If a customer purchases 3 of the products, the probability of getting at least one that is defective is 38.59%.
How to determine the probabilityIn order to determine the probability of getting at least one defective product if a customer purchases three products with a defective rate of 7.5%, we can use the concept of complementary probability.
The probability of getting at least one defective product can be calculated as the complement of the probability of getting none defective products.
So, the probability of getting no defective products is:
P(none defective) = (1 - 0.075)³ = 0.6141
Therefore, the probability of getting at least one defective product is:
P(at least one defective) = 1 - P(none defective) = 1 - 0.6141 = 0.3859 or 38.59%
.So, the probability of getting at least one that is defective is 38.59%.
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If you walk for 1. 5 hours and at 3mph then for the next 0. 5 hours how fast do you run for an average of 4 mph
In order to have an average speed of 4 mph for the entire trip, you would need to run at a speed of 7 mph for the next 0.5 hours after walking for 1.5 hours at 3 mph.
Let's begin by calculating the total distance covered during the entire trip, which is:
distance = (time walked) x (walking speed) + (time ran) x (running speed)
We know that the time walked is 1.5 hours and the walking speed is 3 mph, so the distance covered during walking is:
distance walked = (time walked) x (walking speed) = 1.5 x 3 = 4.5 miles
We also know that the average speed for the entire trip is 4 mph, and that the total time for the trip is 2 hours. Therefore, the distance covered during the entire trip is:
distance = (average speed) x (total time) = 4 x 2 = 8 miles
So the distance covered during running is:
distance ran = distance - distance walked = 8 - 4.5 = 3.5 miles
Now we can use the formula for average speed:
average speed = total distance / total time
To find the running speed, we need to solve for the running time, which is:
time ran = distance ran / running speed
Substituting this expression into the average speed formula, we get:
4 = (distance walked + distance ran) / (1.5 + time ran)
4 = (4.5 + 3.5) / (1.5 + distance ran / running speed)
Simplifying this expression, we get:
1.5 + distance ran / running speed = 2
distance ran / running speed = 0.5
Substituting the values we calculated, we get:
3.5 / running speed = 0.5
Solving for the running speed, we get:
running speed = 3.5 / 0.5 = 7 mph
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A graduated cylinder that is 24 cm tall can hold 1 L of water.
What is the radius of the cylinder? (round to the nearest hundredth)
part c question consider two groups of randomly selected peppers, where neither group has all big or all small peppers. a mean difference of zero indicates the peppers are well distributed between the two groups.a. true b. false
False. A mean difference of zero does not necessarily indicate that the peppers are well distributed between the two groups.
What is number?Number is a mathematical object used to count, measure, and label. It is one of the most fundamental concepts in mathematics and is used to describe a wide variety of physical, logical, and abstract objects. Numbers are used to represent quantities, lengths, temperatures, money, and many other measurements. They are also used to represent abstract ideas, such as the number of days in a year or the number of colors in a rainbow.
Although the mean may be equal between the two groups, the distribution of the peppers could be skewed and not evenly distributed. For example, one group could have all large peppers and the other all small peppers and the mean would be equal. This would not be considered well distributed. To ensure the peppers are evenly distributed, one should check the standard deviation or range of the data to see if the peppers are spread across both groups.
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Answer: A mean difference of zero indicates the peppers
are not well distributed between the two groups.
Why: Part B Answers
What do large differences of the means of each group indicate? Select the correct answer.
The peppers with different weights aren’t properly distributed between the two groups.
So they arent distributed well
Helpppppppppp pleaseeee I really need itttttt
Answer: 96
Step-by-step explanation:
M = 180 - 84 = 96
m<k = 96
It keeps saying "don't use such words so here is the question as a picture I guess.
As a result, 47 degrees of freedom represent the essential t value ().
What's really degree as well as its types?Least. To contrast one thing to another, degrees of comparison are employed. One thing or person is described in the positive degree. When comparing two things or people, utilize the comparative degree. Moreover, the exceptional degree is employed to describe groups, individuals, or more than two objects.
