A volleyball has a volume of 268 cubic inches. Which is the radius of the volleyball
to the nearest inch?
1.4 inches
2.5 inches
3.8 inches
4.9 inches
Help a girl out quick!

Answers

Answer 1
8484844848575858 inch

Related Questions

Find the radius of a circle with a diameter of 12.5 inches.

Answers

Answer:

The radius is 6.25 inches

Explanation:

The radius of a circle is half the diameter of the said circle.

If the diameter of a circle is D, then the radius is D/2.

Given a diameter of 12.5 inches

The radius is 12.5/2 = 6.25

The radius is 6.25 inches.

After the translation, where is A located?

Answers

Answer:

(-3, 13)

Step-by-step explanation:

The transformation that moves a point 4 left and 8 up is ...

 (x, y) ⇒ (x -4, y +8)

The transformation that reflects a point across the y-axis is ...

 (x, y) ⇒ (-x, y)

Applied after the translation, the transformation of ∆ABC becomes ...

 (x, y) ⇒ (-(x -4), y +8) = (4 -x, y +8)

Then point A gets moved to ...

 A(7, 5) ⇒ A'(4 -7, 5 +8) = (-3, 13)

Which of the following is the form factor of a standard Parallel connector? A. DB-9. B. DB-15. C. DB-25. D. DIN-6.

Answers

The form factor of a standard Parallel connector is DB-25.

A Parallel connector is a type of interface commonly used for connecting devices such as printers, scanners, and external storage devices to a computer. It allows for the simultaneous transmission of multiple bits of data through separate data lines. The form factor refers to the physical shape and configuration of the connector.

Option A, DB-9, is incorrect because DB-9 connectors are commonly used for serial communication, not parallel communication. They have 9 pins arranged in two rows and are typically used for applications such as RS-232 serial ports.

Option B, DB-15, is also incorrect as DB-15 connectors are commonly used for video graphics (VGA) connections, not parallel communication. They have 15 pins arranged in three rows and are commonly found on older computer monitors.

Option D, DIN-6, is also not the correct form factor for a standard Parallel connector. DIN-6 connectors are typically used for MIDI (Musical Instrument Digital Interface) connections or other audio applications. They have 6 pins arranged in a circular configuration.

The correct option is C, DB-25. The DB-25 connector, also known as a "D-sub 25" or "D-subminiature 25," is a type of connector commonly used for parallel communication interfaces. It has 25 pins arranged in two rows, with the pins surrounded by a D-shaped metal shield. The DB-25 connector is typically used for printer ports, parallel ports, and other parallel communication applications.

Therefore, the form factor of a standard Parallel connector is DB-25.

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SOMEONE PLZ HELP ME T_T

SOMEONE PLZ HELP ME T_T

Answers

Answer:

7x - 1

Step-by-step explanation:

Whole rectangle:

Length = x + 5

Width = x - 1

Area of the whole triangle = length × width

= (x + 5) (x - 1)

= x² - x + 5x - 5

= x² + 4x - 5

Unshaded rectangle:

Length = x + 1

Width = x - 4

Area of the unshaded rectangle = length × width

= (x + 1) (x - 4)

= x² - 4x + x - 4

= x² - 3x - 4

Area of the shaded region = Area of the whole rectangle - Area of the unshaded rectangle

= (x² + 4x - 5) - (x² - 3x - 4)

= x² + 4x - 5 - x² + 3x + 4

= 7x - 1

Area of the shaded region = 7x - 1

PLEASE HELP, TWENTY POINTS PLUS BRAINLIEST
Write an explicit function to represent the following sequence.
3,9, 27, 81,...

Answers

The equation would be if y is the numbers in the sequence (like 3, 9, 27) and x is the is the number in the order (like 3 is 1, 9 is 2, 81 is 4) the the equation would be y = 3^x.

For a card trick, Barry is using 14 cards from a deck. If each suit he uses has 5 numbered cards and c face cards, which expression shows how many suits he is using?

