a) The electrical potential V (center) at the center C of the circle due to the charges Q1 and Q2 is -2.30 V.
b) The total electrical potential at the point P due to charge Q1 and Q2 , located at a distance of 6.71 cm is -1.78V.
Given that:
Radius of the circle, r = 8.20cm
Charge distributed along one-quarter of the circumference of the circle,
Q1 = +4.20pC
Charge distributed along 3/4th of the circumference of the circle,
Q2 = 25.20pC
Distance of the point from the center, d = 6.71cm
The electric potential at infinity is. V = 0
Using the concept of potential at a point on a thin rod, we can obtain the individual potential due to each charge. The sum of these values can now be used to obtain the desired potential value at the center and at the point, taking into account the distance to the point.
Formula:
The potential due to a point charge at a distance r from the point charge is determined by the equation V = [tex]\frac{1}{4\pi E_0} * \frac{q}{r}[/tex] ---------------- (1)
The potential due to the collection of point charges is determined by formula:
V = ∑[tex]\frac{1}{4\pi E_0} * \frac{q}{r}[/tex] -------------------- (2)
(a) The potential VQ1 at the center C of the circle due to the charge Q1 is determined using equation (i) as:
[tex]V_Q_1 = \frac{1}{4\pi E_0} * \frac{Q_1}{R}[/tex] --------------------- (3)
Here, R is the radius of circle
The potential VQ1 at the center C of the circle due to the charge Q1 is determined using equation (i) as:
[tex]V_Q_1 = \frac{1}{4\pi E_0} * \frac{Q_2}{R}[/tex] ----------------------- (4)
Now the potential V(center) at the center C of the circle due to charges Q1 and Q2 is the sum of the potential due to charge Q1 and the potential due to charge Q2. This is given using equations (a) and (b) in equation (ii) as:
[tex]V_Center = \frac{1}{4\pi E_0} * \frac{Q_1}{R} + \frac{1}{4\pi E_0} * \frac{Q_2}{R}[/tex]
⇒ [tex]V_Center = \frac{1}{4\pi E_0} *( \frac{Q_1}{R} + \frac{Q_2}{R})[/tex]
⇒ 9.0 × 10⁹Nm²/C₂ ([tex]\frac{4.20*10^-12 C}{0.082m} +\frac{-25.2*10^-12C}{0.082m}[/tex])
⇒ -2.30V
The electrical potential V (center) at the center C of the circle due to the charges Q1 and Q2 is -2.30 V.
(b) From the above figure the r between each charged particle and the point P is given as:
[tex]r = \sqrt{R^{2}+D^{2} }[/tex]
In the figure above, r between each charged particle and point P is defined as: ="1662703245948 "
[tex]V_Q_1 = \frac{1}{4\pi E_0} * \frac{Q_1}{\sqrt{R^{2} + D^{2} } }[/tex] ------------------ (5)
Again, the electric potential at point P due to charge Q2 is given using equation (i) as follows:
[tex]V_Q_1 = \frac{1}{4\pi E_0} * \frac{Q_2}{\sqrt{R^{2} + D^{2} } }[/tex] -------------------- (6)
A The potential at point P due to charge Q2 is determined using equation (i): The potential of charge Q1 and the potential of charge Q2. This is given using equations (c) and (d) in equation (ii) as:
[tex]V_P = 9.0 * 10^9Nm^2/C^2 (\frac{4.20*10^-12C-6(4.20*10^-12C)}{\sqrt{(0.082m)^{2} + (0.082m)^{2} } })[/tex]
= -1.78V
The total electrical potential at the point P due to charge Q1 and Q2 , located at a distance of 6.71 cm is -1.78V.
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what is the number of the month august
August is the eighth month of the year. Depending on the region and culture, August is known for a variety of historical and cultural events, as well as natural occurrences.
August is noted in the United States for National Women's Equality Day (August 26th), which honors the enactment of the 19th Amendment to the United States Constitution, which granted women the right to vote.
