By squaring both sides and solving for [tex]\(x\),[/tex] we find [tex]\(x = 4\)[/tex] as the solution, which is not extraneous upon substitution into the original equation.
To solve the equation [tex]\(\sqrt{2x + 1} = 3\),[/tex] we need to isolate[tex]\(x\).[/tex] Here's how:
1. Square both sides of the equation to eliminate the square root:
[tex]\[ (\sqrt{2x + 1})^2 = 3^2 \][/tex]
[tex]\[ 2x + 1 = 9 \][/tex]
2. Subtract 1 from both sides to isolate [tex]\(2x\)[/tex]:
[tex]\[ 2x = 9 - 1 \][/tex]
[tex]\[ 2x = 8 \][/tex]
3. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{8}{2} \][/tex]
[tex]\[ x = 4 \][/tex]
Now, we have [tex]\(x = 4\)[/tex]. To determine if it's an extraneous solution, we need to check if it satisfies the original equation.
Substitute [tex]\(x = 4\)[/tex] into the original equation:
[tex]\[ \sqrt{2(4) + 1} = 3 \][/tex]
[tex]\[ \sqrt{9} = 3 \][/tex]
[tex]\[ 3 = 3 \][/tex]
Since the equation holds true, [tex]\(x = 4\)[/tex]is a valid solution, not an extraneous one.
Therefore, the solution to the equation [tex]\(\sqrt{2x + 1} = 3\) is \(x = 4\),[/tex]and it is not an extraneous solution.
how to solve xy''+2y'+xy=0
Why is the answer to a subtraction problem called the difference...
Box of 12 golf balls costs $45. what is the unit rate
how do you derive the lens maker's equation? ...?
Using Snell’s law ( n1 sin θi = n2 sin θr ) at point A gives
1) sin i1 = n sin r1
2) i1 = n r1
3) i1 = α1 + β1
4) r1 = β1 - γ
5) α1 + β1 = n(β1 - γ)
6) sin i2 = n sin r2
7) i2 = n r2
8) i2 = α2 + β2
9) r2 = β2 + γ
10) α2 + β2 = n(β2 + γ)
11) α1 + α2 = (n - 1)(β1 + β2)
12) 1/s + 1/s' = (n - 1)(1/R1 + 1/R2)The lens maker's equation is derived by considering the image formed by the first refracting surface (left surface) and then using this image as the object for the second refracting surface. The equation takes into account the radii of curvature and the refractive indices of the lens and the surrounding medium.
Explanation:The lens maker's equation is derived by considering the image formed by the first refracting surface (left surface) and then using this image as the object for the second refracting surface.
The equation takes into account the radii of curvature and the refractive indices of the lens and the surrounding medium.
The lens maker's equation can be stated as: 1/f = (n₂ - n₁)((1/R₁) - (1/R₂)), where f is the focal length, n₂ is the refractive index of the lens, n₁ is the refractive index of the surrounding medium (usually air), R₁ is the radius of curvature of the first surface of the lens, and R₂ is the radius of curvature of the second surface of the lens.
What value corresponds to the 60th percentile in the following data set?
{12, 28, 35, 42, 47, 49, 50}
1) 47
2) 49
3) 28
4) 42
...?
To find the 60th percentile, multiply 60/100 by the total number of observations plus one, which equates to 4.8. This falls between the 4th and 5th value, and rounding up means the 60th percentile corresponds to the 5th value, which is 47.
Explanation:To find the value that corresponds to the 60th percentile in the given data set {12, 28, 35, 42, 47, 49, 50}, we need to follow a series of steps.
Arrange the data in ascending order, which is already done in this case.Determine the rank of the percentile using the formula: Rank = (P/100) * (N + 1), where P is the percentile and N is the number of observations in the set.For the 60th percentile, the formula becomes Rank = (60/100) * (7 + 1), which gives us Rank = 4.8. This means the 60th percentile falls between the 4th and 5th values in the set.Since we cannot have a fraction of an observation, we round up to the nearest whole number. The 5th value in the set is 47.Therefore, the value corresponding to the 60th percentile in this data set is 47.
Final answer:
The 60th percentile value in the data set {12, 28, 35, 42, 47, 49, 50} is 47, which is the 5th number in the ordered set after calculating the index.
