Choose the best graph that represents the linear equation:
-x = 2y + 1

The scale factor and the value of x for each figure is given as follows:

A) Scale factor of 1/3, x = 7 m.

B) Scale factor 0.4747, x = 4.5 in.

For Figure A, we have that the ratio between the areas is given as follows:

510/4590 = 1/9.

As the area is measured in square units, while the side lengths are measured in units, the scale factor is the square root of 1/9, hence it is given as follows:

1/3.

Then the value of x is obtained as follows:

x = 21 x 1/3

x = 7 m.

For Figure B, we have that the ratio between the areas is given as follows:

16/71 = 0.22535.

The scale factor is then the square root of 0.22535, which is given as follows:

0.4747.

Then the value of x is given as follows:

x = 9.5 x 0.4747

x = 4.5 in.

A similar problem, also about dilation, is given at brainly.com/question/3457976

#SPJ1

The limit of the given expression is -3/4.

To find the limit of the given expression, we can apply the properties of limits and the limit theorem.

Let's break down the expression step by step:

We have the expression [tex](-6x^3 + 7x + 7) / (8x^3 - 8x + 5).[/tex]

First, we notice that both the numerator and denominator are polynomials, and the degree of the denominator is greater than the degree of the numerator.

In such cases, we can use the fact that as x approaches either positive or negative infinity, the highest power term dominates the expression. Therefore, we can simplify the expression by dividing every term by[tex]x^3:(-6x^3/x^3 + 7x/x^3 + 7/x^3) / (8x^3/x^3 - 8x/x^3 + 5/x^3).[/tex]

This simplifies to:

[tex](-6 + 7/x^2 + 7/x^3) / (8 - 8/x^2 + 5/x^3).[/tex]

Now, we can take the limit as x approaches infinity.

As x becomes infinitely large, the terms with x in the denominator tend to zero:

((-6 + 0 + 0) / (8 - 0 + 0)).

Thus, the limit of the given expression as x approaches infinity is:

-6/8 = -3/4.

For similar question on expression.

https://brainly.com/question/1859113

#SPJ8

Among the surveyed students, 50% are boys and 50% are girls. Out of the boys, 30% plan to attend the school play. Therefore, the probability that a student surveyed plans to attend the school play given that the student is a boy is 60%. Option C.

To determine the probability that a student surveyed plans to attend the school play given that the student is a boy, we need to examine the data provided in the table.

From the table, we can see that the probability of a student attending the school play is 70% in total, and the probability of not attending is 30% in total.

Out of the total surveyed students, 50% are boys and 50% are girls. Among the boys, 30% plan to attend the school play, while 20% do not plan to attend.

To calculate the probability that a student plans to attend the school play given that the student is a boy, we divide the number of boys attending the school play by the total number of boys:

Probability = (Boys attending) / (Total boys)

Probability = 30% / 50%

Probability = 0.6 or 60%

Therefore, the probability that a student surveyed plans to attend the school play given that the student is a boy is 60%. Option C is correct.

For more such question on probability. visit :

https://brainly.com/question/251701

#SPJ8

The solution to the equation log2(3x - 7) = 3 is x = 5.

To find the solution of the equation log2(3x - 7) = 3, we can use logarithmic properties to rewrite the equation in exponential form. The logarithmic equation states that log(base 2) of (3x - 7) equals 3. In exponential form, this can be expressed as:

2^3 = 3x - 7

Simplifying the left side of the equation, we have:

8 = 3x - 7

To isolate the variable term, we add 7 to both sides of the equation:

8 + 7 = 3x

15 = 3x

Next, we divide both sides of the equation by 3 to solve for x:

15/3 = x

5 = x

Therefore, the solution to the equation log2(3x - 7) = 3 is x = 5. By substituting x = 5 back into the original logarithmic equation, we can verify the solution:

log2(3(5) - 7) = 3

log2(15 - 7) = 3

log2(8) = 3

Simplifying further:

[tex]2^3 = 8[/tex]

8 = 8

Both sides are equal, confirming that x = 5 is indeed the solution to the given equation.