The Perseid meteor shower, which happens yearly between late July and late August and is visible from many parts of the world, is also celebrated in August.
August is connected with the harvest season in several cultures and is commemorated with numerous festivals and rituals.
August is also a popular month for vacations and travel, particularly in the Northern Hemisphere, where it corresponds with the summer season.
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Evaluate: E^10n=1 8(1/4)^n-1=[?]
Therefore, the value of the expression E¹⁰ⁿ=1 8(1/4)ⁿ⁻¹ is 27.
What is fraction?A fraction is a way of representing a part of a whole or a ratio between two quantities. It is written as two numbers, one above the other, with a horizontal line separating them. The number above the line is called the numerator and the number below the line is called the denominator. For example, the fraction 3/4 represents three parts out of four equal parts, or the ratio of three to four. Fractions can be used to represent a wide range of values, including numbers between whole numbers (such as 1/2 or 3/5), ratios between quantities (such as 2/3 cups of flour to 1 cup of sugar), and percentages (such as 75/100, which is equivalent to 75%, or three-quarters). Fractions are an important concept in mathematics and are used in many different contexts, such as in cooking, measurement, and finance.
Here,
The expression E¹⁰ⁿ=1 8(1/4)ⁿ⁻¹ means we need to evaluate the sum of the expression 8(1/4)ⁿ⁻¹, as n ranges from 1 to 10.
To do this, we can simply substitute each value of n into the expression and add up the results. Here are the first few terms of the sum:
n = 1: 8(1/4)⁰ = 8
n = 2: 8(1/4)¹ = 2
n = 3: 8(1/4)² = 1/2
n = 4: 8(1/4)³= 1/8
We can continue this pattern for the remaining values of n, and then add up all the terms:
8 + 2 + 1/2 + 1/8 + ... + (1/4)⁹
To simplify this expression, we can use the formula for the sum of a geometric series:
S = a(1 - rⁿ) / (1 - r)
where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.
In this case, the first term is 8, the common ratio is 1/4, and there are 10 terms. So we can substitute these values into the formula to get:
S = 8(1 - (1/4)¹⁰) / (1 - 1/4)
S = 8(1 - 1/1048576) / (3/4)
S = 8(1048575/1048576) * (4/3)
S = 8 * (3495250/1048576)
S = 27
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Look at the picture. Write and solve an equation to model the picture. Show your answer as a multiplication equation with 1/2 as a factor. (6 pears)
Equation for 6 halves of pears is 1/2 x 6 = 3
What is factor?The factor of a number is a number that divides the given number completely without any remainder. The factors of a number can be positive or negative.
Every factor is divisor of that number but every divisor is not a factor of that number.
Given, in the picture,
half of 6 pears
One half of pear can be denoted as = 1/2
Now,
equation of 6 half pears.
= 1/2 x 6
= 4
Hence, equation for 6 half pears can be written as 1/2 × 6 = 3.
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what is polar to rectangular calculator equation
A polar to rectangular calculator equation is an online tool that converts polar coordinates (r, θ) to rectangular coordinates (x, y).
A polar to rectangular calculator equation is a math tool used to convert polar coordinates to rectangular coordinates. Polar coordinates are expressed in terms of a radius r and an angle θ, while rectangular coordinates are expressed in terms of x and y.
To use the calculator, users input the radius and angle in degrees or radians, and the calculator uses the formulas x = r cos(θ) and y = r sin(θ) to calculate the corresponding rectangular coordinates. This tool is commonly used in physics, engineering, and other applications that involve two-dimensional coordinates. It is also useful in graphing polar functions, such as polar curves or polar graphs.
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Turner drew a scale drawing of a city. The scale of the drawing was 5 inches = 1 yard. What is the scale factor of the drawing?
The scale factor of the drawing is 5:36.The scale factor is the amount by which a shape is increased or shrunk.
