Find the average value of f(x) = √81 -x² over the interval [0, 9].

Point K is located at 3.5 on the number line.

To find point K on the number line, we need to apply the ratio LK to KM, which is 6:3. First, we can simplify this ratio to 2:1.

Next, we will find the total distance between points L and M. The distance LM = 6.5 - (-2.5) = 9 units.

Now, we can divide this distance into 3 equal parts, as per the simplified ratio 2:1. Each part will be 9/3 = 3 units long.

Since LK is 2 parts of the ratio, we will multiply the length of one part (3 units) by 2. LK = 3 * 2 = 6 units.

Finally, we'll find the coordinates of point K by adding the length of LK to the coordinate of point L. So, K = -2.5 + 6 = 3.5.

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Answer:

Point K is at 3.5

Step-by-step explanation:

I just took the test

The derivative of the function is solved and f' ( 1 ) = 0.183

Given data ,

Let the function be represented as f ( x )

Now , the value of f ( x ) is

f ( x ) = sinh⁻¹ ( √ (2x + 3 )

On taking the derivative of the function , we get

f' ( x ) = 1/ ( √2x + 4 ) ( √ ( 2x + 3 ) )

On simplifying , we get

when x = 1

f ( 1 ) = 1 / ( √6 ) ( √5 )

f ( 1 ) = 1/√30

f ( 1 ) = 0.18257

f ( 1 ) = 0.183

Hence , the function is solved

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The area of the shaded region, to the nearest hundredth can be calculated as approximately: 1,195.71 mm².

The area of the shaded region can be found using the area of circle, which is: Area = πr²

Therefore, area of the shaded region = area of larger circle - area of smaller circle.

Area of larger circle:

radius (r) = 14 + 6.6 = 20.6 mm

Area of larger circle = 3.14 * 20.6² = 1,332.4904 mm²

Area of smaller circle:

radius (r) = 6.6 mm

Area of larger circle = 3.14 * 6.6² = 136.7784 mm²

Area of the shaded region = 1,332.4904 - 136.7784 = 1,195.71 mm².

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Triangle- m ∠ 2 m ∠ 3 m ∠ 5 =  180 °( by the Triangle Sum Theorem)          

The statements are true. still, grounded on the given options, three statements that are always true regarding any triangle are . The sum of the angles of a triangle is always equal to 180 degrees.

This statement is true because the total degrees of any triangle is 180 degrees.

This is known as the Triangle Sum Theorem.

2. The  surface angle of a triangle is equal to the sum of its interior  contrary angles.  This statement is true for any triangle. The  surface angle is the angle formed by extending one side of the triangle. The  contrary innards angles are the two angles inside the triangle that don't partake a vertex with the  surface angle.  

3. The measure of each angle of an equilateral triangle is 60 degrees.  This statement is true for any equilateral triangle. An equilateral triangle has three equal sides and three equal angles, each measuring 60 degrees.  thus, grounded on the given options,

the three statements that are always true regarding the  illustration could be  

- m ∠ 3 m ∠ 4 m ∠ 5 =  180 °( by the Triangle Sum Theorem)

- m ∠ 5 m ∠ 6 =  180 °( by the sum of the angles of a straight line)

- m ∠ 2 m ∠ 3 m ∠ 5 =  180 °( by the Triangle Sum Theorem)

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The correct statements that apply to the given limits are:

Option A: Multiply by (√(x + 2) + 3)/(√(x + 2) + 3)

Option D: "Divide out a common factor of x - 7."

Option E: The limit of the given expression is 1/6.

Option: We generally multiply by the conjugate of the expression to eliminate square roots in simplification. Thus, option A applies.

Option B: Get x - 1 in the numerator:

This statement is not correct. The expression does not involve x - 1 in the numerator.

Option C: Get (x - 7)(√(x + 2) - 3) in the denominator:

This statement is not correct due to the fact that we want to simplify the expression, and not introduce a more complex denominator.

Option D: Divide out a common factor of x - 7:

This statement is correct because by factoring out (x - 7) from both the numerator and denominator, we can cancel out the common factor.

