5. - 1 5/7 divided by 1/2

The value of k such that y = k cos(3x) + 4sin(x) is a solution to the differential equation y’’ + y = -9 cos(2x) is 9/8.

Given a differential equation,

y’’ + y = -9 cos(2x)

The solution of the equation is,

y = k cos(3x) + 4sin(x)

Now,

y' = -3k sin (3x) + 4 cos(x)

y'' = -9k cos (3x) - 4 sin (x)

Substituting these to the given equation,

-9k cos (3x) - 4 sin (x) + k cos(3x) + 4sin(x) = -9 cos(2x)

-8k cos (3x) = -9 cos (3x).

Comparing,

-8k = -9

k = 9/8.

Hence the value of k is 9/8.

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Population size is a characteristic of the data set and is not used to analyze the data. The correct option is A

The quantity of an organisms belonging to a specific species is referred to as its population size.

Population size is a characteristic of the data collection that is therefore ignored when data analysis is performed. The population under investigation's size is merely described in terms of the number of individuals or data points.

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Answer:

Step-by-step explanation:

The true statements are B, D, E, and H for the given margin of error of 0.017.

The margin of error shows the range of values within which the real value of a population parameter is expected to reside, based on the findings of a sampling of that population.

To find the margin of error in terms of percentage, we need to divide the margin of error by the sample proportion and multiply by 100.

The sample proportion is 284/500 = 0.568, so the margin of error in terms of percentage is (0.017/0.568) × 100 ≈ 2.99%.

Therefore, the true statements are:

B. The margin of error is between 55.1% and 58.5%.

D. The margin of error is between 17,991 and 19,101 subscribers.

E. The results show 56% of the subscribers also subscribe to at least one other newspaper online.

H. It cannot be concluded that over half of the subscribers also subscribe to at least one other newspaper online.

Statement A is incorrect because the margin of error is less than 3%, which is outside the range of 54% to 58%.

Statement C is incorrect because the margin of error is less than 3%, which is outside the range of 17,632 to 18,938 subscribers.

Statement F is incorrect because the sample proportion is 0.568, which is not equal to 56.8%.

Statement G is incorrect because we cannot conclude that over half of the subscribers also subscribe to at least one other newspaper online due to the margin of error.

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Step-by-step explanation:

circular permutations of n objects =(n-1)!

The slope of a linear function represents the rate at which the output variable (y) changes with respect to the input variable (x). The slope is often denoted by the letter "m" and can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are any two points on the line.

To find the slope of Function 1, we can compare the coefficient of x in its equation with the formula for slope. We see that the coefficient of x in y = (4/5)x + 2 is 4/5. Therefore, the slope of Function 1 is 4/5.

To find the slope of Function 2, we can choose any two points from the table and use the slope formula. Let's choose the points (0, 1) and (3, 2.5). Plugging in these values, we get:

m = (2.5 - 1) / (3 - 0) = 1.5 / 3 = 1/2

Therefore, the slope of Function 2 is 1/2.

Comparing the slopes, we can see that the slope of Function 1 (4/5) is greater than the slope of Function 2 (1/2). Therefore, Function 1 has a greater slope than Function 2.

Answer:

Function 1 has the greatest slope.

Step-by-step explanation:

Function 1 is given in slope-intercept form, y = mx + b, where m is the slope (and b is the y-intercept).

Therefore, the slope of function 1 is ⁴/₅.

To find the slope of function 2, use the slope formula.

[tex]\boxed{\begin{minipage}{8cm}\underline{Slope Formula}\\\\Slope $(m)=\dfrac{y_2-y_1}{x_2-x_1}$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.\\\end{minipage}}[/tex]

Let (x₁, y₁) = (0, 1)

Let (x₂, y₂) = (2, 2)

Substitute the values into the formula:

[tex]\implies m=\dfrac{2-1}{2-0}=\dfrac{1}{2}[/tex]

Therefore, the slope of function 2 is ¹/₂.