What does a scale factor look like?Scale factor is the proportion of an original object's scale to a new one that is an representation of it that is a different size (bigger or smaller). For instance, by increasing each side by, say, 2, we can expand overall size of the a rectangle having lengths of 4 cm and 8 cm. The scale discrepancy between two things is contrasted by a scale factor, which is a ratio. You can figure out the scale factor by selecting two comparable or related components on the two items you are comparing. By comparing every one of these two elements, you may determine a ratio.
When we find the scale factor , we obtain:
5 inches = 1 yard
1 inches=[tex]\frac{1}{5}[/tex] yard.
actual , 1 inche = [tex]\frac{1}{36}[/tex] yard
Scale factor= [tex]\frac{\frac{1}{36} }{\frac{1}{5} }[/tex]= [tex]\frac{5}{36}[/tex].
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Exit
Which tool would you use to measure the amount of oil required for a food recipe?
A. measuring spoon
B. ruler
C. hanging scale
D. platform scale
Answer:
Measuring Spoon
Step-by-step explanation:
The tool you would use is a measuring spoon.
Answer:
Measuring spoon
Step-by-step explanation:
Sydney read
50
50 pages in
40
40 minutes. She reads at a constant rate.
How many pages can Sydney read per minute?
The table of values represents a function f(x) How much greater is the average ra
The average rate of change over the interval [9,10] is greater by 6490.
The average rate of change of a polynomial is the slope of the function.
The average rate of change over the interval [7, 9] is the slope between ( 7, f(7) ) and ( 9, f(9) )
where f(7) and f(9) is the value of the function at 7 and 9.
So, the slope between ( 7, 1852 ) and ( 9, 3878 )
= (3878-1852)/(9-7)
= 2026/2
= 1013
Similarly,
The average rate of change over the interval [4, 6]
i.e. slope between ( 4, f(4) ) and ( 6, f(6) )
So, the slope between ( 4, 358 ) and ( 6, 1178 )
= (1178-358)/(6-4)
= 820/2
= 410
and
1013-410 = 603
So, the polynomial is 603 greater than the average rate of change over the interval [7, 9] than the interval [4, 6]
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area is determined by what type of measurement?
Area is calculated using length measurements, specifically measurements of two dimensions of a flat or planar shape, such as length and width. The resulting value represents the amount of space occupied by the shape on a flat surface.
Area is commonly measured in square units such as square metres ([tex]m^{2}[/tex]), square feet ([tex]ft^{2}[/tex]), square inches ([tex]in^{2}[/tex]), and so on.
To calculate the area of a flat shape, multiply its length and width by the appropriate unit of measurement.
Area = Length x Width
Similarly, to find the area of a circular surface, we would use the formula for the area of a circle:
Area = π x (Radius)^2
[tex]Area = \pi . (r)^2[/tex]
where π is the mathematical constant pi, and the radius is half the diameter:
So, area is a measure of the amount of space occupied by a flat or planar shape and is calculated using length measurements.
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25
An expression is shown below.
2/3
3a + 3b + 8
Place a check mark next to each equivalent expression.
A. O 3a + 3b + (8 + 4) − 4
B. O −(−¾a + ½¾⁄b + 8)
c.
(a + b + 8)
(3a + 3b + 8) × ²1/1
E.O a + b + 0 × c + 8
D. 6 x
Answer: To determine which expressions are equivalent to the given expression, we need to simplify the given expression first:
2/3 (3a + 3b + 8) = 2a + 2b + 16/3
Now we can check which expressions are equivalent to this simplified expression:
A. 3a + 3b + (8 + 4) − 4 = 3a + 3b + 8, not equivalent
B. −(−¾a + ½¾⁄b + 8) = 3/4 a - 3/2 b - 8, not equivalent
C. (a + b + 8) (3a + 3b + 8) × ²1/1 = (3a^2 + 9ab + 24a + 3b^2 + 24b + 64)/1, not equivalent
D. 6x = 6x, not equivalent
E. a + b + 0 × c + 8 = a + b + 8, not equivalent
None of the expressions is equivalent to the given expression 2/3 (3a + 3b + 8).