Option E: Calculate the limit as 1/6:

After canceling out the common factor of x - 7, the simplified expression becomes:

lim x -> 7 (√(x + 2) - 3)/(x - 7)

= lim x -> 7 1/(√(x + 2) + 3)

To find the limit, we substitute the value x = 7 into the simplified expression:

lim x -> 7 1/(√(7 + 2) + 3) = 1/(√(9) + 3) = 1/(3 + 3) = 1/6

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Check the picture below.

[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ a^2+o^2=c^2\implies a=\sqrt{c^2 - o^2} \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{40}\\ a=\stackrel{adjacent}{ST}\\ o=\stackrel{opposite}{24} \end{cases} \\\\\\ ST=\sqrt{ 40^2 - 24^2}\implies ST=\sqrt{ 1600 - 576 } \implies ST=\sqrt{ 1024 }\implies ST=32 \\\\[-0.35em] ~\dotfill\\\\ \tan(S )=\cfrac{\stackrel{opposite}{24}}{\underset{adjacent}{32}} \implies \tan(S )=\cfrac{3}{4}[/tex]

The equivalent equations of 4(2-5x)=6-3(1-3x)​ are 4(2 - 5x) = 6 - 3(1 - 3x), 8 - 20x = 6 - 3 + 9x and 29x = 5

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

4(2-5x)=6-3(1-3x)​

Express properly

So, we have

4(2 - 5x) = 6 - 3(1 - 3x)

Open the brackets

So, we have

8 - 20x = 6 - 3 + 9x

Evaluate the like terms

This gives

29x = 5

This means that the equivalent equations are 4(2 - 5x) = 6 - 3(1 - 3x), 8 - 20x = 6 - 3 + 9x and 29x = 5

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1: The required table is described below.

2: The range is (0,36)

3: X-intercepts are (-6,0) and (6,0) and the Y-intercept is (0,36)

A line has length but no width, making it a one-dimensional figure. A line is made up of a collection of points that can be stretched indefinitely in opposing directions. Two points in a two-dimensional plane determine it.

We are given that,

y = -x² + 36

The equation of the rainbow is represented by the parabola,

Now, we are required to find a linear equation which cuts the graph of the parabola at two points.

Let us consider the equation joining the points (-6,0) and (0,36), given by .

y = 6x + 36

So, the corresponding table for the linear equation is given by,

x            y = 6x + 36

-6                      0

0                       36

1                        42

6                       72

Now, we will answer the questions corresponding the functions.

1. Domain and Range of the rainbow.

Since, the equation of the rainbow is

y = -x² + 36

So, from the figure, we get that,

Domain is the set of all real numbers.

Range is the set {y | y ≤ 36}

Here, domain represents the points which are used to plot the path of the rainbow and range represents the points which are form the rainbow.

Not all points make sense in the range as the parabola is opening downwards having maximum point as (0,36).

2. X and Y-intercepts of the rainbow.

As, the 'x and y-intercepts are the points where the graph of the function cuts x-axis and y-axis respectively i.e. where y=0 and x=0 receptively'.

We see that from the figure below,

X-intercepts are (-6,0) and (6,0) and the Y-intercept is (0,36)

Here, these intercepts represents the point where the parabola intersects the individual axis.

3. Is the linear function positive or negative.

As the linear function is  represented by the upward flight of the drone.

So, the linear function is a positive function.

y = 6x + 36

4. The solution of the system of equations is the intersection points of their graphs.

So, from the figure, we see that the equations intersect at the points (-6,0) and (0,36).

Thus, the solution represents the position when both the drone and rainbow intersect each other.

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Answer:

To find the z-score of a data value, we use the following formula:

z = (x - μ) / σ

where z is the z-score, x is the data value, μ is the mean of the distribution, and σ is the standard deviation.

Given a normal distribution with a mean (μ) of 146 and a standard deviation (σ) of 4, and a data value (x) of 153, we can plug these values into the formula:

z = (153 - 146) / 4

z = 7 / 4

z = 1.75

Thus, the z-score of the data value 153 is 1.75.