The following formula must be used to get the freedom degrees for the crucial t value ():
df = n - 1
where the sample size is "μ". The sample size in this instance is 48, so:
df = 48 - 1
df = 47
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PLEASE HELP ME THIS IS PAST DUE AND I NEED TO GET MY GRADE UP
Answer:
2916 cm^2
Step-by-step explanation:
Area = side x side = 54 x 54 or 54^2 = 2916 cm^2
Answer:
The other guy is right, 2916 cm^2 is the answer
Step-by-step explanation:
Do you need help with any other questions, if so I'm available to help you.
what is the area of ABC
The answer of the given question based on the finding the area of the triangle of ABC the answer is the area of triangle ABC is approximately 62.82 square cm.
What is Triangle?A triangle is three-sided polygon with three angles. It is two-dimensional geometric shape, and one of basic shapes in geometry. A triangle can be classified based on length of its sides and measure of its angles. The sum of the interior angles of triangle are 180 degrees. Triangles are used in many fields, like mathematics, engineering, architecture, and art.
To find the area of triangle ABC, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
where the base is one side of the triangle and the height is the perpendicular distance from the base to the opposite vertex.
In this case, we know that AB = 11 cm and AC = 17 cm, and angle A is 45 degrees. To find the height of the triangle, we need to use trigonometry.
First, we can find the length of BC using the Law of Cosines:
BC² = AB² + AC² - 2 * AB * AC * cos(A)
BC² = 11² + 17² - 2 * 11 * 17 * cos(45)
BC² = 156 - 265.42
BC² = 109.42
BC = 10.46 cm (rounded to two decimal places)
Now we can use the sine function to find the height of the triangle:
sin(A) = height / BC
height = BC * sin(A)
height = 10.46 * sin(45)
height = 7.39 cm (rounded to two decimal places)
Finally, we can use the area formula to find the area of the triangle:
Area = (1/2) * base * height
Area = (1/2) * 17 * 7.39
Area = 62.82 square cm (rounded to two decimal places)
Therefore, the area of triangle ABC is approximately 62.82 square cm.
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The data in the table below shows the average temperature in Northern Latitudes:
Estimate to the nearest whole number the average temperature for a city with a latitude of 48.
[___________]
Therefore , the solution of the given problem of mean comes out to be 15 is the solution.
What is mean?The sum of all values divided by all of the values constitutes the result from a collection, also referred to as the arithmetic mean. It is often referred to be known as "mean" as well as is one of the most frequency used main trend indicators. To find the answer, multiply the collection's overall amount of numbers by all of its values. Either the original data or data which has been combined into frequency charts can be used for calculations.
Here,
We can use linear interpolation between the two closest latitude numbers in the table, 45 and 50, to determine the typical temperature for a city with a latitude of 48.
Let T(45) and T(50) represent the typical temperatures at respective latitudes of 45 and 50, respectively. The following algorithm can be used to determine the temperature at 48 degrees latitude:
=> T(48) = T(45) + (T(50) - T(45)) * (48 - 45)/(50 - 45)
Using the numbers from the table as inputs, we obtain:
=> T(48) = 14 + (16 - 14) * (48 - 45)/(50 - 45)
=> T(48) = 14 + 2 * 3/5
=> T(48) = 14 + 1.2
=> T(48) = 15.2
The estimated average temperature for a city with a latitude of 48 is 15, rounded to the closest whole number.
Consequently, 15 is the solution.
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Select THE TWO values of that are solutions to the equation (* - 5) (7x - 21) = 0.
Using the discriminant, how many real solutions does the following quadratic equation have? x^2 +8x+c= 0
The equation has two distinct real roots if 64 - 4c > 0, one real root if 64 - 4c = 0, and no real roots if 64 - 4c < 0.
The discriminant of a quadratic equation of the form [tex]ax^2 + bx + c = 0[/tex] is given by [tex]b^2 - 4ac[/tex]. In the given quadratic equation, a = 1, b = 8, and c = c. Therefore, the discriminant is:
[tex]b^2 - 4ac[/tex]
[tex]= 8^2 - 4(1)(c)[/tex]
[tex]= 64 - 4c[/tex]
Now, we can use the discriminant to determine the nature of the solutions of the quadratic equation. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root (a double root). If the discriminant is negative, the equation has no real roots (two complex conjugate roots).