Step-by-step explanation:
What is a limit that does not exist?
A limit that does not exist is a limit that approaches infinity or negative infinity. The formula is [tex]\lim_{x \to \ > 2}+ f_(x)[/tex].
If the graph has any gap at the x value c, then two sided limits at that point will not exist. If the graph has vertical asymptote and one side of the asymptote goes towards the infinity and other goes towards negative infinity, then the "limit does not exist".
A limit that does not exist is when function does'nt approach a specific value as it approaches a certain point. This is usually due to the fact that the function is discontinuous or that it oscillates between two or more values.
This type of limit is known as an "infinite limit" or an "oscillatory limit" since the function does not converge or diverge, but instead fluctuates or oscillates between values. In this case, the limit does not exist because the function does not approach a single value as it reaches the point in question.
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bryan is performing data entry for an insurance company by taking handwritten records and entering them into the database. thus far, 15.6% of the handwritten records have had missing data that prevented them from being entered into the database. if bryan takes 10 handwritten records, what is the probability that at most three records will be incomplete? use a ti-83, ti-83 plus, or ti-84 calculator to find the probability. round your answer to three decimal places.
The probability that at most three records will be incomplete is approximately 0.650 (rounded to three decimal places).
What is Binomial Distribution ?
The binomial distribution is the discrete probability distribution used in probability theory and statistics that only allows for Success or Failure as the possible outcomes of an experiment.
This is a binomial distribution problem with n=10 and p=0.156. Let X be the number of incomplete records. We need to find P(X ≤ 3).
Using a TI-83, TI-83 Plus, or TI-84 calculator, we can use the binomial cumulative distribution function (binomcdf) to find this probability:
binomcdf(10, 0.156, 3) ≈ 0.650
Therefore, the probability that at most three records will be incomplete is approximately 0.650 (rounded to three decimal places).
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What is the unit rate for 30 millimeters every 4 hours
Answer:
30 X 4=yhhvvihddthjgyujjnbbbvvvukgdddsxgjooi
what is the difference between the desired value of a variable and the controlled (actual) value?
The desired value is the target or goal for the variable, while the controlled value is the actual or measured value.
The difference between the desired value of a variable and the controlled (actual) value is known as the error or deviation. It addresses the sum by which the actual value varies from the desired value, and is normally utilized as criticism in control frameworks to change the framework yield.
The objective of a control framework is to limit this mistake and bring the actual value as close as conceivable to the desired value. The mistake can be positive (assuming the actual value is more noteworthy than the desired value), negative (in the event that the actual value is not exactly the desired value), or zero (in the event that the actual value equivalents the desired value).
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select the correct answer.
a circle has a radius of 6 inches and central angle of 60°. what is the measure of the arc length associated with this angle?
3pi
2pi
pi
6pi inches
Answer:
[tex]2\pi[/tex] inches
Option B
Step-by-step explanation:
The arc length is proportional to the central angle
If L is the arc length and θ is the central angle in degrees
[tex]L = \dfrac{\theta}{360} \times C[/tex]
where C is the circumference of the circle
Given radius r = 6 inches, C = 2πr = 2π·6 = 12π
θ = 60°
So
[tex]L = \dfrac{60}{360} \times 12\pi\\\\= \dfrac{1}{6} \times 12\pi\\\\= 2\pi\;inches[/tex]
i need help with this question
Answer:
Below
Step-by-step explanation:
Volume of a cone = 1/3 pi r^2 h
4125 pi = 1/3 pi r^2 h
4125 = 1/3 r^2 h <===if diameter =30 then radius = 15 units
4125 = 1/3 (15)^2 h
4125 / (1/3 * 15^2) = h = 55 units
Given the figure below find the values of x and z.
