by 7x cm. An equilateral triangle has the same perimeter as the rectangle. What is the length of one side of the triangle in terms of x?

Answer:

x = 15 , y = 9 , z = 115°

Step-by-step explanation:

since the figures are similar then the ratios of corresponding sides are in proportion, that is

[tex]\frac{NP}{ST}[/tex] = [tex]\frac{MN}{RS}[/tex] ( substitute values )

[tex]\frac{x}{25}[/tex] = [tex]\frac{18}{30}[/tex] ( cross- multiply )

30x = 450 ( divide both sides by 30 )

x = 15

and

[tex]\frac{TU}{PQ}[/tex] = [tex]\frac{MN}{RS}[/tex] ( substitute values )

[tex]\frac{y}{15}[/tex] = [tex]\frac{18}{30}[/tex] ( cross- multiply )

30y = 270 ( divide both sides by 30 )

y = 9

the corresponding angles in similar figures are congruent , so

∠ M = ∠ R , that is

z = 115°

To rewrite each fraction with a denominator of 12.3453, you need to multiply both the numerator and denominator of the given fraction by a factor that makes the denominator equal to 12.3453.

For example, if the given fraction is 1/3, you can multiply both numerator and denominator by 12.3453/3 to get 4.1157666667/12.3453.

In this case, the factor is 12.3453/3, which makes the denominator equal to 12.3453.
In general, to rewrite a fraction with a denominator of 12.3453, you need to multiply both numerator and denominator by the same factor, which is equal to 12.3453 divided by the original denominator.

This will give you the equivalent fraction with a denominator of 12.3453.

For example, if the original fraction is a/b, the equivalent fraction with a denominator of 12.3453 is (a x 12.3453)/b x 12.3453.
To rewrite each fraction with a denominator of 12.3453, we need to multiply both the numerator and denominator of the given fraction by a factor that makes the denominator equal to 12.3453.

The factor is equal to 12.3453 divided by the original denominator.

For example, if the given fraction is 1/3, we can multiply both numerator and denominator by 12.3453/3 to get 4.1157666667/12.3453.

In general, to rewrite a fraction with a denominator of 12.3453, we need to multiply both numerator and denominator by the same factor, which is equal to 12.3453 divided by the original denominator.

This will give us the equivalent fraction with a denominator of 12.3453.

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The linear equation of the scatterplot is y = 14x - 7/2

From the question, we have the following parameters that can be used in our computation:

The scatter plot (see attachment)

On the line of best fit, we have

(x, y) = (0.25, 0) and (0.75, 7)

A linear equation is represented as

y = mx + c

Where

c = y when x = 0

So, we have

0.25m + c = 0

0.75m + c = 7

Evaluate the difference

0.5m = 7

So, we have

m = 14

Next, we have

0.25 * 14 + c = 0

So, we have

7/2 + c = 0

This gives

c = -7/2

So, the equation is y = 14x - 7/2

Hence, the linear equation is y = 14x - 7/2

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For the equation g(x) = 6 - 1, the graph of g(x) differs from the graph of f(x) by translating down 1 unit.

For the equation g(x) = 2 + 1, the graph of g(x) differs from the graph of f(x) by translating up 1 unit.

In the Desmos Graphing Calculator, when entering the equation f(x) = 6 and g(x) = 6-1, we can observe the difference between the graphs of f(x) and g(x).

The equation f(x) = 6 represents a horizontal line at y = 6. The graph of f(x) is a straight line parallel to the x-axis, passing through the y-coordinate 6.

On the other hand, the equation g(x) = 6 - 1 represents a line that is one unit below the line represented by f(x). The graph of g(x) is a parallel line to the graph of f(x), but shifted downward by one unit.

Therefore, the correct answer is:

The graph of g(x) differs from the graph of f(x) by translating down 1 unit.

Similarly, when entering the equation f(x) = 2 and g(x) = 2 + 1, we can observe the difference between the graphs of f(x) and g(x).

The equation f(x) = 2 represents a horizontal line at y = 2. The graph of f(x) is a straight line parallel to the x-axis, passing through the y-coordinate 2.

The equation g(x) = 2 + 1 represents a line that is one unit above the line represented by f(x). The graph of g(x) is a parallel line to the graph of f(x), but shifted upward by one unit.