To determine which function has the greatest slope, rewrite both slopes so that the denominator of the fractions are the same.

[tex]\textsf{Slope of function 1}=\dfrac{4}{5}=\dfrac{4 \cdot 2}{5 \cdot 2}=\dfrac{8}{10}[/tex]

[tex]\textsf{Slope of function 2}=\dfrac{1}{2}=\dfrac{1 \cdot 5}{2 \cdot 5}=\dfrac{5}{10}[/tex]

As 8 is greater than 5, the slope of function 1 is greater than the slope of function 2.

Answer:

The height of the squirrel is 31 feet.

Step-by-step explanation:

You need to know your Right Triangle Trigonometry to do this problem.

Rt Triangle trig is all about ratios. Angles and ratios.

In your question, there is a rt triangle. The side measure given, 28 is next to the angle. The math word for "next to" is "adjacent" You know the adjacent side. The squirrel's height is the opposite side. The ratio that puts together adjacent and opposite is tangent.

tan Angle = opposite/adjacent

tan 48° = x/28

multiply both sides by 28

28•tan48° = x



You have to use a calculator that has trig functions. It will have buttons that say "sin", "cos", and "tan"



Enter 28 × tan48° =

It will return 31.0971504152

Your question asks you to round to the nearest whole.

x = 31

The squirrel's height in the tree is 31ft.

Answer:

To solve the system of linear equations:

4x + 3y = 18

4x - 2y = 18

we can use the method of elimination or substitution.

Method 1: Elimination

In this method, we eliminate one of the variables by adding or subtracting the equations in a way that eliminates one of the variables. Here's how we can do it:

Multiply the first equation by 2, and the second equation by 3 to eliminate x:

8x + 6y = 36

12x - 6y = 54

Now, subtract the second equation from the first:

8x + 6y - (12x - 6y) = 36 - 54

8x + 6y - 12x + 6y = -18

-4x + 12y = -18

Divide both sides by -4 to isolate x:

-4x/(-4) + 12y/(-4) = -18/(-4)

x - 3y = 4.5

Now, we can substitute this value of x into one of the original equations, let's use the first equation:

4(4.5) + 3y = 18

18 + 3y = 18

3y = 18 - 18

3y = 0

y = 0

So, the solution to the system of equations is x = 4.5 and y = 0.

Method 2: Substitution

In this method, we solve one of the equations for one variable and then substitute that expression into the other equation to solve for the other variable. Here's how we can do it:

Solve the first equation for x:

4x + 3y = 18

4x = 18 - 3y

x = (18 - 3y)/4

Now, substitute this expression for x into the second equation:

4[(18 - 3y)/4] - 2y = 18

18 - 3y - 2y = 18

-5y = 18 - 18

-5y = 0

y = 0

Now, substitute this value of y back into the expression for x:

x = (18 - 3(0))/4

x = 18/4

x = 4.5

So, the solution to the system of equations is x = 4.5 and y = 0, which is consistent with the solution obtained using the elimination method.

Answer:

(4.5,0)

Step-by-step explanation:

The point that makes this system of equations true is (4.5,0.

When x=4.5, and y=0 both of these equations equal each other.

You can find this point either by using your calculator, or graphing the equations and seeing where they intersect.

Answer:

30 meters after 0.5 seconds

Step-by-step explanation:

To find the values of t for which the rocket's height is 30 meters, we can set h = 30 in the given equation and solve for t:

h = 65t - 5t

30 = 65t - 5t

30 = 60t

t = 30/60

t = 0.5

Therefore, the rocket's height is 30 meters after 0.5 seconds.

A dilation can be defined as a transformation that multiplies the distance between every point in an object and a fixed point, called the center of dilation, by a constant factor called the scale factor.

The lengths of the corresponding segments are given as follows:

Hence the scale factor is given as follows:

k = 4/2

k = 2.

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a.

x + y = 28

x - y = 14

b.

Add the two equations:

2x = 42

x = 21

Substitute back into either original equation:

y = 28 - 21 = 7

Therefore, the two numbers Tyra picked are x = 21 and y = 7.