(ANY IMPROPER ANSWERS WILL BE REPORTED)
Answer:
x = 14°
z = 74°
Step-by-step explanation:
10x - 34° = 106° ( A.I.A )
x = 72°
106° + z = 180° ( linear pair axiom )
z = 74°
MARK ME AS BRAINILIST
Answer:
x=14°
z= 74°
Step-by-step explanation:
Here
(10x-34)°= 106°
since the Vertically Opposite Angle are equal.
so solving for it.
First, we can add 34° to both sides of the equation to get:
10x° - 34°+ 34° = 106° + 34°
Simplifying the left side gives:
10x° = 140°
Then we can divide both sides of the equation by 10 to get:
x = 14°
Again
z+106°=180°
since the sum of angle in the linear pair is equal to 180°
solving for it
we can subtract 106° from both sides of the equation to get:
z + 106° - 106° = 180° - 106°
Simplifying the left side gives:
z = 74°
A random sample of 240 doctors revealed that 108 are satisfied with the current state of US health care.
The conditions for inference are met.
A 90% confidence interval for the true proportion of all US doctors who are satisfied with the current state of US health care is (0.397, 0.503).
Interpret this interval.
90% of all doctors are satisfied with the current state of US health care.
Between 39.7% and 50.3% of the sample of doctors are satisfied with the current state of US health care.
We are 90% confident that the interval from 0.397 to 0.503 captures p = the true proportion of all US doctors who are satisfied with the current state of US health care.
A majority of the doctors in the sample are satisfied with the current state of US health care.
answer is C
The correct interpretation of the given confidence interval is: We are 90% confident that the interval from 0.397 to 0.503 captures the true proportion (p) of all US doctors who are satisfied with the current state of US health care.
What is confidence interval?In statistics, a confidence interval is a range of values that is likely to contain an unknown population parameter, such as a population mean or proportion, based on a sample from that population. A confidence interval is constructed with a particular level of confidence, which represents the probability that the interval will contain the true population parameter. The confidence level is typically expressed as a percentage, such as 90%, 95%, or 99%.
Here,
This means that if we were to repeat the process of taking a random sample of 240 US doctors and calculating a 90% confidence interval for the proportion of doctors who are satisfied with the current state of US health care, 90% of those intervals would contain the true proportion.
It is not correct to say that 90% of all doctors are satisfied with the current state of US health care, because the confidence interval only provides information about the range of plausible values for the true proportion, not the exact value.
It is also not correct to say that a majority of the doctors in the sample are satisfied with the current state of US health care, because the confidence interval includes values both above and below 0.5, which means that we cannot conclude whether a majority or a minority of doctors are satisfied with the current state of US health care.
Therefore, the correct answer is: We are 90% confident that the interval from 0.397 to 0.503 captures the true proportion of all US doctors who are satisfied with the current state of US health care.
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Answer:
Step-by-step explanation:
what is 67 kg in pounds?
The value of weight of 67 kg when converted to pounds is 147.70954 pounds.
kilogram is the standard unit of mass in the International System of Units (SI), while pounds are a common unit of mass used in the United States and other countries.
To Know how to convert between different units of measurement is essential for accurate calculations and comparisons. convert 67kg to pounds, you can use the formula: 1 kilogram is equal to 2.20462 pounds.
Therefore, to convert 67kg to pounds, you would multiply 64 by 2.20462.
67 x 2.20462 = 147.70954 pounds
So, weight of 64kg is equivalent to 141.09568 pounds.
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According to the Empirical Rule, approximately _____% of the data in a normal distribution will fall within ±1 standard deviation of the mean.a. 34b. 68c. 95d. 99.7
According to the Empirical Rule, approximately 68% of the data in a normal distribution will fall within ±1 standard deviation of the mean. The correct option is (d) 68%
The Empirical Rule is a statistical rule of thumb that describes the approximate proportions of data that fall within a certain number of standard deviations from the mean in a normal distribution.