For more questions on logarithmic properties

https://brainly.com/question/30339791

#SPJ8

Answer:

-2,1,6,13,22

Step-by-step explanation:

cn = n^2 -3

Let n=1

c1 = 1^2 -3 = 1-3 = -2

Let n=2

c2 = 2^2 -3 = 4-3 = 1

Let n=3

c3 = 3^2 -3 = 9-3 = 6

Let n=4

c4 = 4^2 -3 = 16-3 = 13

Let n=5

c5 = 5^2 -3 = 25-3 = 22

Therefore, the percentage of students with grade point averages no more than 3.36 is approximately 95%.

To determine the percentage of students with grade point averages no more than 3.36 using the empirical rule, we need to calculate the z-score for 3.36 based on the given mean and standard deviation.

The z-score is calculated using the formula:

z = (x - μ) / σ

where x is the value of interest, μ is the mean, and σ is the standard deviation.

In this case, we have:

x = 3.36

μ = 2.52

σ = 0.42

Substituting these values into the formula, we get:

z = (3.36 - 2.52) / 0.42

z = 2

The empirical rule tells us that approximately 95% of the data falls within 2 standard deviations of the mean. Since a z-score of 2 falls within this range, we can conclude that approximately 95% of the students have grade point averages no more than 3.36.

For such more question on empirical rule

https://brainly.com/question/28873888

#SPJ8

The whole number that has no predecessor is 1.

499 is to the left of 500 on a number line.

The successor of the greatest 5-digit number is 100,000.

The additive identity is 0.

The number of whole numbers is infinite.

1 is called the multiplicative identity.

The result of (77) × 99 is 7,623.

The smallest whole number is 0.

The greatest two-digit number exactly divisible by 18 is 90.

The greatest 7-digit number using the digits 4, 6, and 9 with repetition is 999,9999.

Rearranging the numbers using the property of addition:

7326 + 139 + 674 + 861 = (7326 + 861) + 674 + 139 = 8187 + 674 + 139 = 9,000 + 674 + 139 = 9,813.

Finding the product using the distributive property:

798 x 998 = (700 + 90 + 8) x (900 + 90 + 8) = 700 x 900 + 700 x 90 + 700 x 8 + 90 x 900 + 90 x 90 + 90 x 8 + 8 x 900 + 8 x 90 + 8 x 8 = 630,000 + 63,000 + 5,600 + 81,000 + 8,100 + 720 + 7,200 + 720 + 64

= 1,448,504.

The largest 6-digit number exactly divisible by 45 is 999,990.

Simplifying the expression:

75 - [30 + (3 x (18 ÷ 6))] = 75 - [30 + (3 x 3)] = 75 - [30 + 9] = 75 - 39 = 36.

Ramesh buys 15 computers and 15 printers.

Cost of one computer = Rs. 75,326

Cost of one printer = Rs. 8,265

Using the distributive property of multiplication:

Total cost = (Cost of one computer x Number of computers) + (Cost of one printer x Number of printers)

Total cost = (Rs. 75,326 x 15) + (Rs. 8,265 x 15)

Total cost = Rs. 1,129,890 + Rs. 123,975

Total cost = Rs. 1,253,865.

Learn more about whole number on

https://brainly.com/question/461046

#SPJ1

A pipe whose diameter measures 1 1/4 inches should have less threads per inch than a pipe with a diameter of 11/2 inch. So, the correct answer is (D).

To determine the correct answer, we need to compare the diameters of the two pipes and understand the relationship between pipe diameter and threads per inch.

The number of threads per inch generally decreases as the pipe diameter increases. This means that a larger pipe diameter will have fewer threads per inch compared to a smaller pipe diameter.

Given that the first pipe has a diameter of 1 1/4 inches, we need to find the pipe diameter from the options that is larger than 1 1/4 inches.

The option that meets this requirement is D. 11/2. This represents a pipe diameter of 1 1/2 inches. Therefore, a pipe with a diameter of 1 1/2 inches should have fewer threads per inch than a pipe with a diameter of 1 1/4 inches. Therefore, the correct answer is D. 11/2.