Answer:

Correct question:-

Prove that tan9.tan17.tan45.tan73.tan81=1

LHS

Step-by-step explanation:

Answer:

wouldn't it be 5/16

Step-by-step explanation:

honestly not sure but isn't this just a probability question?

The required it would take John 26.3 minutes to run 5 kilometers.

Since there are 1.60934 kilometers in a mile, we can first convert 2 miles to kilometers:

2 miles = 2 * 1.60934 kilometers = 3.21869 kilometers

Now we can use the formula for speed:
speed = distance/time.

time = distance/speed

We know the distance (5 kilometers), and we can find the speed by dividing the distance by the time it takes to run 2 miles:

speed = 3.21869 kilometers / 18 minutes

speed ≈ 0.19 kilometers per minute

Now we can use this speed to find the time it would take to run 5 kilometers:

time = 5 kilometers / 0.19 kilometers per minute

time ≈ 26.3 minutes

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hola te hice tres los otros no tuve tiempo espero te sirvan de ayuda :)

If the distance between Anaheim and Sacramento in real life is 420 miles, then 1 cm on map represents 21 miles, so, Greg's scale is correct.

To determine what 1 cm on the map represents in miles, we calculate the scale-factor, which tells us the ratio of distance on the map to distance in the real world.

The scale factor is = (distance on map)/(distance in real world),

The distance between, "Anaheim" and "Sacramento" in real life is = 420 miles, and on map is 20cm,

So, Scale factor is = 420/20 = 21 miles,

So, we can say that 1 cm on the map represents 21 miles in real-life.

Now, we can compare the scale given by Greg and Grace to the actual scale,

⇒ Greg's scale is 1 cm = 21 miles, which is the same as the actual scale we just calculated. So, Greg is correct.

⇒ Grace's scale is 1 cm = 19 miles, which is not the same as the actual scale we just calculated. So, Grace is incorrect.

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The given question is incomplete, the complete question is

The distance between Anaheim and Sacramento is 420 miles. A map shows the distance to be 20 cm. What does 1 cm on the map represent in miles? Greg says that the scale is 1 cm = 21 miles. Grace says that the scale is 1 cm = 19 miles.​ Who is correct.

If the distance between Anaheim and Sacramento in real life is 420 miles, then 1 cm on map represents 21 miles, so, Greg's scale is correct.

To determine what 1 cm on the map represents in miles, we calculate the scale-factor, which tells us the ratio of distance on the map to distance in the real world.

The scale factor is = (distance on map)/(distance in real world),

The distance between, "Anaheim" and "Sacramento" in real life is = 420 miles, and on map is 20cm,

So, Scale factor is = 420/20 = 21 miles,

So, we can say that 1 cm on the map represents 21 miles in real-life.

Now, we can compare the scale given by Greg and Grace to the actual scale,

⇒ Greg's scale is 1 cm = 21 miles, which is the same as the actual scale we just calculated. So, Greg is correct.

⇒ Grace's scale is 1 cm = 19 miles, which is not the same as the actual scale we just calculated. So, Grace is incorrect.

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The given question is incomplete, the complete question is

The distance between Anaheim and Sacramento is 420 miles. A map shows the distance to be 20 cm. What does 1 cm on the map represent in miles? Greg says that the scale is 1 cm = 21 miles. Grace says that the scale is 1 cm = 19 miles.​ Who is correct.

An expression for the length of the hypotenuse z is [tex]\sqrt{x^2 + y^2}[/tex] = z.

What is Pythagoras' theorem?

A fundamental relationship in Euclidean geometry between a right triangle's three sides is known as the Pythagorean theorem or Pythagoras' theorem. According to this rule, the area of the square with the hypotenuse side is equal to the sum of the areas of the squares with the other two sides.

Here, we have

Given: A right triangle with legs of lengths x and y has a hypotenuse of length z.

We have to write an expression for the length of the hypotenuse z.