The Empirical Rule states that for a normal distribution:
About 68% of the data will fall within 1 standard deviation of the mean
About 95% of the data will fall within 2 standard deviations of the mean
About 99.7% of the data will fall within 3 standard deviations of the mean
Therefore, option (b) 68 is the correct answer.
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A therapist wanted to determine if yoga or meditation is better for relieving stress. The therapist recruited 100 of her high-stress patients. Fifty of them were randomly assigned to take weekly yoga classes, and the other 50 were assigned weekly meditation classes. After one month, 30 of the 50 patients in the yoga group reported less stress, and 35 of the 50 patients in the meditation group reported less stress. Assuming the conditions for inference are met, what is the 95% confidence interval for the difference in proportions of patients experiencing stress relief from the yoga and meditation groups?
Find the z-table here.
(0.40 minus 0.30) plus-or-minus 1.65 StartRoot StartFraction 0.40 (1 minus 0.40) Over 100 EndFraction + StartFraction 0.30 (1 minus 0.30) Over 100 EndFraction EndRoot
(0.60 minus 0.70) plus-or-minus 1.96 StartRoot StartFraction 0.60 (1 minus 0.60) Over 100 EndFraction + StartFraction 0.70 (1 minus 0.70) Over 100 EndFraction EndRoot
(0.60 minus 0.70) plus-or-minus 1.65 StartRoot StartFraction 0.60 (1 minus 0.60) Over 50 EndFraction + StartFraction 0.70 (1 minus 0.70) Over 50 EndFraction EndRoot
(0.60 minus 0.70) plus-or-minus 1.96 StartRoot StartFraction 0.60 (1 minus 0.60) Over 50 EndFraction + StartFraction 0.70 (1 minus 0.70) Over 50 EndFraction EndRoot
answer is D
The correct answer to this question is option D: (0.60 minus 0.70) plus-or-minus 1.96 StartRoot StartFraction 0.60 (1 minus 0.60) Over 50 EndFraction + StartFraction 0.70 (1 minus 0.70) Over 50 EndFraction EndRoot.
To find the 95% confidence interval for the difference in proportions of patients experiencing stress relief from the yoga and meditation groups, we need to use the formula:
CI = (p1 - p2) ± z*√[(p1(1-p1)/n1) + (p2(1-p2)/n2)]
Where p1 and p2 are the proportions of patients experiencing stress relief in the yoga and meditation groups, respectively, n1 and n2 are the sample sizes of the yoga and meditation groups, respectively, and z is the critical value from the z-table corresponding to the 95% confidence level.
Plugging in the given values:
p1 = 30/50 = 0.60
p2 = 35/50 = 0.70
n1 = 50
n2 = 50
z = 1.96 (from the z-table for a 95% confidence level)
CI = (0.60 - 0.70) ± 1.96*√[(0.60(1-0.60)/50) + (0.70(1-0.70)/50)]
CI = (-0.10) ± 1.96*√[(0.24/50) + (0.21/50)]
CI = (-0.10) ± 1.96*√[0.009]
CI = (-0.10) ± 1.96*(0.095)
CI = (-0.10) ± 0.186
Therefore, the 95% confidence interval for the difference in proportions of patients experiencing stress relief from the yoga and meditation groups is (-0.286, 0.086).
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What is bell shape distribution?
The bell shaped distribution is the continuous probability distribution with a specific shape resembling a symmetrical
Bell-shaped distribution is a statistical term that describes a continuous probability distribution with a specific shape resembling a symmetrical, bell-shaped curve. This distribution is also known as the normal distribution or Gaussian distribution.
The bell-shaped distribution is characterized by its mean, which is the center of the curve, and its standard deviation, which measures the spread of the distribution. The curve is symmetric, meaning that the probabilities of values to the left and right of the mean are equal.