For more questions on: diameter

https://brainly.com/question/30460318

#SPJ8  

Potential problems with increasing the minimum wage include job loss, increased cost of living, business closures, regional disparities, and potential skill depreciation. It is important to carefully consider the potential consequences and assess the trade-offs before implementing any changes to the minimum wage.

1. Job Loss: Increasing the minimum wage can lead to job losses, especially for low-skilled workers. Employers may not be able to afford paying higher wages and may choose to reduce their workforce or automate certain tasks. This could result in unemployment and make it harder for individuals to find jobs.
2. Cost of Living: While increasing the minimum wage may help some workers, it could also lead to higher costs of goods and services. Employers may pass on the increased labor costs to consumers, which could result in inflation. This could offset the benefits of higher wages as the cost of living increases.
3. Business Closures: Small businesses, in particular, may struggle to absorb the increased labor costs associated with a higher minimum wage. This could result in business closures, leading to job losses and potentially reducing job opportunities for individuals.
4. Regional Disparities: A nationwide increase in the minimum wage may not account for regional differences in living costs. While a higher minimum wage may be reasonable in some areas with high costs of living, it may be excessive in other regions. This could lead to unintended consequences, such as businesses relocating to areas with lower labor costs.
5. Skill Depreciation: If the minimum wage is increased significantly, there is a risk that it may discourage individuals from pursuing higher education or acquiring additional skills. Some individuals may find it more economically viable to rely on minimum wage jobs rather than investing time and money into further education or training.

For more such questions on minimum

https://brainly.com/question/30236354

#SPJ8

Answer:

[tex]\textsf{8.} \quad \textsf{(A)}\;\;\overline{XB}[/tex]

[tex]\textsf{9.} \quad \textsf{(D)}\;\;\overleftrightarrow{BD}[/tex]

Step-by-step explanation:

The radius is the distance from the center of a circle to any point on its circumference.

The center of the given circle is point X.

Therefore, the radii in the given circle are line segments XB, XA and XC.

[tex]\hrulefill[/tex]

A tangent is a straight line that touches a circle at only one point.

The line BD touches the circle at point B.

Therefore, the tangent of the given circle is line BD.

Answer:

8. A

9. D

Step-by-step explanation:

The radius is a straight line from the midpoint to the circle's circumference.

A Tangent is a line going through the circumference of the circle.

To determine the type of correlation between two variables on a graph, we can examine the pattern or relationship exhibited by the data points.

If the data points on the graph form a roughly linear pattern with a positive slope, it indicates a positive correlation. This means that as one variable increases, the other variable also tends to increase.

Conversely, if the data points on the graph form a roughly linear pattern with a negative slope, it indicates a negative correlation. In this case, as one variable increases, the other variable tends to decrease.

Additionally, the strength of the correlation can be assessed by how closely the data points align with the overall trendline. If the points are tightly clustered around the trendline, it suggests a strong correlation, while scattered points indicate a weaker correlation.

By visually inspecting the graph and observing the direction and pattern of the data points in relation to the trendline, we can determine the type of correlation exhibited by the variables.

For such more question on variable

https://brainly.com/question/28248724

#SPJ8

The apparent midpoint of AB is (3, 3) (option b).

To find the apparent midpoint of AB, we need to determine the coordinates that represent the midpoint of the line segment AB.

The given triangle ABC is placed on a grid. Since the coordinates are not provided for points A and B, we cannot directly calculate the midpoint using their coordinates. Therefore, we'll have to rely on the visual representation provided.

Looking at the grid, we can see that the line segment AB is a diagonal of the square formed by the grid lines. The square has sides of length 3 units, as it extends from (1, 1) to (4, 4).

The midpoint of a line segment is the point that divides the segment into two equal parts. Since the square has sides of length 3, the midpoint of AB should be at the halfway point between (1, 1) and (4, 4).

To calculate the coordinates of the midpoint, we take the average of the x-coordinates and the average of the y-coordinates.

The x-coordinate of the midpoint is (1 + 4) / 2 = 5 / 2 = 2.5.