The value of the composite function (f . g)(1) when evaluated is 6

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

The graph of the function f(x)

Also, we have

The graph of the function g(x)

The composite function (f . g)(1) is calculated as

(f . g)(1) = f(1) * g(1)

From the graph, we have the following values

f(1) = -2

Also, we have

g(1) = 3

So, we have

(f . g)(1) = 2 * 3

Evaluate

(f . g)(1) = 6

Hence, the composite function (f . g)(1) when evaluated is 6

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Using a calculator, the trigonometric ratio of sin 98 is approximately 0.9848.

One of the trigonometric ratio we have in mathematics is the sine ratio. We can use calculator to find the sine of an angle without making use of tables. To do this on your calculator, enter the sine function followed by the degree of the angle you want to find its sine. You will get your answer.

Using a calculator, we can find the sine of 98 degrees as follows:

sin 98° ≈ 0.9848

Rounding this to four decimal places, we get:

sin 98° ≈ 0.9848

Therefore, sin 98° is approximately equal to 0.9848.

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If in a large population, about 70% of families have a pet. A researcher takes a random sample of 6 families and surveys whether they have a pet. The probability of families that have a pet is: 0.0595.

Number of trial=  6

Probability of success = 0.7

Probability of failure = 1 - 0.7 = 0.3

Probability that exactly 2 of the families have a pet

Let make us of binomial distribution formula to determine the probability

P(X = 2) = (6 choose 2) * 0.7² *[tex]0.3^{4}[/tex]

= 15 * 0.49 * 0.0081

= 0.0595

Therefore the probability  is 0.0595.

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The description of the given bearing with the towns and a similar drawing of bearing is given below:

The directions to draw the diagram that shows the relative position of the towns is given below:

Town R is at the center of the coordinate system.

Town C is 27km away on a bearing of 060°. Start at R, move 27km at 060° to find C towards east-southeast.

Town T is 19 km away from R on a ring of 3400.

T is on a circular path of radius of 3400 units.

To plot T, draw a circle at R with a 3400 unit radius. Put T on the circle, 19 km from R. Diagram: R center, C ESE, T on a circular path with a 3400 unit radius around R.

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Michio answer was wrong. The correct answer for x in the expression 2(3)ˣ = 34 is 2.5789

To know if Michio is correct or not, we shall determine the value of x in the algebraic expression. This is shown below:

2(3)ˣ = 34

Divide both sides by 2

3ˣ = 34 / 2

3ˣ = 17

Take the Log of both sides

Log 3ˣ = Log 17

Recall

Log Aⁿ = nLogA

Thus, we have

xLog 3 = Log 17

Divide both sides by Log 3

x = Log 17 / Log 3

x = 2.5789

From the above calculation, we see that Michio was wrong as the correct value of x in the expression is 2.5789

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The area of the object in the image shown in the attachment below is calculated as: 62 cm².

The area of the object can be found by decomposing the figure into two rectangles - larger rectangle and smaller rectangle.

Area of the larger rectangle = length * width

Length of the larger rectangle = 5 cm

Width of the larger rectangle = 4 cm

Area of the larger rectangle = 5 * 4 = 20 cm²

Area of the smaller rectangle = length * width

Length of the larger rectangle = 7 cm

Width of the larger rectangle = 6 cm

Area of the larger rectangle = 7 * 6 = 42 cm²

Area of the object = 20 + 42 = 62 cm².

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The desired sketch is attached below.

To sketch the scenario where BC intersects KT at point F, with F being the midpoint of BC but not KT, you can follow these steps:

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The equation of line that passes through and is perpendicular toy = (7/5)x + 6 is  [tex]y = -\frac{5}{7}x - \frac{32}{7}[/tex].

The slope-intercept form is expressed as;

y = mx + b

Where m is slope and b is the y-intercept.

Given the equation of the original line:
y = (7/5)x + 6

Using the slope intercept, slope m is 5/2

Slope m = 7/5

Note: the equation of a perpendicular line to y = (7/5)x + 6 must have a slope that is the negative reciprocal of the original slope.