In this case, we do not have enough information about the value of c to determine the nature of the roots of the equation. All we know is that the discriminant is 64 - 4c.
Hence, if 64 - 4c > 0, we can state that the equation has two separate real roots, one real root if 64 - 4c = 0, and no real roots if 64 - 4c < 0.
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Given the price R64,00 (excluding VAT), find the total cost to the buyer (including VAT at 15%).
Therefore, the total cost to the buyer, including VAT at 15%, is R73,60.
What is vat?The term "VAT" usually refers to "Value Added Tax," which is a type of consumption tax that is levied on the value added to goods and services at each stage of production and distribution. It is commonly used by governments around the world as a way to generate revenue and to shift the tax burden from income and sales taxes onto consumption.
by the question.
To calculate the total cost to the buyer including VAT at 15%, you need to add the VAT amount to the original price. Here's how to do it:
Calculate the VAT amount:
VAT = 15% of R64,00
VAT = 0.15 x R64,00
VAT = R9,60
Add the VAT amount to the original price to get the total cost to the buyer:
Total cost = R64,00 + R9,60
Total cost = R73,60
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If the lateral area of a square pyramid is 800 in² and the slant
height is 16 in, what is the length of the base edge?
a 30 in
b 25 in
C 21 in
d 32 in
Answer:
b. 25 in
Step-by-step explanation:
the lateral area is the area of the outside walls (not including the base area).
on a square pyramid there are 4 such outside walls in the form of triangles.
so, each such triangle has an area of 800/4 = 200 in².
the area of a triangle is
baseline × height / 2
the slant height is the height of a wall triangle (not the inner height of the pyramid).
a base edge is nothing else but the baseline of such a wall triangle.
so, we have
200 = baseline × 16 / 2 = baseline × 8
baseline = 200/8 = 25 in
place the publication of three major books on race in chronological order, from earliest to most recent. Start by clicking the first item in the sequence or dragging it here Drag the items below into the box above in the correct order, starting with the first item in the sequence. Tomas Almaguer wrote about historical race relations in California in Racial Fault Lines Michael Omi and Howard Winant wrote about the social construction of race in Racial Formation in the United States. Ta-Nehisi Coates wrote about race and the African American experience in Between the World and Me.
The publication of three major books on race in chronological order, from earliest to most recent is:
- Micheal Omi and Howard Winant wrote about the social construction of race in Racial Formation in the United States.- Tomas Almaguer wrote about historical race relations in California in Racial Fault Lines.- Ta- Nehisi Coates wrote about race and the African American experience in Between the World and Me.Chronological order is the listing, description, or discussion of when events occurred in relation to time. Essentially, it is similar to looking at a chronology to see what happened initially and what happened after that. For example, if teachers asked their pupils to recount their first day of school, they would expect students to begin by waking up that morning and getting ready. If pupils begin from the time they enter the school, significant information is lost and the listener may become confused due to a lack of knowledge.
Helping pupils grasp what chronological order is and how to use the skill correctly can benefit students ranging from kindergarten to collegiate levels. The concept may appear simple, yet failing to master chronological sequence can cause kids to struggle academically and lack a solid educational foundation.
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The developers of a training program designed to improve manual dexterity claim that people who complete the 6-week program will increase their manual dexterity. A random sample of 12 people enrolled in the training program was selected. A measure of each person's dexterity on a scale from 1 (lowest) to 9 (highest) was recorded just before the start of and just after the completion of the 6-week program. The data are shown in the table below. Person Before Program After ProgramA 6.7 7.8 B 5.4 5.9 C 7.0 7.6 D 6.6 6.6E 6.9 7.6 F 7.2 7.7 G 5.5 6.0 H 7.1 7.0 J 7.9 7.8 I 5.9 6.4 K 8.4 8.7 L 6.5 6.5 TOTAL 81.1 85.6 Can one conclude that the mean manual dexterity for people who have completed the 6-week training program has significantly increased? Support your conclusion with appropriate statistical evidence.