By Pythagoras' theorem

x²+y² = z²

[tex]\sqrt{x^2 + y^2}[/tex] = z

Hence, an expression for the length of the hypotenuse z is [tex]\sqrt{x^2 + y^2}[/tex] = z.

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If each year the population grows by 2.5%, then the "exponential-function" which shows the relationship between "y" and "t" is y = 300,000 × [tex](1.025)^t[/tex].

The "Exponential-Growth" is defined as a type of growth where the rate at which something grows is proportional to its current value. This results in a rapid and increasingly faster growth over time.

The relationship between "y" (population) and "t" (time in years) can be modeled by an "exponential-function" ;

⇒ y = a × [tex](1+r)^{t}[/tex],

where "a" is = initial population, "r" is = annual growth-rate (in decimal), and "t" is = time;

In this case, the "initial-population" is = 300,000, and

The annual growth rate is = 2.5% or 0.025,

So, we can write exponential function as : y = 300,000 × [tex](1+0.025)^{t}[/tex],

Simplifying the expression,

We get,

⇒ y = 300,000 × [tex](1.025)^t[/tex],

Therefore, the required exponential-function is y = 300,000 × [tex](1.025)^t[/tex].

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The expression "log₃(3/8)" can be expressed in the form of "difference-of-logarithms" is log₃(3) - log₃(8), and it's simplified value is -0.89278.

The "Quotient-Property" states that logarithm of quotient of "two-numbers" is equal to the difference of logarithms of individual numbers. It is expressed as : logₐ(b/c) = logₐ(b) - logₐ(c);

Using the quotient property, we can write log₃(3/8) as:

⇒ log₃(3) - log₃(8)

We know that log₃(3) =   and  log₃(8) =   ,

So, ⇒ log₃(3) - log₃(8),

⇒ 1 - 1.89278

⇒ -0.89278,

Therefore, the value of log₃(3/8) is -0.89278.

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The given question is incomplete, the complete question is

Use the Quotient Property of Logarithms to write the logarithm as a difference of logarithms, and simplify if possible: log₃(3/8).

You would save a total of $272 in one week if you continued along the same pattern.

What is pattern?

A pattern is a repeating or predictable sequence of events, numbers, shapes, or objects. In the context of the question you asked earlier, the pattern is the sequence of the amounts saved on each day, which follows a predictable rule: the amount saved doubles each day, starting with $2 on day 1. So the pattern is: $2, $22, $2³, $2⁴, $2⁵, $2⁶, $2⁷. Recognizing and understanding patterns is an important skill in many areas of life, from mathematics and science to language and music.

Based on the given pattern, on day 4, the amount saved would be $2⁴ = $16. On day 5, the amount saved would be $2⁵ = $32. On day 6, the amount saved would be $2⁶ = $64. And on day 7, the amount saved would be $2⁷ = $128.

To calculate the total amount saved for the week, we simply add up the amounts saved on each day:

Total amount saved = $2 + $22 + $2³ + $16 + $32 + $64 + $128

= $2 + $22 + $8 + $16 + $32 + $64 + $128

= $272

Therefore, you would save a total of $272 in one week if you continued along the same pattern.

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Answer:  a

Step-by-step explanation:

Theres 5 times that we could get at least 3 heads and were tossing the coin 4 times so 5/4. 5 divided 4 = 1.25 x100 =12.5 percent

The probability of P(at least 3 heads) is 31.25%

Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event.

From the information given, we have that;

{HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT}.

Count the number of events with 3 H's

Now, to determine the P(at least 3 heads).

We get;

P(at least 3 heads) = 5/16 × 100/1

Divide the values, we get;

P(at least 3 heads) = 0.3125×100/1

Multiply the values we get;

P(at least 3 heads) = 31.25%

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The coordinates of point N that divides the line segment formed joining points M(-6,-3) and P(26,13) in the ratio 3:5, evaluated using section-formula is (14,7)

What is section-formula?