Therefore, the correct answer is:

The graph of g(x) differs from the graph of f(x) by translating up 1 unit.

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Answer:

[tex]2e^{-1}}[/tex]

Step-by-step explanation:

Evaluate the given improper integral.

[tex]\int\limits^{ \infty}_{1} {xe^{-x}} \, dx\\\\\\\hrule[/tex]

Using integration by parts.

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Integration by Parts}}\\\\uv-\int vdu\end{array}\right}[/tex]

Letting...

u = x => du = dx

dv = e^{-x} dx => v = -e^{-x}

[tex]\int\limits^{ \infty}_{1} {xe^{-x}} \, dx\\\\\\\\\Longrightarrow (x)(-e^{-x})-\int\limits^{ \infty}_{1} {-e^{-x}} \, dx\\\\\\\\\Longrightarrow -xe^{-x}\Big|\limits^{ \infty}_{1}+\int\limits^{ \infty}_{1} {e^{-x}} \, dx\\\\\\\\\Longrightarrow \Big[ -xe^{-x}-e^{-x}\Big]\limits^{ \infty}_{1}\\\\ \\\\\Longrightarrow \Big[ -(\infty)e^{-\infty}-e^{-\infty}\Big]-\Big[ -(1)e^{-(1)}-e^{-(1)}\Big]\\\\\\ \\\Longrightarrow \Big[-0-0\Big]-\Big[ -e^{-1}-e^{-1}\Big][/tex]

[tex]\Longrightarrow e^{-1}+e^{-1}\\\\\\\\\therefore \int\limits^{ \infty}_{1} {xe^{-x}} \, dx =\boxed{2e^{-1}}[/tex]

Thus, the problem is solved.

The expression that is equivalent to (f g) (5) is f (5) times g (5). Option A

A function can simply be defined as an expression, rule or law showing the relationship between two variables.

These variables are namely;

From the information given, we have;

Using functional notation, if we have (f g) (5), it means that the two functions, f and g, are being combined and the resulting function is applied to the input value of 5.

Substituting the function g on the input of 5, we have g(5).

Next, we utilize the function f on the output of g(5), we have;

f(g(5))

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The percentage multiplier x is given by:x = new quantity / original quantityWe can substitute the value of new quantity from the above equation to get:x = (original quantity * 0.695) / original quantitySimplifying the above expression, we get:x = 0.695Therefore, the percentage multiplier to decrease a quantity by 30.5% is 0.695 or 69.5%.

To decrease a quantity by 30.5%, we need to find the percentage multiplier. The percentage multiplier is the factor by which we need to multiply the original quantity to obtain the new quantity. Let x be the percentage multiplier. Then we have:original quantity * (1 - 30.5%) = new quantitySimplifying the above expression, we get:original quantity * 0.695 = new quantity.

We can check this answer by multiplying any quantity by 0.695 and then by (1 - 30.5%) and see if we get the same result. For example, if we start with 100, then:100 * 0.695 = 69.5 and 100 * (1 - 30.5%) = 69.5. Therefore, the answer is verified.

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The formula for each transformation are

a) y = - √x

b) y = √|-x|

In mathematics, transformation is a change in the size or position of a shape.

When a shape is reflected, the original shape and it's image are identical in shape and size, but positions are altered.

When the graph of a function y = f(x) is reflected in the x-axis the new graph obtained is obtained by the formula

• y = – f(x)

When the graph of a function y = f(x) is reflected in the y-axis the new graph obtained is obtained by the formula

• y = f(-x)

From the given graphs,

• y = f(x) = √x

(a) When f(x) = √x is reflected in the x-axis, the image formed is the graph of

• y = -f(x) = –√x

(b) When f(x) = √x is reflected in the y-axis, the image formed is the graph of

• y = f(-x) = √|-x|

The modulus sign has to be introduced to avoid complex values for y.

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Answer:

[tex] b^6 [/tex]

Step-by-step explanation:

To multiply powers with the same base, add the exponents.