There is evidence to support the claim of the developers that people who complete the 6-week program will increase their manual dexterity.
To determine if the mean manual dexterity has significantly increased after completing the 6-week training program, we can perform a paired t-test. The null hypothesis is that the mean difference between the before and after the program is zero, while the alternative hypothesis is that the mean difference is greater than zero.
Using the given data, the mean difference is (85.6-81.1)/12 = 0.375, with a standard deviation of 0.753. The t-statistic is then calculated as (0.375-0)/(0.753/sqrt(12)) = 2.35, with 11 degrees of freedom (df = n-1).
Using a significance level of 0.05 and a one-tailed test, the critical t-value is 1.796. Since the calculated t-value (2.35) is greater than the critical t-value (1.796), we can reject the null hypothesis and conclude that the mean manual dexterity has significantly increased after completing the 6-week training program.
However, it is important to note that this conclusion is based on a small sample size and that further research may be needed to confirm these findings.
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Eddie est discutiendo con Tana sobre las probabilidades de los distintos resultados al lanzar tres monedas. Decide lanzar una moneda de un centavo, una de cinco centavos y una de die centavos. ¿ Cuál es la probabilidad de que las tres monedas salgan cruz?
The probability of getting tails in the three coins would be 0.125 or 12.5%.
How to calculate the probability?To calculate the probability of an event happening, first, we need to identify the rate of the desired outcome versus the total possible outcomes. Moreover, to determine the total probability of two or more events happening we need to calculate the probability of each event and then multiply the results.
Probability of getting tails in any of the three coins:
1 / 2 = 0.5
Total probabilityy:
0.5 x 0.5 x 0.5 = 0.125 or 12.5%
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mai has a jar of quarters and dimes. she takes at least 10 coins out of the jar and has less than $2.00. write a system of inequalities that represents the number of quarters, `x`, and the number of dimes, `y`, that mai could have.
The system of inequalities that represents the number of quarters, x, and the number of dimes, y, that Mai could have is given by:
x + y ≥ 10 and 0.25x + 0.1y < 2
These are the two systems of inequalities that represent the number of quarters, x, and the number of dimes, y, that Mai could have.
Let x be the number of quarters and y be the number of dimes that Mai has. Then, the system of inequalities can be represented as:
Thus, the first inequality is x + y ≥ 10.
Also, Mai has less than $2.00, therefore, the second inequality is 0.25x + 0.1y < 2. The value of x and y are assumed to be non-negative integers.+
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Find the trigonometric ratios and the measures of the angles or the unknown lengths of these right tiangles, Round off angles to the nearest whole degree, trigonometric values to four decimal places, and lengths of sides to two decimal places.)
The value of x= 10.2.
What is trignometric ratios?This is the boundary or contour length of a 2D geometric shape.
Depending on their size, multiple shapes may have the same circumference. For example, imagine a triangle made up of wires of length L.
The same wire can be used to create a square if all sides are the same length.
The length covered by the perimeter of the shape is called the perimeter. Therefore, the units of circumference are the same as the units of length.
As we can say, the surroundings are one-dimensional. As a result, you can measure in meters, kilometers, millimeters, etc.
Inches, feet, yards, and miles are other globally recognized units of circumference measurement.
Hence, The value of x= 10.2.
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A sector subtends an angle of 42° at the centre of a circle of radius 2.8 cm. Calculate the perimeter of the sector.
[tex]\textit{arc's length}\\\\ s = \cfrac{\theta \pi r}{180} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ \theta =42\\ r=2.8 \end{cases}\implies s=\cfrac{(42)\pi (2.8)}{180}\implies s=\cfrac{49\pi }{75}\implies s\approx 2.05 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{ \textit{Perimeter of the sector} }{2.8~~ + ~~2.8~~ + ~~2.05} ~~ \approx ~~ \text{\LARGE 7.65}[/tex]
let's recall that the sector's perimeter includes the arc plust the radii.