The coordinates of the point that splits a specific line segment into two halves are given by the section formula. The line may be split at the point either internally or externally. Their lengths are divided in a way that keeps them in the m: n ratio.  The ratio in which the line segment is divided and the coordinates of the points connecting it allow us to calculate the coordinates.

According to section-formula:

(x,y) = ( [tex]\frac{c.m + a. n}{m + n}[/tex] , [tex]\frac{d.m + b. n}{m + n}[/tex])    {where (x,y)are coordinates of point which divides & (a,b) ; (c,d) are coordinates of points forming line segment.

Given P=(a,b) = (26,13)

M=(c,d) = (-6.-3)

m:n = 3 : 5

N=(x,y) = ?

N=(x,y) = ( [tex]\frac{c.m + a. n}{m + n}[/tex] , [tex]\frac{d.m + b. n}{m + n}[/tex])

             = {[tex]\frac{(-6)(3)+(26)(5)}{3+5}[/tex] ; [tex]\frac{(-3)(3)+(13)(5)}{3+5}[/tex] }

             ={ [tex]\frac{-18+130}{8}[/tex] ; [tex]\frac{-9+65}{8}[/tex] }

             = {[tex]\frac{112}{8}[/tex]  , [tex]\frac{56}{8}[/tex]  }

             ={14 , 7}

The coordinates of point N are (14,7) which divides line joining P(26,13) and M(-6,-3) in ratio 3:5.

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Answer:

1. 4 tens 2 ones

2. 6 tens 5 ones

3. 3 tens 7 ones

4. 2 tens 6 ones

5. 7 tens 4 ones

6. 5 tens 9 ones

7. 8 tens 1 ones

8. 9 tens 9 ones

9. 1 tens 3 ones

Step-by-step explanation:

not sure if I did it right, lmk if im wrong. Think ur just suppsoed to count the number of boxes that make 10, and those that dont count up to 10, depending on the number, are seen as "ones".

The calculated value of the perimeter of △JKL is 88 units

Assuming that segments that appear to be tangent are tangent, we have

7y - 9 = 2y + 11

8x - 35 = 5x - 8

When the expressions are evaluated, we have

y = 4

x = 9

So, we have the following side lengths

OK = 2(4) + 11

OK = 19

Also, we have

JM = 5(4) - 8

JM = 12

Lastly, we have

LO = 32 - 19

LO = 13

The perimeter of △JKL is then calculated as

Perimeter = 2 * (OK + JM + LO)

Substitute the known values in the above equation, so, we have the following representation

Perimeter = 2 * (19 + 12 + 13)

Evaluate

Perimeter = 88

Hence, the perimeter is 88 units

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Answer:

y = 2x + 12.

Step-by-step explanation:

To find the equation of a line perpendicular to y=-1/2x+4 and passing through the point (-2,8), we can first determine the slope of the perpendicular line.

Recall that two lines are perpendicular if and only if their slopes are negative reciprocals of each other. Therefore, the slope of the line we want to find is the negative reciprocal of the slope of y=-1/2x+4.

The slope of y=-1/2x+4 is -1/2, so the slope of the line perpendicular to it is 2 (since the negative reciprocal of -1/2 is 2).

Next, we can use the point-slope form of the equation of a line to write the equation of the line perpendicular to y=-1/2x+4 that passes through the point (-2,8):

y - 8 = 2(x + 2)

Simplifying and putting the equation in slope-intercept form, we get:

y = 2x + 12

Therefore, the equation of the line perpendicular to y=-1/2x+4 that passes through the point (-2,8) is y = 2x + 12.

[tex]\sf y =2x+12.[/tex]

The slope of any linear equation can be found just by taking a look at the equation when it's solved for "y".

[tex]\sf y=-\dfrac{1}{2} x+4[/tex]

Looking at the given equation, we can clearly tell that the value of slope is [tex]-\dfrac{1}{2}[/tex].

The slope of any linear equation that is perpendicular to another can be found by writting the multiplicative reciprocal of the original equation's slope, and changing the sign on it.