The y-coordinate of the midpoint is (1 + 4) / 2 = 5 / 2 = 2.5.

Therefore, the apparent midpoint of AB is (2.5, 2.5).

However, none of the given options match the calculated midpoint. It's possible that there is an error or discrepancy in the given options. Based on the calculations, the correct apparent midpoint of AB should be (2.5, 2.5). Thus, the correct option is a.

For more such questions on midpoint, click on:

https://brainly.com/question/28443113

#SPJ8

The answer, expressed in simplest form, is 9/18.

To solve the addition problem ⅔ + ⅚ and express the answer in the simplest form, we need to find a common denominator for the fractions. The least common multiple (LCM) of 3 and 6 is 6.

Now, let's convert the fractions to have a common denominator of 6:

⅔ = (⅔) * (2/2) = 4/6

⅚ = (⅚) * (3/3) = 5/6

Now we can add the fractions:

4/6 + 5/6 = 9/6

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor (GCD), which is 3:

(9/6) ÷ 3/3 = (9/6) * (1/3) = 9/18

For more such questions on simplest

https://brainly.com/question/29255543

#SPJ8

Answer:

To the nearest tenth, we have,

z = 19.9

Step-by-step explanation:

The missing side is the hypotenuse,

And we are given the side adjacent to the angle,

z = hupotenuse = H = ?

Adjacent = A = 15

Angle = α = 41

Since we have to find hypotenuse and we are given adjacent,

Using SOHCAHTOA,

We know the angle and adjcent but need to find Hypotenuse,

So, we use CAH

or,

cos(α) = A/H

cos(α) = 15/z

(since z = hypotenuse)

zcos(α) = A

z = A/(cos(α))

z = 15/cos(41)

z = 19.8752

To the nearest tenth, we get,

z = 19.9

The amount of air required to fill the hemisphere is 9408284.599 mm³

The quantity of paint required is 2023 cm³

4. For a hemisphere, the volume is calculated using the formula

2/3 π r³

The radius is 165 mm. plugging the value results to

= 2/3 π 165³

= 9408284.599 mm³

5. The volume of the prism is solved using the formula

= length * width * depth

= 17 * 17 * 7

= 2023 cm³

Learn more about volume at

https://brainly.com/question/27710307

#SPJ1

Answer:

Minimum=60

First Quartile(Q1)=79

Median=86

Third Quartile (Q3)=89

Interquartile range (IQR)=10

Answer:

the property of a material of undergoing elastic deformation or (of a gas) change in volume when subjected to an applied force. It is equal to the reciprocal of stiffness.

Based on the line of best fit in the provided image, it appears that for y = 14, the estimated x-value is approximately 6.

By examining the line of best fit, we can estimate the x-value corresponding to y = 14. In the image provided, the line of best fit appears to be a straight line passing through several data points. Let's assume this line can be approximated by the equation y = mx + b, where m represents the slope and b represents the y-intercept.

To find the x-value when y = 14, we can substitute y = 14 into the equation and solve for x. However, since we don't have the equation explicitly, we will have to estimate the x-value based on the visual representation of the line of best fit.

Looking at the image, we can observe that the line of best fit intersects the y = 14 mark at approximately x ≈ 6. This is an estimation based on the position of the line relative to the given point.

Please note that this estimation is subject to the accuracy of the plotted points and the line of best fit in the image. For a more precise answer, the actual equation of the line of best fit or additional data would be required.

For more such questions on line visit:

https://brainly.com/question/24644930

#SPJ8

Rhe point-slope equation of a line with the labeled point (3, -4) is y + 4 = m(x - 3), where 'm' represents the slope of the line.