A. 14 ÷ (5 + c)
B. 14 x 5 + c
C. 14 ÷ 5 + c
D. 14 x (5 + c)​

Answers

The correct answer is option A. 14 ÷ (5 + c).

Barry exists spreading the cards among the suits, then he needs enough suits to hold the total of 14 cards he has. Hence, we need to divide the total number of cards by the total number of cards,

What is meant by expression ?

An expression, often known as a mathematical expression, is a finite collection of symbols that are well-formed in accordance with context-dependent principles.

A number, a variable, or a mix of numbers, variables, and operation symbols make up an expression. Two expressions joined by an equal sign form an equation.

If we assume Barry exists spreading the cards among the suits, then he needs enough suits to hold the total of 14 cards he has. Hence, we need to divide the total number of cards by the total number of cards.

Therefore, the correct answer is option A. 14 ÷ (5 + c)

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which of the following is one of the requirements for using a one-way, between-subjects anova?

Answers

The following is one of the requirements for using a one-way, between-subjects Anova is the data must be measured on a continuous scale.

Between-subjects ANOVA

One of the requirements for using a one-way, between-subjects ANOVA is that the data must be measured on a continuous scale. This means that the variable being measured (such as time, weight, or temperature) should be able to take on any value within a certain range.

Another requirement for using a one-way, between-subjects ANOVA is that the data must be independent. This means that the observations for each group should not be related to one another in any way. For example, if the groups being compared are different treatment groups in a medical study, it would not be appropriate to use a one-way, between-subjects ANOVA if the patients in each group were related to one another (such as siblings or close friends).

The third requirement is that the data should be approximately normally distributed in each group. A normal distribution is a probability distribution that has a bell shape, which means that the data is symmetric and the mean, median, and mode are equal. The ANOVA assumes that the data is normally distributed and if the data is not normally distributed, the results may be unreliable. This requirement can be checked by using normality test such as Shapiro-Wilk test.

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determine the unit normal vector to the curve with vector equation \vec r \, (t) =<1^t,\ln(4/t^2), \, \ln(e/(t^4)) > r (t)=<1 t ,ln(4/t 2 ),ln(e/(t 4 ))> at the point where t=2t=2.

Answers

the unit normal vector to the curve with vector equation \vec r \, (t) =<1^t,\ln(4/t^2), \, \ln(e/(t^4)) > r (t)=<1 t ,ln(4/t 2 ),ln(e/(t 4 ))> at the point where t=2t=2 is \vec n = <0,0,-1>

A unit normal vector is a normal vector with magnitude 1, and it points in the direction that is perpendicular to the tangent to the curve at a given point.

To determine the unit normal vector to a curve, it's important to find the tangent first and then find the normal. In this case, we're looking for the unit normal vector to the curve with the vector equation \vec r (t) =<1^t,\ln(4/t^2), \ln(e/(t^4)) > at the point where t = 2.

To find the unit normal vector to the curve with the vector equation \vec r (t) =<1^t,\ln(4/t^2), \ln(e/(t^4)) > at the point where t = 2, it is necessary to find the tangent to the curve and then find the normal. This is a crucial step in understanding the geometry of curves in 3D space.

To find the tangent to the curve, we'll differentiate the vector equation with respect to t and evaluate it at t = 2. The derivative of \vec r (t) is given by:

d\vec r/dt = <1, -2t^-3, -4e^-t^-4>

So the tangent to the curve at t = 2 is:

d\vec r/dt = <1, -2(2)^-3, -4e^-2^-4> = <1, 1.5, -1.648721>

Next, we'll find the normal to the tangent by taking the cross product of the tangent with a fixed vector, for example, the unit vector i = <1,0,0>. The cross product of two vectors results in a vector that is perpendicular to both of them.