Let's do it step by step:

a) Write the slope of the original equation.

[tex]-\dfrac{1}{2}[/tex]

b) Write the multiplicative reciprocal.

For this, you just need to change the order in the fraction. In other words, switch places between the numerator and denominator.

[tex]-\dfrac{1}{2}\Longrightarrow-\dfrac{2}{1}=-2[/tex]

[tex]\sf -2\Longrightarrow2[/tex]

Therefore, the slope of the new equation will be 2, and it is perpendicular to the original equation ([tex]\sf y=-\dfrac{1}{2} x+4[/tex]).

With the given ordered pair (-2, 8) and the slope (2) we can calculate the formula of the new equation.

Formula to use: [tex]\sf y-y_{1} =m(x-x_{1} )[/tex]

[tex]\sf x_{1} =-2\\ \\\sf y_{1} =8\\ \\m=2[/tex]

Now we substitute the variables in the equation by the identified values in step 3.

[tex]\sf y-(8) =(2)(x-(-2))\\ \\y-8 =(2)(x+2)\\ \\[/tex]

Use the distributive property of multiplication on the right side of the equation (check the attached image).

[tex]\sf y-8 =(2)(x)+(2)(2)\\\\y-8 =2x+4\\ \\y-8+8 =2x+4+8\\ \\y =2x+12[/tex]

a) Is it perpendicular?

According to the theory explained in step 2, it is, because the slope is 2.

b) Does it pass through point (-2, 8)?.

For this, simply substitute "x" by "-2" in the calculated equation. If y= 8, then the function also meets this requirement.

[tex]\sf y =2(-2)+12\\ \\y=-4+12\\ \\y=8[/tex]

That's correct. We have found the correct answer.

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Answer:

1) c

2) d

3) a

4) b

Step-by-step explanation:

1)

[tex] \frac{13400}{13400 + 52100} = 20.5\%[/tex]

2)

[tex] \frac{9200}{9200 + 25800} = 26.3\% [/tex]

3)

[tex] \frac{52100 + 25800}{52100 + 25800 + 13400 + 9200} = 77.5\%[/tex]

4)

[tex] \frac{25800 + 9200}{25800 + 9200 + 52100 + 13400} = 34.8\%[/tex]

Answer:

1) c

2) d

3) a

4) b

Step-by-step explanation:

1)

[tex] \frac{13400}{13400 + 52100} = 20.5\%[/tex]

2)

[tex] \frac{9200}{9200 + 25800} = 26.3\% [/tex]

3)

[tex] \frac{52100 + 25800}{52100 + 25800 + 13400 + 9200} = 77.5\%[/tex]

4)

[tex] \frac{25800 + 9200}{25800 + 9200 + 52100 + 13400} = 34.8\%[/tex]

The following steps we have to follow to make the graph of the given function:

[tex]f(x) = x^2 + 6x + 4[/tex]

Start with the graph of [tex]y = x^2[/tex], which is a parabola that opens upwards and passes through the origin (0, 0).

To shift the parabola to the left by 3 units (since 6/2 = 3), we subtract 3 from x inside the function. This gives us

[tex]y = (x - 3)^2[/tex],

which is the same parabola shifted 3 units to the right.

To shift the parabola up by 4 units, we add 4 to the function. This gives us f(x) = (x - 3)^2 + 4, which is the final function we want to graph.

To graph f(x), we can plot a few points and sketch the resulting parabola. For example, when x = -2, we have

[tex]f(-2) = ( -2 - 3)^2 + 4 = 9[/tex],

so one point on the graph is (-2, 9). Similarly,

when x = -1, we have

[tex]f(-1) = (-1 - 3)^2 + 4 = 8[/tex], so another point on the graph is (-1, 8). We can continue this process to get more points and sketch the parabola.