Explanation:
The 60th percentile value in a data set is found by first arranging the data in ascending order, which is already done in the provided data set. Then calculate the index using the formula Index = P(N + 1), where P is the percentile in decimal form and N is the number of data points. In this case, the 60th percentile index for our 7-point data set is 0.60 × (7 + 1) = 4.8. Since the index is not a whole number, we round up to the next whole number, which here would be 5. Therefore, we take the 5th number in the ordered set, which is 47.
What are the least common factors of 8 and 12
Julia can type 150 words in five minutes at this rate how many words can she type in one minute
Julia can type 30 words per minute.
Julia can type 150 words in five minutes.
To find out how many words she can type in one minute, divide 150 by 5, which equals 30.
[tex]\frac{150}{5} = 30[/tex]
So she can type 30 words per minute.
PLEASE HELPPP!!!
1. Choose the equation that could be used to solve the given number problem.
The sum of two consecutive integers is 141. Find the integers.
n + 1= 141
2n + 2 = 141
2n = 141
2n + 1 = 141
2. Choose the equation that could be used to solve the given number problem.
The sum of two consecutive odd integers is 112. Find the integers.
2n + 3 = 112
2n + 2 = 112
2n + 1 = 112
n + 2 = 112
if 6-x=2(x-6) what is the value of x? need steps!!
The value of x in the equation 6 - x = 2(x - 6) is 6. To convert this to feet, the calculation shows that 6 miles is equivalent to 31,680 feet.
Explanation:To solve the equation 6 - x = 2(x - 6), follow these steps:
Distribute the 2 on the right side to both terms inside the parenthesis: 6 - x = 2x - 12.Add x to both sides to get x terms on one side: 6 = 3x - 12.Add 12 to both sides to isolate the x terms: 18 = 3x.Divide both sides by 3 to solve for x: x = 6.So, the value of x is 6.
To convert 6 miles to feet, using the equation 1 mile/5280 feet = 6 miles/x feet, we cross multiply to get:
1x = 6 × 5280.x = 31,680 feet.Therefore, 6 miles is equivalent to 31,680 feet.
Yolanda paid for her movie ticket using 28 coins, all nickels and quarters. The ticket cost $4. Which system of linear equations can be used to find the number of nickels, n, and the number of quarters, q, Yolanda used?
Yolanda paid for her movie ticket using 28 coins, all nickels and quarters. The ticket cost $4.
Let n be the number of nickels
and q be the number of quarters
1 nickels = $0.05 , so n nickels cost = 0.05n
1 quarter = $0.25, so q quarters cost = 0.25q
Total number of coins = 28
number of nickels + number of quarters = 28
n + q = 28 ------- equation 1
Total ticket cost =$4
cost of n nickels + cost of q quarters = $4
0.05n + 0.25q = 4 -------> equation 2
system of linear equations can be used to find the number of nickels, n, and the number of quarters, q, Yolanda used are
n + q = 28
0.05n + 0.25q = 4
Which system has no solutions?
x < 5
x > 1
x < 5
x < 1
x > 5
x > 1
x > 5
x < 1
HELP meh plz!!!
To determine which system has no solutions, we analyze a set of inequalities to identify the contradictory statement.
To determine which system has no solutions, we need to analyze the given inequalities:
x < 5
x > 1
x < 5
x < 1
x > 5
x > 1
x > 5
x < 1
From the list, the system 'x < 1' and 'x > 1' has no solutions as it creates a contradictory statement.
A quadratic equation is shown below:
9x2 - 36x + 36 = 0
Part A: Describe the solution(s) to the equation by just determining the discriminant. Show your work. (3 points)
Part B: Solve 2x2 - 9x + 7 = 0 using an appropriate method. Show the steps of your work and explain why you chose the method used.(4 points)
Integer is a special name for _____. positive whole numbers. positive or negative whole numbers. negative whole numbers. positive whole or mixed numbers.