The cross product of the tangent and the unit vector i is given by:

\vec T x \vec i = <1, 1.5, -1.648721> x <1,0,0> = <0,0,-1.5>

Since the magnitude of the normal is not 1, we'll normalize it by dividing it by its magnitude:

\vec n = \vec T x \vec i / ||\vec T x \vec i|| = <0,0,-1.5> / ||<0,0,-1.5>|| = <0,0,-1>

So the unit normal vector to the curve with the vector equation \vec r (t) =<1^t,\ln(4/t^2), \ln(e/(t^4)) > at the point where t = 2 is given by:

\vec n = <0,0,-1>

This is the unit normal vector that points in the direction that is perpendicular to the tangent to the curve at t = 2. The magnitude of 1 ensures that it's a unit vector and the direction points towards the normal direction.

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How much the statistic varies from one sample to another is known as the ____ _____ of a statistic.

Answers

How much a statistic varies from one sample to another is known as the sampling variability of a statistic.

Testing variability arises due to the reality that exceptional samples of the same size can produce special values of a statistic, although the samples are described from the same populace. the quantum of slice variability depends on the scale of the sample, the variety of the populace, and the unique statistic being calculated.

The end of statistical conclusion is to apply the data attained from a sample to make consequences about the population, while counting for the slice variability of the statistic. ways similar as thesis checking out and tone belief intervals do not forget the sampling variability of a statistic to make redundant accurate consequences about the populace.

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3 ways to make whole 2.8

Answers

Step-by-step explanation:

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3 ways to make whole 2.8

Answer:

2.8 as a whole is 2 4/5. Take out the fraction, 4/5.2.8 can also be 28/10 which is the improper number of 2 8/10.2.8 can also be simplified to 1.4, 1.4 is 1 2/5.

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Nadia has a $15 gift card to an online music store. Each song at the store costs $0.99. The equation m = 15 - 0.99s represents the amount of money, m, Nadia has left on the gift card after buying 5 songs. a.


The independent variable is b.


How does the amount of money left on the card change for each extra song Nadia buys?


Show your work. and the dependent variable is​

Nadia has a $15 gift card to an online music store. Each song at the store costs $0.99. The equation

Answers

Answer:

All I know is she had 10.05 left

Determine if the given vector is in the range of t(x) = ax where a = [1 -2 0 3 2 1] y = [-3 6] y = [1 -4] y = [2 7] Suppose that a linear transformation T satisfies

Answers

The vector b = [7 0 3 3 3 3]T is in the range of the transformation T defined by the matrix A.

Given the vectors y1 = [-3 6]T, y2 = [1 -4]T, and y3 = [2 7]T , we will determine whether the vector b = [7 0 3 3 3 3]T is in the range of the transformation T defined by the matrix A = [ 1 -2 0 3 2 1 ; 2 1 2 -1 1 2 ; 0 0 1 0 2 1 ].

In order to determine whether a vector b is in the range of a linear transformation T, we need to solve the equation T(x) = b for some vector x. In this case, the linear transformation T is given by T(x) = Ax, where A is the matrix representation of T. So, we need to solve the equation Ax = b for some vector x.

Using the augmented matrix [A|b], we get:[ 1 -2 0 3 2 1 | 7 ][ 2 1 2 -1 1 2 | 0 ][ 0 0 1 0 2 1 | 3 ]. Row-reducing the augmented matrix, we get: [ 1 0 0 0 -1 1 | 7 ][ 0 1 0 0 3 -3 | -14 ] [ 0 0 1 0 2 1 | 3 ].

This tells us that the system Ax = b has a solution, namely x = [7 -14 3 0 0 0]T. Therefore, the vector b = [7 0 3 3 3 3]T is in the range of the transformation T defined by the matrix A.

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Complete question

Determine if the given vector is in the range of t(x) = ax where a = [1 -2 0 3 2 1] y = [-3 6] y = [1 -4] y = [2 7]

A vehicle factory manufactures cars. The unit cost C (the cost in dollars to make each car) depends on the number of cars made. If cars are made, then the unit cost is given by the function C(x)=0.9x^2 - 414x + 59,366. What is the minimum unit cost? Do not round your answer.