In a bell-shaped distribution, the majority of data points cluster around the mean, with fewer points as you move further away from the mean. The tails of the distribution extend infinitely in both directions, although the probability of observing values far from the mean becomes increasingly small.
The normal distribution is used in many fields, including science, engineering, and finance, to model real-world phenomena that are influenced by many small, random factors.
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What is the result of 4000 x 12 ?
Answer:
I'm positive the answer is 56000
The solution of equation 4000 x 12 is,
4000 x 12 = 48,000
We have to given that,
Expression to solve,
⇒ 4000 x 12
We can multiply it,
⇒ 4000 x 12
⇒ 48,000
Therefore, The solution of equation 4000 x 12 is,
4000 x 12 = 48,000
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An arch can be modeled by y=-9/50(x-2)(x-16). Where x and y are measured in meters. The x-axis represents the ground. Find the width in meters
Answer:
+x+=+3+
+y+=+-.9%2A%28+3+%2B+3+%29%2A%28+3+-+3+%29+
+y+=+-.9%2A6%2A0+
+y+=+0+
----------------------
The distance between ( -3, 0 ) and ( 3, 0 )
is +3+-%28-3%29+=+6+
The width of the arch at ground level is 6 ft
-----------------------
Here's the plot of the equation:
Step-by-step explanation:
what is 101 kg to lbs?
101 kg can be converted to into lbs by simply multiply the kg by 2.2046 so the value will be 222.67 lbs
Unit conversion is the process of converting the measurement of a given amount between various units, often by multiplicative conversion factors that alter the value of the measured quantity without altering its effects.
There are 2.204622622 pounds per kilogramme and 0.45359237 kilogrammes in a lbs. Hence, there are two ways to translate 101 kilogrammes into lbs. Either divide 101 by 0.45359237 or increase it by 2.204622622. The calculation is as follows: multiply 101 kg by 2.204622622 to obtain the answer.
101 divided by 2.204622622 equals 222.666884822 101 kilogram or 222.67 lbs.
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let f, g,∈ f(r, r) be the functions defined by f(t) = ert and g(t) = est, where r '= s. prove that f and g are linearly independent in f(r, r).
For the defined function f(t)=e^rt and g(t)=e^st where r is not equal to s it was proved that the f and g are linearly independent function in F(r,r).
For the defined function f and g we have,
f(t)=e^rt
g(t)=e^st
Where r is not equal to s.
To prove f and g are linearly independent function we need,
f( t) ± g(t) = 0
Substitute the value of f(t) and g(t) we get,
⇒ e^rt ± e^st = 0
⇒ e^rt ( 1 ± e^( s - r )t = 0
This implies ,
e^rt = 0 or ( 1 ± e^( s - r )t = 0
But e^rt is always greater than zero for all the values of r and t.
e^rt = 0 is not possible.
If ( 1 ± e^( s - r )t = 0
This is possible if and only if r = s .
Which is against the given condition.
Therefore, f(t) and g(t) are linearly independent function in f( r , r).
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The above question is incomplete, the complete question is:
Let f, g, element F(r,r) be the function defined by f(t)=e^rt and g(t)=e^st, where r cannot equal s. prove that f and g are linearly independent in F(r,r).
Which number should you multiply the first equation by
to eliminate the x-variable term?
O 1
O
-1
3
To eliminate the x-variable term from an equation, we need to multiply that equation by the opposite of the coefficient of the x-variable. If the first equation is of the form ax + by = c, then we need to multiply it by -a to eliminate the x-variable. Therefore, the number we should multiply the first equation by to eliminate the x-variable term is -3 if the first equation is of the form 3x + 2y = 7.
find the area of the surface generated when the given curve is revolved about the given axis.
The area of the surface generated when the given curve is revolved about the given axis 5x^1/3 is [tex]\frac{\pi }{675} [34\sqrt{34} - 125][/tex].