Which of the following equations makes the most sense for the following situation: Jerry works twice as fast as Kenny. If Jerry works for 2 hours and is then joined by Kenny...and they work an additional 3 hours...how fast could they each do the job alone?
a. 5/x + 3/2x =1
b. x/3 + 2x/5 = 1
c. 5/2x + 3/x = 1
d. 5/x + 3/(x-2) = 1
The correct equation for the given situation is D: 5/x + 3/(x-2) = 1.
Let's assume that Kenny takes x hours to complete the job alone. As per the given information, Jerry works twice as fast as Kenny, which means he can do the job alone in x/2 hours.
Now, Jerry works for 2 hours, which means he completes 2/x of the job. After that, both Jerry and Kenny work together for 3 more hours and complete the remaining (1-2/x) part of the job.
Since they are working together, we can add their rates of work. Jerry's work rate is 1/(x/2) = 2/x and Kenny's work rate is 1/x. Therefore, their combined work rate is 2/x + 1/x = 3/x.
Using the formula "work = rate × time," we get the equation:
2/x × 2 + 3/x × 3 = 1
Simplifying the above equation, we get 5/x + 3/(x-2) = 1.
Therefore, option D is the correct equation for the given situation.
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bobo the clown always likes to wear one white sock and one black sock. if he changes his socks every day, and at the end of several days finds that he has used exactly $\frac15$ of his white socks and $\frac58$ of his black socks, what fraction of all of his socks has he used?
The required fraction of socks that Bobo the clown has used can be[tex]$\frac{\frac{1}{5}W+\frac{5}{8}B}{W+B}[/tex]
What is the fraction?Bobo likes to wear one white sock and one black sock.
Let the number of white socks be W and the number of black socks be B. Bobo changes his socks every day. At the end of several days, he finds that he has used exactly [tex]\frac15[/tex] of his white socks.
So, the number of white socks he has used [tex]=\frac{1}{5}W[/tex]
Also, he has used [tex]\frac58[/tex] of his black socks. So, the number of black socks he has used [tex]=\frac{5}{8}B[/tex]
The total number of socks Bobo has used = number of white socks used + number of black socks used
[tex]=\frac{1}{5}W+\frac{5}{8}B[/tex]
The fraction of all of Bobo's socks he has used [tex]=\frac{\text{Total number of socks used}}{\text{Total number of socks}}[/tex]
So, we need to find the total number of socks.
Total number of white socks = W
Total number of black socks = B
The total number of socks Bobo has is = Total number of white socks + Total number of black socks [tex]= W + B[/tex]
The fraction of all of Bobo's socks he has used [tex]= \frac{\frac{1}{5}W+\frac{5}{8}B}{W+B}[/tex]
Hence, the required fraction of socks that Bobo the clown has used can be calculated as [tex]$\frac{\frac{1}{5}W+\frac{5}{8}B}{W+B}[/tex]
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What is the area of the base, B,
of the prism?
B = 12 ft²
What is the volume, V, of the prism?
V= ? ft³
4 ft
6 ft
3 ft
1. The area of the base, B, of the prism is: B = 12 ft².
2. The volume, V, of the prism is: V = 72 ft²
What is a prism?A prism is a transparent object with flat, polished surfaces that refract or bend light as it passes through. The most common type of prism is a triangular prism, which has two triangular faces and three rectangular faces. When light enters the prism at an angle, it is refracted or bent as it passes through the first surface of the prism.
Area of the base, B:
As given in the question, we can see that the prism is a rectangular prism. Thus, the shape of the base is rectangle.
∴ Length of the base of a rectangular prism = 4 ft
Width of the base of a rectangular prism = 3ft
⇒ Area of the base of a rectangular prism is = ( length of the base ) × ( Width of the base)
= ( 4 × 3 ) ft²
= 12 ft²
∴ B = 12 ft²
Volume of a prism: V = BH
Where B = Base of the prism = 12 ft²
H = Height of the prism = 6 ft²
∴ 12 × 6 = 72 ft²
Thus, V = 72 ft²
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ab +3a² - 7a+ab+a² simplify this algebraic expression
Simplifying the algebraic expression ab + 3a² - 7a + ab + a² = a(4a + 2b - 7)
What is an algebraic expression?An algebraic expression is a mathematical expression in which letters are used to represent the variables.