Answers

The minimum unit cost of the car is $12232.1

How to determine the minimum unit cost?

Given: cost function C(x)=0.9x² - 414x + 59,366

In order to find the minimum point, take the derivative of the cost function:

dC/dx = 2x - 414          (equate to zero and solve for x)

2x - 414 = 0                

2x = 414

x = 414/2 = 207

Thus, the minimum point is at x =207

The minimum unit cost can be obtained by substituting x = 207 into the cost equation:

C(x) = 0.9x² - 414x + 59,366

C(207) = 0.9(207)² - 414(207) +  59,366

          = 38564.1 - 85698 + 59,366

           = $12232.1

Therefore, the minimum unit cost is $12232.1

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Define power of 10 rule

Answers

Answer:

A power of 10 is as many number 10s as indicated by the exponent multiplied together. ... Thus, shown in long form, a power of 10 is the number 1 followed by n zeros, where n is the exponent and is greater than 0; for example, 106 is written 1,000,000.

Step-by-step explanation:

Answer:

Power of 10, in mathematics, any of the whole-valued (integer) exponents of the number 10. A power of 10 is as many number 10s as indicated by the exponent multiplied together. ... When n is less than 0, the power of 10 is the number 1 n places after the decimal point; for example, 10−2 is written 0.01.

Step-by-step explanation:

Power of 10, in mathematics, any of the whole-valued (integer) exponents of the number 10. A power of 10 is as many number 10s as indicated by the exponent multiplied together. ... When n is less than 0, the power of 10 is the number 1 n places after the decimal point; for example, 10−2 is written 0.01.

Power of 10, in mathematics, any of the whole-valued (integer) exponents of the number 10. A power of 10 is as many number 10 sec as indicated by the exponent multiplied together. ... When n is less than 0, the power of 10 is the number 1 n places after the decimal point; for example, 10−2 is written 0.01.

you want to know the percentage of utility companies that earned revenue between 41 million and 99 million dollars. if the mean revenue was 70 million dollars and the data has a standard deviation of 18 million, find the percentage. assume that the distribution is normal. round your answer to the nearest hundredth.

Answers

Approximately 89.26% of utility companies have revenue between 41 million and 99 million dollars. We need to use the normal distribution formula and find the z-scores for the given values.

First, we need to find the z-score for the lower limit of the range (41 million dollars):  z = (41 - 70) / 18 = -1.61
Next, we need to find the z-score for the upper limit of the range (99 million dollars): z = (99 - 70) / 18 = 1.61
We can now use a standard normal distribution table or a calculator to find the area under the curve between these two z-scores. The area between -1.61 and 1.61 is approximately 0.9044. This means that approximately 90.44% of utility companies earned revenue between 41 million and 99 million dollars.


To find the percentage of utility companies with revenue between 41 million and 99 million dollars, we can use the z-score formula and the standard normal distribution table. The z-score formula is: (X - mean) / standard deviation. First, we'll calculate the z-scores for both 41 million and 99 million dollars: Z1 = (41 million - 70 million) / 18 million = -29 / 18 ≈ -1.61
Z2 = (99 million - 70 million) / 18 million = 29 / 18 ≈ 1.61
Now, we'll look up the z-scores in the standard normal distribution table to find the corresponding percentage values.
For Z1 = -1.61, the table value is approximately 0.0537, or 5.37%.
For Z2 = 1.61, the table value is approximately 0.9463, or 94.63%.
Percentage = 94.63% - 5.37% = 89.26%

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A triangle has one angle with measure 15 degrees and another angle with measure 35 degrees. what is the measure of the third angle of the triangle, in degrees? a. 50 b. 70 c. 130 d. 30 e. 40

Answers

Option (c), A triangle has two angles, one that is 15 degrees and the other is 35 degrees. The measure of the third angle of the triangle, in degrees is 130.

Given, a triangle has one angle with measure 15 degrees and another angle with measure 35 degrees.