The curve x = f(y)
The area of the surface around the y-axis from y = a → y = b is:
[tex]\int\limits^a_b {2\pi x\sqrt{1+(\frac{dx}{dy})^2 } } \, dx[/tex]
From the given curve:
[tex]y = 5x^{1/3}[/tex] ; assuming the region bounded by the curve is 0 ≤ y ≤ 1
So;
[tex]y = 5x^{1/3}[/tex]
⇒ 5x = y³
⇒ x = 1/5 y³
The differential of the above equation Is:
dx/dy = 1/5 (3y²)
⇒ dx/dy = 3/5 (y²)
Now, we have the area of the surface produced around the curve [tex]y = 5x^{1/3}[/tex] through the y axis from the region y = 0 to y = 1
∴ [tex]\int\limits^1_0 {2\pi \frac{1}{5}y^3\sqrt{1+(\frac{3}{5} y^2)^2} } \, dx[/tex]
= [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{1+(\frac{9}{25} y^4)} } \, dy[/tex]
= [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{\frac{25+9y^4}{25})} } \, dy[/tex]
= [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{25+9} y^{4} )} } \, dy[/tex]
Let make u = [tex]\sqrt{25+9y^{4} }[/tex]
It implies that:
[tex]u^3 = (25+9y^4) \sqrt{25+9y^4}[/tex]
and, [tex]u = \sqrt{25+9y^{4} }[/tex]
⇒ [tex]du = \frac{1}{2\sqrt{25+9y^4} } (36y^3) dy[/tex]
⇒ [tex]y^3dy = \frac{1}{18} \sqrt{25+9y^{4} }du[/tex]
when y = 0 ;
[tex]u = \sqrt{25+9y^{4} }[/tex]
[tex]u = \sqrt{25+9{(0)^4} }[/tex]
[tex]u = \sqrt{25 }[/tex]
u = 5
when y = 1;
[tex]u = \sqrt{25+9{(1)^4} }[/tex]
[tex]u = \sqrt{34} }[/tex]
∴The equation [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{25+9} y^{4} )} } \, dy[/tex] can be written as:
= [tex]\frac{\pi }{225} [\frac{u^3}{3} ]\left \{ {{y=\sqrt{34} } \atop {x=5}} \right.[/tex]
= [tex]\frac{\pi }{675} [34\sqrt{34} - 125][/tex]
Therefore,
The area of the surface generated when the given curve is revolved about the given axis 5x^1/3 is [tex]\frac{\pi }{675} [34\sqrt{34} - 125][/tex].
Complete Question:
Find The area of the surface generated when the given curve is revolved about the given axis 5x^1/3.
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A person deposited 4000 lei at a bank with annual interest of 8%. To find out how much he has after one year
Answer:4320
Step-by-step explanation:8% interest =108% which is equivalent to x1.08
4000 x 1.08 = 4320
15. Prove inductively and deductively that the sum of six consecutive numbers is divisible by 3. (4
marks)
Deductive Reasoning
The required proof is inductively and deductively that the sum of six consecutive numbers is divisible by 3 in the solution part.
What is deductive reasoning?Deductive reasoning is a method of reasoning in which conclusions are drawn based on premises that are known or assumed to be true.
Let's assume that the six consecutive numbers are n, n+1, n+2, n+3, n+4, and n+5.
We have to show that the sum of these numbers is divisible by 3.
First, we can express the sum of the six numbers as follows:
n + (n+1) + (n+2) + (n+3) + (n+4) + (n+5)
Simplify the expression by combining like terms:
6n + 15
Next, factor out 3 from the expression:
6n + 15 = 3(2n + 5)
Since 2n + 5 is an integer, we can see that 3(2n + 5) is divisible by 3. Therefore, the sum of six consecutive numbers is divisible by 3.
This is an example of deductive reasoning because we started with a premise (the assumption that the six consecutive numbers are n, n+1, n+2, n+3, n+4, and n+5) and used logical steps to arrive at a conclusion (that the sum of the six numbers is divisible by 3).
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