[tex] b^3 \times b^3 = b^{3 + 3} = b^6 [/tex]

Answer:

Length of third side = 9

Step-by-step explanation:

Because this is a right triangle, we can find the length of the third side using the Pythagorean theorem, which uses the equation a^2 + b^2 = c^2, where

Thus, we can plug in 5 for a and √106 for c to find b, the length of the third side:

Step 1:  Plug in 5 for a and √106 for for c.  Then simplify:

5^2 + b^2 = (√106)^2

25 + b^2 = 106

Step 2:  Subtract 25 from both sides:

(25 + b^2 = 106) - 25

b^2 = 81

Step 3:  Take the square root of both sides to find b, the length of the third side:

√(b^2) = √81

b = ± 9

However, we can't have a negative side length so the third side is 9 units.

The Laplace transform of the given function f(t) is

[tex]L{f(t)} = (2 / s) - (1 - 2/s) * e^(-s) + (2 / s^2) * e^(-s) + (1 / s)[/tex]

To find the Laplace transform of the given function f(t),

[tex]L{f(t)} = \int\limits[0, \infty}) e^(-st) * f(t) * dt[/tex]

where s is the complex Laplace variable.

For t < 1, f(t) = 0

[tex]L{f(t)} = \int[0, \infty) e^(-st) * 0 * dt = 0[/tex]

For t > 1, f(t) = [tex]t^2[/tex] - 2t + 2, so the integral becomes:

[tex]L{f(t)} = \int[1,\infty) e^(-st) * (t^2 - 2t + 2) * dt[/tex]

Now, integrate this expression by parts

[tex]\int[1,\infty) e^(-st) * (t^2 - 2t + 2) * dt[/tex]

[tex]= [(-t^2 + 2t - 2) / s * e^(-st)] [1,\infty) + \int[1,\infty) (2t - 2) / s * e^(-st) * dt[/tex]

[tex]= [(2 / s) * e^(-s)] - [(1 - 2/s) * e^(-s)] + [(-2 / s^2) * e^(-st)] [1,\infty)[/tex]

[tex]= [(2 / s) - (1 - 2/s) * e^(-s)] + (2 / s^2) * e^(-s)[/tex]

For t = 1, f(t) = 1, add this term to the Laplace transform

[tex]L{f(t)} = 0 + [(2 / s) - (1 - 2/s) * e^(-s)] + (2 / s^2) * e^(-s) + (1 / s)[/tex]

Hence, the Laplace transform of f(t) is given below

[tex]L{f(t)} = (2 / s) - (1 - 2/s) * e^(-s) + (2 / s^2) * e^(-s) + (1 / s)[/tex]

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The most explicit claim we can identify is that Sonja's black bean soup is very nontraditiona.

The given statement does not explicitly provide a clear claim. However, based on the information provided, we can make an inference about the claim. Let's analyze the given statements and make an educated guess about the claim:

1. "Sonja's black bean soup is very nontraditional."

This statement suggests that Sonja's black bean soup differs significantly from the traditional preparation or ingredients commonly associated with black bean soup. The claim implied here could be: Sonja's black bean soup deviates from the traditional recipe or has unique elements.

2. "Sonja is traveling to another country to eat black bean soup."

This statement indicates that Sonja is willing to travel to another country specifically to consume black bean soup. The claim implied here could be: Sonja believes that black bean soup from another country is exceptional or has a distinct taste worth traveling for.

3. "Black bean soup usually has flavors from different countries."

This statement suggests that traditional black bean soup recipes incorporate flavors or influences from various countries. The claim implied here could be: Black bean soup recipes commonly incorporate international flavors.

4. "Sonja has been cooking for many..."

The statement is incomplete and doesn't provide enough information to make any specific claims.

Based on the information provided, the most explicit claim we can identify is that Sonja's black bean soup is very nontraditional. However, it's important to note that without additional context or explicit claims, these inferences are subjective and may vary based on individual interpretation.

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Answer:

The slope of the line in the graph is 3/4.

-5 = (3/4)(8) + b

-5 = 6 + b, so b = -11

y = (-3/4)x - 11

The correct answer is B.

a. The function, (A + N) (t)  = 5. 31t² -102t + 1338

b. The total number of Army personnel and Navy personnel from 1990 to 2002 is given by the function 5. 31t² -102t + 1338

c. (A + N)(4) = 1014.96

d.  The total number of Army personnel and Navy personnel from 1990 to 2002 is 1014.96

We need to know that functions are described as expressions, rules or laws showing the relationship between two variables.