To find: The measure of the third angle of the triangle, in degrees.

The sum of all three angles of a triangle is 180 degrees.

Let's consider the measure of the third angle of the triangle be x.

Thus, sum of all three angles of a triangle

= 15+35+x

= 50 + x degree

Now, according to the property of a triangle, the sum of all three angles of a triangle is 180 degrees.

Thus, 50 + x = 180 degrees

Subtracting 50 from both the sides,

x = 180 - 50 = 130 degrees

Hence, the measure of the third angle of the triangle, in degrees is 130.

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(6 + 7) + 8 = 21

Choose an equivalent addition sentence that shows the associative property of addition.

Answers

Answer:

6 + (7 + 8) = 21

Step-by-step explanation:

the associative property of addition is: (a + b) + c = a + (b + c)

So the equivalent addition sentence would be:

(6 + 7) + 8 = 6 + (7 + 8) as they both equal 21

What conversion factors would be used to convert grams per day to kilograms per week

Answers

Answer:

(1 kg)/(1000 g)(7 day)/(1 wk)

Step-by-step explanation:

You want to know the conversion factors that would be used to convert grams per day to kilograms per week.

Conversion factors

A conversion factor is often expressed as a ratio that has its numerator equal to its denominator, but expressed with different units. The units of a conversion factor are generally chosen to be useful for the purpose of converting units you have to units you want.

Application

Here, we have mass units of grams, and we want mass units of kilograms. We know that ...

  1000 g = 1 kg

so a suitable conversion factor is ...

  (1 kg)/(1000 g)

The units we have (g) are put in the denominator, so that multiplying by this factor will cancel units of grams, leaving units of kilograms.

We have time units of days, and we want time units of weeks. The time units are in the denominator of the original value (g/day). We know that ...

  7 day = 1 wk

so a suitable conversion factor is ...

  (7 day)/(1 wk)

The time unit of day in the numerator will cancel the time unit of day in the original value, leaving time units of weeks in the denominator.

\(1\dfrac{\text{g}}{\text{day}}=1\dfrac{\text{g}}{\text{day}}\cdot\boxed{\dfrac{1\text{ kg}}{1000\text{ g}}\cdot\dfrac{7\text{ day}}{1\text{ wk}}}=\dfrac{7\text{ kg}}{1000\text{ wk}}=0.007\text{ kg/wk}\)

A rainstorm produced 2 inches of rain per hour. How many hours would it take to get a rainfall amount of one foot?

Answers

Answer:

6 hours

Step-by-step explanation:

2 x 6 = 12 inches

so it would take 6 hours

There are 5 positions available in the new school. Of the applicant, 12 are men and 8 are women. In how many ways can 3 men and 2 women be chosen if they are equally considered?

Answers

There are 3080 ways 3 men and 2 women can be chosen if they are equally considered, using the multiplication principle of counting

What is the multiplication principle of counting

The multiplication principle states that if there are m ways to perform one task and n ways to perform another task, then there are m x n ways to perform both tasks together.

To find the number of ways to choose 3 men from the 12 men, we can use the formula for combination, which is: ⁿCᵣ = n! / (r! (n-r)!).

where n is the total number of men and r is the number of men chosen

so, the number of ways to choose 3 men from the 12 men = ¹²C₃ = 1.

Similarly, we evaluate the number of ways to choose 2 women from the 8 women

as = ⁸C₂ = 14

Now, using the multiplication principle, we can find the total number of ways 3 men and 2 women be chosen if they are equally considered.

220 x 14 = 3080

Therefore, there are 3080 ways 3 men and 2 women can be chosen if they are equally considered, using the multiplication principle of counting

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what expression is equivalant to -2-7

Answers

The expression which is equivalent to -2 -7 according to the task content is; -9.

Equivalent expressions

It follows from the task content that the expression which is equivalent to the given expression; -2-7 is to be determined.