Slope of perpendicular line = -1 / ( 7/5 )

Slope of perpendicular line = -5/7

Plug the slope m = -5/7 and point (2,-6) into point-slope formula and simplify.

( y - y₁ ) = m( x - x₁ )

[tex]( y - (-6) = -\frac{5}{7}( x - 2 ) \\\\y + 6 = -\frac{5}{7}x + \frac{10}{7} \\\\y = -\frac{5}{7}x + \frac{10}{7} - 6\\\\y = -\frac{5}{7}x - \frac{32}{7}[/tex]

Therefore, the equation of the line is [tex]y = -\frac{5}{7}x - \frac{32}{7}[/tex].

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125 workers can produce approximately 102.94 cars in 7 hours.Additionally, this assumes that the constant of Proportionality remains the same for different numbers of workers and hours worked.

We are given that the number of cars manufactured at an automobile plant varies jointly as the number of workers and the time they work. This can be expressed as:

C = kWT

where C is the number of cars produced, W is the number of workers, T is the time they work, and k is a constant of proportionality

We can use this formula to solve the problem. We are given that 85 workers can produce 136 cars in 8 hours, so we can set up an equation using this information:

136 = k(85)(8)

Solving for k, we get:

k = 2/17

Now we can use the formula to find the number of cars that 125 workers can produce in 7 hours:

C = (2/17)(125)(7)

C = 102.94

Therefore, 125 workers can produce approximately 102.94 cars in 7 hours.

It's important to note that this is an approximation since we have rounded the answer to two decimal places. Additionally, this assumes that the constant of proportionality remains the same for different numbers of workers and hours worked.

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answer :

median or 37

explanation :

median is not influenced by an outlier like 65 in the dataset

steps :

to get the median

arrange the weights in order from smallest to largest:

34, 35, 35, 37, 37, 37, 38, 39, 65

median = middle value = 37

37 has :

four weights before it &

four weights after it

so the median weight is 37 pounds.

Answer:

[tex]13.5cm[/tex]

Step-by-step explanation:

You just need to subtract to find your answer!

The information we are given:

We have a 16cm glue stick.

Christina used 2.5cm of it.

Subtracting:

16 - 2.5 = 13.5

Now, dont forget your units :D

[tex]13.5 -- > 13.5cm[/tex]

~~~Harsha~~~

Answer:

Step-by-step explanation:

5.6

The numerical value of the logarithm expression that is [tex]log\frac{a^2}{b^4c^4}[/tex] is 10 when log a = -1, log b = -6 and log c = 3.

Given that,

We have to find the numerical value of the logarithm expression that is [tex]log\frac{a^2}{b^4c^4}[/tex] when log a = -1, log b = -6 and log c = 3.

We know that,

The use of a logarithm can be applied to solve issues that cannot be resolved using the concept of exponents simply. A logarithm is simply a different way to define exponents.

So,

Take the logarithm expression that is [tex]log\frac{a^2}{b^4c^4}[/tex]

= [tex]log\frac{a^2}{b^4c^4}[/tex]

= log a² - log(b⁴c⁴)            [ from [tex]log\frac{x}{y}[/tex] = log x-log y ]

= log a² - (log b⁴ + log c⁴)     [ from log (xy) = log x + log y ]

= 2log a  - 4log b - 4log c       [ from log xⁿ = nlog x ]

= 2( -1) - 4(-6) - 4(3)

= -2 + 24 - 12

= 24 - 14

= 10

Therefore, The numerical value of the logarithm expression that is [tex]log\frac{a^2}{b^4c^4}[/tex] is 10.

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Applying the inscribed angle theorem, the indicated measures in the circle are: a) m<JIK = 51°; b) m<KJL = 51°.

Recall that the measure of an intercepted arc is equal to twice the measure of an inscribed angle, based on the inscribed angle theorem.

We are given the following:

m(IJ) = 78°

m<IJL = 39°

Therefore, we have:

a. m<JIK = 1/2(m(JK)) [based on the inscribed angle theorem]

m(JK) = 180 - m(IJ)

m(JK) = 180 - 78 = 102°

Thus:

m<JIK = 1/2(102)

m<JIK = 51°

b. m<KJL = 1/2(m(KL) [based on the inscribed angle theorem]

m(IL) = 2(IJL)

m(IL) = 2(39) = 78°

m(KL) = 180 - m(IL) = 180 - 78

m(KL) = 102°

m<KJL = 1/2(m(KL) = 1/2(102)

m<KJL = 51°

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Complete Question:

In the circle below, IK is a diameter. Suppose m(IJ) = 78° and m(IJL) = 39°. Find the following.

a. m<JIK

b. m<KJL

Missing image is attached below.

The value of the given limit expression is 1.

Given is limit expression [tex]\lim _{x\to \infty }\left(x^{\frac{1}{3x}}\right)[/tex], we need to use L’Hôpital’s Rule to solve it,

So,

[tex]\lim _{x\to \infty \:}\left(x^{\frac{1}{3x}}\right)[/tex]

Using the exponent rule,

[tex]=\lim _{x\to \infty \:}\left(e^{\frac{1}{3x}\ln \left(x\right)}\right)[/tex]

Applying the chain rule,

[tex]\lim _{x\to \infty \:}\left(e^{\frac{1}{3x}\ln \left(x\right)}\right) = 1[/tex]

Hence the value of the given limit expression is 1.

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The mean score of the test is 7.13.

The median score of the test is 7.

To get mean score, we have to multiply each test score by its corresponding frequency, sum up products and then divide by the total frequency.

Mean score = EF/ n

Mean score = (34 + 41 + 50 + 63 + 73 + 85 + 91 + 106) / (4 + 1 + 0 + 3 + 3 + 5 + 1 + 6)

= (12 + 4 + 0 + 18 + 21 + 40 + 9 + 60) / 23

= 164 / 23

= 7.13

To find the median score, we will arrange the scores in ascending order: [tex]3, 4, 4, 4, 4, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10.[/tex]

Since there are 23 scores, the median will be the 12th score (in the middle). So, the median score = 7.

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Answer:

1/4, 4/7, 2/3, 1/2

Step-by-step explanation:

The solution to the system of linear equations is (14, 16/3).

To find the solution to the given system of linear equations, we can use the method of substitution. First, we can solve the second equation for y by dividing both sides by 3, which gives us y = 16/3. .

We can then substitute this value of y into the first equation and solve for x. This gives us x - 3(16/3) = -2, which simplifies to x = 14. This point represents the intersection of the two lines, where they cross over each other on the coordinate plane.

The other answer choices given in the question are not valid solutions to the system of equations.

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Answer:

C

Step-by-step explanation:

Bill's share = $56 Therefore, Bill received $56.

Let's start by setting up some equations based on the given information:

David's share = 1/7 * (Alex's share + Bill's share + Carl's share)

Alex's share = 3/4 * (Bill's share + Carl's share)

Bill's share = 2/5 * Carl's share

The sum of all four shares is $168.

We can use the fourth equation to solve for one of the shares in terms of the others:

Alex's share + Bill's share + Carl's share + David's share = $168

Alex's share + Bill's share + Carl's share = $168 - David's share

Substituting the other equations into this expression, we get:

Alex's share + (2/5 * Carl's share) + Carl's share + (1/7 * (Alex's share + (2/5 * Carl's share) + Carl's share)) = $168

Simplifying this equation, we get:

(8/7) * Alex's share + (22/35) * Carl's share = $168

Now we can use the other equations to solve for Carl's share, and then for Bill's share:

From Bill's share = 2/5 * Carl's share, we get Carl's share = (5/2) * Bill's share

From Alex's share = 3/4 * (Bill's share + Carl's share), we get Alex's share = (3/4) * ((7/2) * Bill's share) = (21/8) * Bill's share

Substituting these expressions into the equation we derived earlier, we get:

(8/7) * (21/8) * Bill's share + (22/35) * (5/2) * Bill's share = $168

3 * Bill's share = $168

Bill's share = $56

Therefore, Bill received $56.

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