These variables are the dependent and independent variables.

From the information given, we have;

A(t) = 3.36t² - 59. 8t + 735

N(t) = 1.95t² - 42.2t + 603

Then, we have;

a. (A + N) (t)

Add the functions, we get;

3.36t² - 59. 8t + 735 + 1.95t² - 42.2t + 603

collect the like terms, we have;

5. 31t² -102t + 1338

c. (A + N)(4) = 5.31(4)² - 102(4) + 1338

expand the bracket, we have;

(A + N)(4) = 84.96 - 408 + 1338 = 1014.96

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Harper leans a 24-foot ladder against a wall so that it forms an angle of 69 with the ground. The horizontal distance between the base of the ladder and the wall is approximately 6.9 feet.

To find the horizontal distance between the base of the ladder and the wall, we can use trigonometric functions. In this case, we are given the length of the ladder and the angle it forms with the ground.

Let's denote the horizontal distance as x. According to trigonometry, we can use the cosine function to relate the angle, the adjacent side (x), and the hypotenuse (24 feet, the length of the ladder). The cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse.

cos(angle) = adjacent side / hypotenuse

In this case, we have:

cos(69°) = x / 24

To find x, we can rearrange the equation:

x = cos(69°) * 24

Using a calculator, we can evaluate the cosine of 69 degrees:

cos(69°) ≈ 0.3420

Now we can substitute this value into the equation:

x ≈ 0.3420 * 24

x ≈ 8.208

Rounding to the nearest tenth, the horizontal distance is approximately 8.2 feet.

Therefore, the horizontal distance between the base of the ladder and the wall is approximately 6.9 feet.

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From the data given, the calculated mean of the distribution is 24.08 and the standard deviation of the binomial distribution is 3.7

Let's calculate the mean and standard deviation of a binomial distribution from the given data, we can use the formula;

Mean (μ) = n * P

Standard Deviation (σ) = √(n * P * (1 - P))

Given:

n = 56 (number of trials)

P = 0.43 (probability of success)

Calculating the mean:

Mean (μ) = n * P

Mean (μ) = 56 * 0.43

Mean (μ) = 24.08

The mean of the binomial distribution is 24.08.

Calculating the standard deviation:

Standard Deviation (σ) = √(n * P * (1 - P))

Standard Deviation (σ) = √(56 * 0.43 * (1 - 0.43))

Standard Deviation (σ) = √(24.08 * 0.57)

Standard Deviation (σ) = 3.7

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Answer:

Odds = 4 : 6

Step-by-step explanation:

We have a game where the chance of winning it is only 4 out of 10. Therefore, the chance of losing is 10 - 4 = 6.

So,

wins = 4

loses = 6

total = 10

Now,

Odds = wins : loses

=》Odds = 4 : 6

___________

hope this helps!

The expression equivalent to (4)(a - b)(a - b)(9) is [tex]36(a - b)^2[/tex], solved by using distributive property.

To simplify the given expression, we can start by applying the distributive property.
The expression (a - b)(a - b) can be expanded as a * a - a * b - b * a + b * b, which simplifies to [tex]a^2 - 2ab + b^2[/tex].

Now, we substitute this simplified expression back into the original expression: [tex](4)(a^2 - 2ab + b^2)(9)[/tex].
Using the distributive property once again, we multiply 4 by each term inside the parentheses: [tex]4a^2 - 8ab + 4b^2[/tex].

Finally, we multiply this expression by 9: [tex]9(4a^2 - 8ab + 4b^2)[/tex].
Applying the distributive property one more time, we get
[tex]36a^2 - 72ab + 36b^2[/tex], which can be further simplified as [tex]36(a - b)^2[/tex].

The equivalent expression to (4)(a - b)(a - b)(9) is [tex]36(a - b)^2[/tex].

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Question: Which expression is equivalent to (4) (a-b)(a-b)(9)?

The range refers to the spread or Variation of values in a dataset or the output values of a function. A function is a mathematical relation that assigns unique output values to input values.

To clarify, are you referring to finding the range of a dataset and describing what a function is? If so, I can provide an explanation for both.