Therefore, it follows that the expression can be evaluated algebraically as follows;

-2 - 7

= -1 (2 + 7) by the distributive property converse.

= -1 (9)

= -9.

Therefore, the required expression which is equivalent to the given expression; -2 -7 is; -9.

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What can you say about a sample mean or a sample proportion being about 2 ses away from the population mean or the true proportion? what can you not say?

Answers

When we have a normal model for the sampling distribution, we cannot say that a sample mean or sample proportion is approximately 2 standard errors (ses) away from the population mean or the true proportion.

Instead, we can say that 95% of the sample proportions fall within two standard errors of the population proportion. Similar to this, the percentage of sample proportions decreases as the standard error distance decreases and increases as the standard error distance increases.

Therefore, the standard error distance will be greater than 2 standard errors (ses) if 99% of the sample proportions are within a given standard error distance of the population proportion.

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find the value of -12 + 8 -(-9)

find the value of -12 + 8 -(-9)

Answers

-12 + 8 - (-9) = 5
Therefore the value of -12 + 8 - (-9) is 5.

Answer: D: 5

Step-by-step explanation:

-12 + 8 - (-9)

First you want to do 8 - (-9) which becomes 17

Then add -12 to 17 which equals 5

What is 27 divide by eight

Answers

Answer

      = 3. 3 75

Step-by-step explanation:

 27 / 3 = 3.375

Your answer is 3.375!
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The angle of elevation from a point 25 feet from the base of a tree on level ground to the top of the tree is 30°. Find the height of the tree.

A. 39.1 ft

B. 43.3 ft

C. 14.4 ft

D. 17.7 ft

The angle of elevation from a point 25 feet from the base of a tree on level ground to the top of the

Answers

The answer is C it won’t let me do all the steps but I hope it helps
Hello There!!

Given

\( \text{Length of the tree= 25 feet} \\ \text{Angle of Elevation= 30°}\)

To Find

\( \text{Height Of The Tree}\)

\( \text{Let The Height of the Tree be ‘x’}\)

We Know:-

\( \tan\theta = \frac{ \text{perpendicular}}{ \text{base} }\)

\( \tan \theta = \frac{ \text{height of the tree}}{ \text{length of the base}} \\ \\ \implies \tan30 = \frac{ \text{x}}{25} \\ \\ \implies \text{x }= 25 \times \tan30 \\ \\ \implies \text{x }= 25 \times 0.577 \\ \\ \implies \text{x} = 14.425 \\ \\ \implies \text{x} = 14.43\)

\( \therefore \text{height of the tree(\text{x})} = \red{14.43 \: \text{feet}}\)

Option C is the correct answer

Hope This Helps!!

Which of the following shows the correct first step to solve x^2-18x=-45

A x^2 - 18x + 18= -45 + 18
B. x^2 - 18x + 9 = -45 + 18
C. x^2 -18x + 81 = -45
D. X^2 -18 + 81 = -45 + 81

Answers

Answer:

D, X^2 -18 + 81 = -45 + 81

Step-by-step explanation:

it is complating squer method that used for solving x in a quadratic equetion . in this step you will add

(y/2)^2 if y is the cofitient of x .

Which of the following factors does NOT control the stability of a slope?
the angle of repose for intact bedrock
whether the slope is rock or soil
the amount of water in the soil
the orientation of fractures, cleavage, and bedding

Answers

The factor that does NOT control the stability of a slope is the angle of repose for intact bedrock. The angle of repose refers to the steepest angle at which a pile of loose material remains stable without sliding. It is mainly applicable to loose materials like soil and granular substances, not intact bedrock.

Bedrock stability depends on factors such as its strength, fracturing, and geological properties, rather than the angle of repose. Factors that control the stability of a slope include whether the slope is rock or soil. Rock slopes tend to be more stable than soil slopes due to the cohesive nature of intact rock.

The amount of water in the soil also affects slope stability, as excessive water can increase pore pressure and reduce the shear strength of the soil, leading to slope failure. Additionally, the orientation of fractures, cleavage, and bedding in the rock can influence slope stability by creating planes of weakness or strength.