Since we have the algebraic expression ab + 3a² - 7a + ab + a², and we want to simplify it. We proceed as follows
ab + 3a² - 7a + ab + a²
First, we collect the like terms together
ab + 3a² - 7a + ab + a² = 3a² + a²- 7a + ab + ab
Adding the similar terms together, we have
= 3a² + a²- 7a + ab + ab
= 4a²- 7a + 2ab
= 4a²- 7a + 2ab
Factorizing out a, we have that
= a(4a - 7 + 2b)
= a(4a + 2b - 7)
So, ab + 3a² - 7a + ab + a² = a(4a + 2b - 7)
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Using the standard domain for the sine function, which of the following expressions corresponds to the (1 point) composite function
sin(arccosx)
? Review Guidelines: If you guessed the answer to this question, or did not answer it correctly, go back and review the composition of trigonon Composition of Trigonometric Functions lesson.
(0pts) x 2
+1
(0pts) x 2
−1
(0pts) 1−x
(1pt)
1−x 2
The composite function -11-x^2 can be written in terms of the standard domain for the sine function as sin[−11−x2]. This function can be evaluated by substituting the value of x into the expression.
To do this, simply substitute the value of x into the equation:[tex]sin[-11-x^2][/tex]and evaluate the result. For example, if x=5, the equation would be sin[-11-5^2], which would evaluate to sin(-26).
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replacement times for cd players are normally distributed with a mean of ten years and the variance of two. find the replacement time separating the bottom 30% to the top 70%.
The replacement time separating the bottom 30% to the top 70% of CD players is 9.27 to 10.73 years.
What is meant by normally distributed data?A normal distribution refers to a continuous probability distribution in statistics that is symmetric and bell-shaped. It is a statistical distribution that is ideal for data that are continuous and normally distributed in a population.
The central limit theorem states that any large number of independent random variables, each with its own distribution, have their sum distribution approach normality as the number of variables grows large enough.
The formula for z-score is given as:
z = (x - μ) / σ
where x is the random variable, μ is the mean, σ is the standard deviation, and z is the z-score.
Substituting the given values of the mean, standard deviation, and percentiles in the above formula, we get:
For the bottom 30%, the percentile rank is 0.3.
Hence, the z-score is given as:
z = invNorm(0.3) = -0.5244
Substituting the z-score formula we get:
-0.5244 = (x - 10) / √2
Rearranging the above formula we get:
x = -0.5244 * √2 + 10 = 9.27 years
For the top 70%, the percentile rank is 0.7.
Hence, the z-score is given as:
z = invNorm(0.7) = 0.5244
Substituting the z-score formula we get:
0.5244 = (x - 10) / √2
Rearranging the above formula we get:
x = 0.5244 * √2 + 10 = 10.73 years
Therefore, the replacement time separating the bottom 30% to the top 70% of CD players is 9.27 to 10.73 years.
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how many square tiles are shaded and not shaded for the 8th figure?
Alexis and her little sister Emma went to Gold Trust National Bank. When they got there, Alexis had three times as much money in her account as Emma. After Alexis withdrew $100 to buy some new clothes and Emma deposited $25 that she got for her birthday, they had the same amount in their accounts.
How much money did Emma have in her account when she and Alexis got to the bank?
Answer:
Emma had $62.50 in her account when she and Alexis got to the bank.
Step-by-step explanation:
Let's start by assigning variables to the unknowns in the problem. Let's call the amount of money in Emma's account before she made her deposit "E", and the amount of money in Alexis's account before she made her withdrawal "A".