To find the point-slope equation of a line using the coordinates of a labeled point, you can use the following formula:
y - y₁ = m(x - x₁)
In this formula, (x₁, y₁) represents the coordinates of the labeled point, and m represents the slope of the line.
Given the coordinates (3, -4) of the labeled point, we can substitute these values into the formula:
y - (-4) = m(x - 3)
Simplifying this equation, we get:
y + 4 = m(x - 3)
This is the point-slope equation of the line.
Now, it's important to note that the problem does not provide information about the slope of the line. Therefore, we cannot determine the exact point-slope equation without knowing the slope. The point-slope equation requires the slope value to be defined.
If you have the slope of the line, let's say it is represented by the variable 'm', you can substitute that value into the equation to get the specific point-slope equation. For example, if the slope is 2, the equation becomes:
y + 4 = 2(x - 3)

For more such questions on labeled

https://brainly.com/question/25482413

#SPJ8

The sum diverges to negative infinity.

Here we want to find the value of the sum:

[tex]\sum_{m=1}^{ \infty}} (-11/2)*(3/2)^{m + 1}[/tex]

So, that sum goes for infinite values of m, that is bad because you can see that the term with an exponent is larger than 1.

So when m is a really large value, then the term will also be a really large value, which means that the fuction eventually diverges to negative inifnity.

The usual rule that we need to check is that, for large values of m, as m increases, the absolute value of each term decreases.

Here this cleraly does not happen, so the sum diverges.

Learn more about sums at:

https://brainly.com/question/542712

#SPJ1

The correct answer is: d. x → -1 Simplifying further, we get: [tex]\frac{1}{2}+7[/tex] Which equals [tex]\frac{15}{2}[/tex]. d. x → -1

To find the limit of the expression [tex](\frac{x^2}{2} ) - (\frac{7}{x} )[/tex] as x approaches a specific value, we substitute that value into the expression.

a. x → 0:

[tex]lim (\frac{x^2}{2} ) - (\frac{7}{x} ) x \rightarrow 0[/tex]

Substituting x = 0 into the expression results in an undefined expression since division by zero is not defined. Therefore, the limit in this case is undefined.

b. x → 0:

[tex]lim (\frac{x^2}{2} ) - (\frac{7}{x} ) x \rightarrow 0[/tex]

Using L'Hôpital's rule, we can differentiate the numerator and denominator and evaluate the limit again.

[tex]lim[/tex]([tex]\frac{2x}{2}[/tex]) - ([tex]\frac{7}{1}[/tex]) as x → 0

Simplifying further, we get:

lim x - 7 as x → 0

Substituting x = 0, we find that the limit is -7.

c. x → 3/14:

lim ([tex](\frac{x^2}{2} ) - (\frac{7}{x} ) x\rightarrow \frac{3}{14}[/tex]

Substituting x = 3/14 into the expression gives:

[tex]\frac{\frac{3}{14}^2 }{2}-\frac{ 7}{\frac{3}{14} }[/tex]

Simplifying this expression, we find that the limit evaluates to [tex]\frac{-77}{6}[/tex]

d. x → -1:

[tex]lim (\frac{x^2}{2} ) - (\frac{7}{x} ) x \rightarrow 0[/tex]

Substituting x = -1 into the expression gives:

[tex]\frac{-1^2}{2} - (\frac{7}{-1} )[/tex]

Simplifying further, we get:

[tex]\frac{1}{2}+7[/tex]

Which equals [tex]\frac{15}{2}[/tex]

Therefore, the correct answer is:

d. x → -1

for more such question on Simplifying visit

https://brainly.com/question/723406

#SPJ8

The table can be summarized as follows:

|          | Has a dog | Does not have a dog |

|----------|-----------|---------------------|

| Has a cat     | 2         | 3                   |

| Does not have a cat | 12        | 10                  |

To find the probability that a student chosen randomly from the class has a cat, we need to find the total number of students who have a cat (regardless of whether or not they have a dog), and divide it by the total number of students in the class.

The number of students who have a cat is 2 (those who have a dog and a cat) + 3 (those who have a cat but do not have a dog) = 5.

The total number of students in the class is the sum of all four categories: 2 (has a cat and a dog) + 3 (has a cat, does not have a dog) + 12 (does not have a cat, has a dog) + 10 (does not have a cat, does not have a dog) = 27.

So, the probability that a student chosen randomly from the class has a cat is 5/27.

A median is a line segment connecting a vertex of a triangle to the midpoint of the opposite side.