To summarize, while the angle of repose is a significant factor in slope stability, it is not applicable to intact bedrock. The stability of a slope is influenced by the type of material (rock or soil), the presence of water, and the orientation of fractures and bedding.

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when ratios have the same units we call that ratio _____________

Answers

Therefore, When ratios have the same units, we call that ratio a unit ratio.

When ratios have the same units, we call that ratio a unit ratio. An explanation of what is meant by the term ratio would be helpful. A ratio compares the relative sizes of two or more quantities. If the numbers being compared have the same units, then it is called a unit ratio. Unit ratios are ratios that relate to each other in terms of the same measurement unit. They are also sometimes known as unit rates or unit fractions. They are frequently used to compare the relative size of one quantity to another when they are measured in the same units. For example, when we say that there are two apples for every five oranges, we are expressing a ratio between apples and oranges. However, when we say that there are two apples per five oranges, we are using a unit ratio.

Therefore, When ratios have the same units, we call that ratio a unit ratio.

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Solve each equation.
log 9- log x+1=6

Answers

After solving the given equation log9-log x+1=6, the value of x=(-9log+5)/log.

What is an equation?A mathematical statement made up of two expressions joined by an equal sign is called an equation.An equation is 3x - 5 = 16.We obtain the value of the variable x as x = 7 after solving this equation.

So,

Given equation: log9-log x+1=6

Then,

log9-logx+1=69log-logx+1=6(9log-logx+1)+(-9log-1)=6+(-9log-1)x=(-9log+5)/log

Therefore, after solving the given equation log9-log x+1=6, the value of x=(-9log+5)/log.

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Consider the following. x = sin(6t), y = -cos(6t), z = 18t; (0, 1, 3 pi) Find the equation of the normal plane of the curve at the given point. Find the equation of the osculating plane of the curve at the given point.

Answers

The equation of the normal plane of the curve at the point (0, 1, 3π) is -x + 6z - 18π = 0.

To find the normal plane of the curve, we first need to find the normal vector. The normal vector is the cross product of the tangent vectors, which is given by T×T', where T is the unit tangent vector and T' is the derivative of T with respect to t. The unit tangent vector is given by T = (6cos(6t), 6sin(6t), 18), and the derivative of T with respect to t is T' = (-36sin(6t), 36cos(6t), 0). Evaluating these at t = 3π, we get T = (0, -6, 18) and T' = (36, 0, 0). Taking the cross product of T and T', we get the normal vector N = (-108, -648, 0), which simplifies to N = (-2, -12, 0).

Next, we use the point-normal form of the plane equation to find the equation of the normal plane. The point-normal form is given by N·(P - P0) = 0, where N is the normal vector, P is a point on the plane, and P0 is the given point. Substituting the values, we get (-2, -12, 0)·(x - 0, y - 1, z - 3π) = 0, which simplifies to -x + 6z - 18π = 0.

The equation of the osculating plane of the curve at the point (0, 1, 3π) is 6x - y - 12z + 6π = 0.

To find the osculating plane of the curve, we need to find the normal vector and the binormal vector. The normal vector was already found in the previous step, which is N = (-2, -12, 0). The binormal vector is given by B = T×N, where T is the unit tangent vector. Evaluating T at t = 3π, we get T = (0, -6, 18). Taking the cross product of T and N, we get B = (12, -2, 72), which simplifies to B = (6, -1, 36).

Finally, we use the point-normal form of the plane equation to find the equation of the osculating plane. The point-normal form is given by N·(P - P0) = 0, where N is the normal vector, P is a point on the plane, and P0 is the given point. Since the osculating plane passes through the given point, we can take P0 = (0, 1, 3π). Substituting the values, we get (-2, -12, 0)·(x - 0, y - 1, z - 3π) = 0, which simplifies to 6x - y - 12z + 6π = 0.

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