Here is a rough sketch of the graph of [tex]f(x) = x^2 + 6x + 4:[/tex]

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The following steps we have to follow to make the graph of the given function:

[tex]f(x) = x^2 + 6x + 4[/tex]

Start with the graph of [tex]y = x^2[/tex], which is a parabola that opens upwards and passes through the origin (0, 0).

To shift the parabola to the left by 3 units (since 6/2 = 3), we subtract 3 from x inside the function. This gives us

[tex]y = (x - 3)^2[/tex],

which is the same parabola shifted 3 units to the right.

To shift the parabola up by 4 units, we add 4 to the function. This gives us f(x) = (x - 3)^2 + 4, which is the final function we want to graph.

To graph f(x), we can plot a few points and sketch the resulting parabola. For example, when x = -2, we have

[tex]f(-2) = ( -2 - 3)^2 + 4 = 9[/tex],

so one point on the graph is (-2, 9). Similarly,

when x = -1, we have

[tex]f(-1) = (-1 - 3)^2 + 4 = 8[/tex], so another point on the graph is (-1, 8). We can continue this process to get more points and sketch the parabola.

Here is a rough sketch of the graph of [tex]f(x) = x^2 + 6x + 4:[/tex]

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The expression -3x^4 + 9x² + 6 is a trinomial because it has three terms: -3x^4, 9x^2, and 6.

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

-3x^4 + 9x² + 6

A polynomial is an expression consisting of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and exponentiation.

In this expression, we have three terms:

This means that it is a trinomial

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The expression -3x^4 + 9x² + 6 is a trinomial because it has three terms: -3x^4, 9x^2, and 6.

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

-3x^4 + 9x² + 6

A polynomial is an expression consisting of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and exponentiation.

In this expression, we have three terms:

This means that it is a trinomial

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The total personnel costs of the cleanup would be $8,625.

Between 9AM and 5PM, a total of 8 hours, the workers are present during the interval t=0 to t=8. During this interval, the number of workers is given by:

f(t) = 30 + 10sin((2π/16)t)

We can calculate the number of workers for each hour using this formula:

f(0) = 30 + 10sin(0) = 30

f(1) = 30 + 10sin(π/8) ≈ 38.66

f(2) = 30 + 10sin(π/4) = 40

f(3) = 30 + 10sin(3π/8) ≈ 38.66

f(4) = 30 + 10sin(π/2) = 40

f(5) = 30 + 10sin(5π/8) ≈ 38.66

f(6) = 30 + 10sin(3π/4) = 30

f(7) = 30 + 10sin(7π/8) ≈ 21.34

f(8) = 30 + 10sin(π) = 20

So, during the first 8 hours, the number of workers fluctuates between 20 and 40.

Between 5PM and 9AM the next day, a total of 16 hours, the workers are present during the intervals t=8 to t=12, t=20 to t=36, and t=44 to t=48. During these intervals, the number of workers is given by:

Between t=8 and t=12, the number of workers is decreasing from 40 to 20. We can approximate this by a linear function:

f(t) = 40 - 5t

Between t=20 and t=36, the number of workers is constant at 20.

Between t=44 and t=48, the number of workers is increasing from 20 to 40. We can use the same linear function as before, but with t shifted by 44:

f(t) = 40 - 5(t-44)

We can now calculate the total personnel costs by multiplying the number of workers at each time interval by the appropriate hourly rate:

From 9AM to 5PM (8 hours): 30 workers on average, at a rate of $15 per hour:

8 × 30 × 15 = $3,600

From 5PM to 9AM the next day (16 hours):

From t=8 to t=12 (4 hours): average of 30 workers, at a rate of $22.5 per hour:

4 × 30 × 22.5 = $2,700

From t=20 to t=36 (16 hours): constant 20 workers, at a rate of $22.5 per hour:

16 × 20 × 22.5 = $7,200

From t=44 to t=48 (4 hours): average of 30 workers, at a rate of $22.5 per hour:

4 × 30 × 22.5 = $2,700

The total cost is the sum of the costs for each time interval:

$3,600 + $2,700 + $7,200 + $2,700 = $16,200

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