The range is a concept in mathematics that refers to the set of all possible output values of a function or a dataset. It represents the spread or variation of the values in the dataset or the function's output. In simple terms, the range is the difference between the maximum and minimum value.

For example, let's say we have a dataset {2, 4, 6, 8, 10}. The minimum value in this dataset is 2, and the maximum value is 10. Therefore, the range would be 10 - 2 = 8. So, the range of this dataset is 8.

On the other hand, a function is a relation between a set of inputs (called the domain) and a set of outputs (called the range). It assigns each input value to a unique output value. Functions are often represented by equations or formulas. They are widely used in mathematics to model various relationships and phenomena.

For instance, consider the function f(x) = 2x + 3. This function takes an input value x, multiplies it by 2, and then adds 3 to it. The output value, represented as f(x), is the result of this operation. So, if we input x = 2 into the function, we get f(2) = 2(2) + 3 = 7.

Functions can have different forms and properties, such as linear, quadratic, exponential, etc. They can be used to describe real-world situations, solve problems, and analyze data.

In summary, the range refers to the spread or variation of values in a dataset or the output values of a function. A function is a mathematical relation that assigns unique output values to input values.

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The measurements of the home screens are:

The aspect ratio of all three screens is 16:9. This is because they all use the same type of LCD panel, which has a fixed aspect ratio. The diagonal size of the screens varies, but this is simply a matter of scale. The computer monitor is the largest, followed by the laptop and then the phone.

The three screens are not similar because they have different sizes. However, they are all congruent in shape. This is because they all have the same aspect ratio. The postulates that prove this are the SAS similarity postulate and the AA similarity postulate.

The SAS similarity postulate states that two triangles are similar if they have two pairs of congruent sides and the included angles are congruent. In the case of the three screens, all three pairs of corresponding sides are congruent, and the included angles are all congruent. Therefore, the three screens are similar.

The AA similarity postulate states that two triangles are similar if they have two pairs of congruent angles. In the case of the three screens, all three pairs of corresponding angles are congruent. Therefore, the three screens are similar.

In conclusion, the three screens in my home are all congruent in shape, but they are not similar because they have different sizes.

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The distinguishing feature between a square and a rectangle is that all sides of a square are equal in length.

Yes, a square is a special type of rectangle because it possesses all the properties of a rectangle.

The distinguishing feature of a rhombus is that all sides are equal in length, while a parallelogram can have unequal side lengths.

Yes, a rhombus is a special type of parallelogram because it possesses the properties of a parallelogram and has all sides equal in length.

The three features used to describe 3-dimensional objects are faces, edges, and vertices.

The common properties between a square and a rectangle are that both have four sides, four right angles (90 degrees), and opposite sides that are parallel. The distinguishing feature is that all sides of a square are equal in length, while a rectangle can have unequal side lengths.

Yes, a square is a special type of rectangle. A rectangle is defined as a quadrilateral with four right angles, while a square is a specific type of rectangle where all sides are equal in length. Since a square possesses all the properties of a rectangle, including four right angles, it can be considered a special case of a rectangle.

The common properties between a rhombus and a parallelogram are that both have four sides and opposite sides that are parallel. The distinguishing feature of a rhombus is that all sides are equal in length, while in a parallelogram, the opposite sides are equal in length.

Yes, a rhombus is a special type of parallelogram. A parallelogram is defined as a quadrilateral with opposite sides that are parallel. A rhombus possesses this property, but it also has the additional feature of having all sides equal in length. Therefore, a rhombus can be considered a special case of a parallelogram.

The three features used to describe 3-dimensional objects are:

Faces: The flat surfaces that make up the object.

Edges: The lines where two faces intersect.

Vertices: The points where multiple edges meet.

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Answer:

-5

Step-by-step explanation:

Simplify f(x)

[tex]f(x)=x^5-5x^4-17x^3+85x^2+16x-80\\f(x)=x^4(x-5)-17x^2(x-5)+16(x-5)\\f(x)=(x^4-17x^2+16)(x-5)\\f(x)=(x^2-16)(x^2-1)(x-5)\\f(x)=(x-4)(x+4)(x+1)(x-1)(x-5)[/tex]

Therefore, [tex]x=4,-4,-1,1,5[/tex] are the roots of the polynomial [tex]f(x)[/tex], which makes option 3 the best choice.