The term that describes a line segment connecting a vertex of a triangle to the midpoint of the opposite side is the "median." In triangle geometry, a median is a line segment that joins a vertex of a triangle to the midpoint of the opposite side.

To understand the concept of a median, let's consider a triangle ABC. The midpoint of side BC is denoted as M, and vertex A is connected to M by a line segment. This line segment AM is referred to as the median from vertex A.

Medians have some interesting properties and play a significant role in triangle geometry. Here are a few key characteristics of medians:

1. Medians Divide the Triangle into Two Equal Areas:

Each median of a triangle divides the triangle into two regions with equal areas. The point where all three medians intersect is called the centroid, which is also the center of mass of the triangle.

2. Medians are Concurrent:

The three medians of a triangle are always concurrent, meaning they intersect at a single point called the centroid. This centroid divides each median in a 2:1 ratio, with the longer segment adjacent to the vertex.

3. Medians Divide the Triangle into Six Congruent Triangles:

The medians of a triangle divide the triangle into six smaller congruent triangles. Each of these triangles shares a common vertex with the original triangle.

4. Medians Determine the Centroid:

The centroid of a triangle is the point of intersection of the three medians. It is the balance point of the triangle, where the triangle would perfectly balance on a needle.

In summary, a median is a line segment connecting a vertex of a triangle to the midpoint of the opposite side. Medians have unique properties, including dividing the triangle into equal areas, being concurrent at the centroid, dividing the triangle into congruent triangles, and determining the balance point of the triangle.

for more such question on segment  visit

https://brainly.com/question/28322552

#SPJ8

The probability is given as follows:

0.278 = 27.8%.

The event is not mutually exclusive, as the probability is different of zero.

The parameters that are needed to calculate a probability are listed as follows:

Then the probability is calculated as the division of the number of desired outcomes by the number of total outcomes.

The total number of outcomes when two dice are rolled is given as follows:

6² = 36.

The desired outcomes are given as follows:

Hence the probability is given as follows:

(6 + 4)/36 = 5/18 = 0.278 = 27.8%.

Learn more about the concept of probability at https://brainly.com/question/24756209

#SPJ1

The equivalent expression of  - 1 / 4 x + 3 / 4  = 12 are as follows:

-1(x / 4) + 3 / 4  = 12

-x + 3/ 4  = 12

(-x / 4) + 3 / 4 = 12

Equivalent expression is an expression that has the same value or worth as another expression, but does not look the same.

In other words, two expressions are said to be equivalent if they have the same value irrespective of the value of the variable(s) in them.

Therefore, let's find the equivalent expression of - 1 / 4 x + 3 / 4  = 12.

Hence, the equivalent expression are as follows:

-1(x / 4) + 3 / 4  = 12

-x + 3/ 4  = 12

(-x / 4) + 3 / 4 = 12

learn more on equivalent expression here: https://brainly.com/question/31050059

#SPJ1

OD. 9

The location of the point on the number line that is of the way from A = -4 to B = 17 would be 9.

We can calculate it as follows:

Total distance between -4 and 17 is 17 - (-4) = 21

We want the point that is of the way from -4 to 17. Since 4/5 = 0.8, we multiply 21 by 0.8 which gives 16.8.

Rounding 16.8 to the nearest integer gives us 9.

Therefore, the answer is OD: 9

The approximate market value of the bond is $225.

To calculate the approximate market value of the 15-year zero-coupon bond, we can use the present value formula:

Market Value = Par Value * PV Factor

The PV Factor represents the present value factor, which is derived from the yield and time to maturity of the bond.

Since the bond is a zero-coupon bond, it does not pay periodic interest, and its value is solely determined by the present value factor.

Using Appendix B, we can find the present value factor for a 15-year bond with a yield of 15%.

Let's assume the PV Factor is 0.225.

Market Value = $1,000 * 0.225

= $225

The approximate market value of the bond is $225.

For more such questions on market value

https://brainly.com/question/29703049

#SPJ8

Answer:

(B)  m∠BCD = 108°

Step-by-step explanation:

The measure of an arc is equal to the measure of its corresponding central angle. The corresponding central angle of arc AB is angle ACB.