From the problem, we know that:
A = 3E (because Alexis had three times as much money as Emma before making her withdrawal)
After Alexis withdrew $100 and Emma deposited $25, they had the same amount of money in their accounts. So we can set up another equation:
A - 100 = E + 25
Simplifying this equation, we get:
A = E + 125
Now we can substitute the first equation into the second equation, because we know that A is equal to 3E:
3E = E + 125
Subtracting E from both sides, we get:
2E = 125
Dividing both sides by 2, we get:
E = 62.5
So Emma had $62.50 in her account when she and Alexis got to the bank.
Answer:62.50
Step-by-step explanation:
she had 62.50
How do you do this I need help please
Answer:
30,000 grams
Step-by-step explanation:
multiply the 30KG by 1,000 (that is the conversion) and you get 30,000g
Answer:
hi I'm really sorry I can't help
When solving the equation 3(x-2)=-12, what is a possible first step?
A) Distributing the 3 into each term of the parenthesis.
B) She could either divide by 3 or distribute 3 into the parenthesis
C) Adding 2 to each side of the equation
D) Subtracting 2 on each side of the equation
E) Dividing each side of the equation by 3
F) none of the answer choices tell what the first step could be.
Answer:
A
Step-by-step explanation:
Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. (If the quantity diverges, enter DIVERGES.) an = sin(√n)/√n
The given sequence an = sin(√n)/√n converges to limit 0 as n approaches infinity
The mentioned nth term of the sequence is an = sin(√n)/√n. To determine the convergence or divergence of the sequence and find its limit, we can use the limit comparison test, which is based on comparing the given sequence with a known sequence whose convergence or divergence is already known.Suppose bn is a known sequence whose convergence or divergence is already known. Then, by the limit comparison test, the given sequence converges or diverges according as the sequence bn converges or diverges.
To apply the limit comparison test, we need to find a suitable sequence bn whose convergence or divergence is known. For this, we observe that sin x ≤ x for all x > 0. Hence, we have 0 ≤ sin(√n)/√n ≤ 1/√n, where the inequality follows by dividing both sides of sin x ≤ x by √n and substituting x = √n. Also, we know that the sequence bn = 1/√n converges to 0 as n approaches infinity. Therefore, by the limit comparison test, the given sequence an = sin(√n)/√n also converges to 0 as n approaches infinity.
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approximate the definite integral using the trapezoidal rule and simpson's rule. compare these results with the approximation of the integral using a graphing utility. (round your answers to four decimal places.) 3 1 ln(x) dx, n=4
Trapezoidal Simpson's Graphing Utility
From the graphing utility, we get the value of the integral as, Integral = 0.7206 Comparing the values of the integrals obtained from Trapezoidal rule, Simpson's rule and graphing utility, we find that the integral value obtained from the graphing utility is closest to the Simpson's rule.
We are given the definite integral ∫31ln(x)dx and we are required to approximate the integral using the trapezoidal rule and Simpson's rule. Also, we are supposed to compare these results with the approximation of the integral using a graphing utility using n=4.Trapezoidal Rule. The trapezoidal rule for numerical integration is a method to approximate the definite integral using linear interpolation.
This rule approximates the definite integral by dividing the total area into trapezoids. The formula for trapezoidal rule is given by:(Image attached)Here, a = 3 and b = 1The value of h can be calculated as follows;h=(b−a)/nh=(3−1)/4=0.5We need to calculate the values of f(1), f(1.5), f(2), f(2.5), f(3) using n = 4(Image attached)The value of integral using the trapezoidal rule is,Integral = 0.7201.
Simpson's rule is used to approximate the value of definite integral. Simpson's rule involves approximating the integral under the curve using the parabolic shape. This is done by dividing the area under the curve into small sections and then approximating each section with a parabolic shape. The formula for Simpson's rule is given as:(Image attached)Here, a = 3 and b = 1
The value of h can be calculated as follows;h=(b−a)/nh=(3−1)/4=0.5We need to calculate the values of f(1), f(1.5), f(2), f(2.5), f(3) using n = 4(Image attached)The value of integral using the Simpson's rule is,Integral = 0.7200Comparing with Graphing UtilityIntegral = 0.7201 (from Trapezoidal rule)Integral = 0.7200 (from Simpson's rule).
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