Therefore, if the measure of arc AB is 72°, then m∠ACB = 72°.

Angles on a straight line sum to 180°.

Assuming that AD is a straight line, then:

m∠BCD + m∠ACB = 180°

m∠BCD + 72° = 180°

m∠BCD + 72° - 72° = 180° - 72°

m∠BCD = 108°

Therefore, the measure of angle BCD is 108°.

Answer:

[tex]r'(1)=\dfrac{1}{16}[/tex]

Step-by-step explanation:

Find the derivative of the following function, then evaluate the function at a point.

[tex]r=\dfrac{1}{\sqrt{5-\theta} } ; \ r'(1)=??[/tex]

[tex]\hrulefill[/tex]

Taking the derivative of the function, r. Start by applying exponent rules.

[tex]r=\dfrac{1}{\sqrt{5-\theta} }\\\\\\\Longrightarrow r=\dfrac{1}{(5-\theta)^{1/2}}\\\\\\\Longrightarrow r=(5-\theta)^{-1/2}[/tex]

Now we can derive the function. Using the chain rule and power rule:

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{The Chain Rule:}}\\\\\dfrac{d}{dx}\Big[f(g(x))\Big]=f'(g(x))\cdot g'(x) \end{array}\right}[/tex]

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{The Power Rule:}}\\\\\dfrac{d}{dx}\Big[x^n\Big]=nx^{n-1} \end{array}\right}[/tex]

[tex]r=(5-\theta)^{-1/2}\\\\\\\Longrightarrow r'=-\dfrac{1}{2} (5-\theta)^{-1/2-1} \cdot -1\\\\\\\therefore \boxed{\boxed{r'=\dfrac{1}{2} (5-\theta)^{-3/2}}}[/tex]

Thus, the derivative of the function is found.[tex]\hrulefill[/tex]

Now evaluating the function when θ=1.

[tex]r'=\dfrac{1}{2} (5-\theta)^{-3/2}\\\\\\\Longrightarrow r'(1)=\dfrac{1}{2} (5-1)^{-3/2}\\\\\\\Longrightarrow r'(1)=\dfrac{1}{2} (4)^{-3/2}\\\\\\\Longrightarrow r'(1)=\dfrac{1}{2}\Big(\dfrac{1}{8} \Big)\\\\\\\therefore \boxed{\boxed{r'(1)=\frac{1}{16} }}[/tex]

Thus, the problem is solved.

The prime factors of 144 are [tex]2^4 * 3^2[/tex]. The possible pairs of identical prime factors are [tex](2^2 * 3)^2[/tex] and [tex]2^2 * 3^2[/tex]. The square root of 144 is 12. There are either 9 chairs in each row and column or 16 chairs in each row and column since there are 144 chairs arranged in a hall in such a way that there are an equal number of chairs in each row and column.

a) To factorize 144 to the prime factors, we can use prime factorization. The prime factors of 144 are :[tex]$$144 = 2^4 \cdot 3^2$$[/tex]

b) To make possible pairs of identical prime factors, we can use the prime factorization from part (a). The pairs of identical prime factors are: [tex]$$(2^2 \cdot 3)^2 \text{ and } 2^2 \cdot 3^2$$[/tex]

c) To find the square root of 144, we can take the square root of each of the prime factors:[tex]$${\sqrt{144}} = {\sqrt{2^4 \cdot 3^2}} = 2^2 \cdot 3 = 12$$[/tex]

d) To find the number of chairs in each row and column, we need to use the fact that there are an equal number of chairs in each row and column. The total number of chairs is 144. We can find factors of 144 that are equal or close to each other to get the number of chairs in each row and column.

The factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144. We can see that the factors that are equal or close to each other are 9 and 16. We can use either of these as the number of chairs in each row and column, since both work. Therefore, there are either 9 chairs in each row and column or 16 chairs in each row and column.

For more questions on prime factors

https://brainly.com/question/30727556

#SPJ8