Kellies mom takes a 400 gm dose of aspirin . Each hour the amount of aspirin in a persons system decreases by about 29%. How much aspirin is left in her system after 6 hours
how many times can 18 go into 357
which fraction below represents the following statement "in the third game of the season Martha made one of 5 free throws"
Answer: Which fraction below represents the following statement?“In the third game of the season, Martha made one out of five free throws.” The answer for this questions would be 1/5
Step-by-step explanation: I got it correct on Edge.
The fraction that represents the statement "in the third game of the season Martha made one of 5 free throws is 1/5.
What is a fraction?A fraction is written with a numerator and a denominator where the numerator is less than the denominator.
Example: 1/3 is a fraction.
We have,
Total number of free throws = 5
Number of free throws Martha made = 1
The number of free throws Martha made can be written in fraction form as:
= Number of free throws Martha made / Total number of free throws
= 1 / 5
Thus,
The fraction that represents the statement "in the third game of the season Martha made one of 5 free throws is 1/5.
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Jenna and her friend, Khalil, are having a contest to see who can save the most money. Jenna
has already saved $110 and every week she saves an additional $20. Khalil has already saved $80
and every week he saves an additional $25. Let x represent the number of weeks and y represent
the total amount of money saved.
So far I've got 110+20x= ??? for jenna because I don't know what goes after the equal sign and 80+25x=???
Explain the difference between f and f(x).
4c-7.8 Evaluate the expression if c=4
If 800 feet of fencing is used to enclose a rectangular plot of land that borders a river, what is the maximum area that can be enclosed? Answer to the nearest square foot without commas. For example, if the answer is 1,000, write 1000.
Answer:
80000
Step-by-step explanation:
The maximum area that can be enclosed with 800 feet of fencing and the river bordering one side is 0 square feet.
Explanation:To find the maximum area that can be enclosed with 800 feet of fencing, we need to maximize the area of the rectangular plot. A rectangle with one side bordering the river will have two sides of equal length and two sides of different length. Let's denote the length of the side parallel to the river as x and the length of the other side as y. Since there are two sides of equal length, the total length of those two sides would be 2x. The other two sides would be y each, giving a total length of 2y. The equation we can form based on the given information is:
2x + y + y = 800
Simplifying the equation, we get:
2x + 2y = 800
Dividing both sides by 2, we have:
x + y = 400
Now, we need to express one of the variables in the equation in terms of the other variable. Let's solve for x:
x = 400 - y
To find the maximum area, we need to express it in terms of a single variable. The area of a rectangle is given by length multiplied by width. So, the area A can be expressed as:
A = x * y = (400 - y) * y
To find the maximum value of A, we can use calculus. Let's take the derivative of A with respect to y:
A' = -y + 400
Setting A' to 0 and solving for y, we get:
0 = -y + 400
y = 400
Substituting this value back into the expression for A, we have:
A = (400 - 400) * 400 = 0
Therefore, the maximum area that can be enclosed is 0 square feet.
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Let f be the function defined by f(x) = 3x^5-5x^3+2
a) On what intervals is f increasing
b) On what intervals is the graph of f concave upward.
c) Write the equation of each horizontal tangent line to the graph of f.
Please provide some detail to your response. ...?
Answer:
a) [-1, ∞-) and [1, ∞+)
b) (√2/2,∞) and (0,-√2/2)
c) y=0, y=2 and y=4
Step-by-step explanation:
a) On what intervals is f increasing?
We need to proceed some steps, to answer it properly. What are the critical points? So
1) Differentiate the original function
[tex]f(x)=3x^{5}-5x^3+2\\ f'(x)=15x^4-15x^{2}[/tex]
2) Turn this into an equation
[tex]15x^4-15x^2=0\\ u=x^{2} and \\u^{2}=x^{4} \\ Then\\ 15u^{2} -15u=0\\ \\15u(u^2 -1)=0\\Therefore\\u=0,u=1[/tex]
Substitute back to x
[tex]x^{2} =u\\ x^{2} =u^4\\ x^{2}=1\\ x= 1,x=-1\\ x^{2} =0\\ x=0[/tex]
Those are x-coordinates of the critical points of [tex]f(x)=3x^{5}-5x^3+2[/tex]
Using the critical points x, to find y-coordinates of those critical points: namely, maximum point, saddle point and minimum point
x=-1 [tex]y=3x^{5}-5x^3+2[/tex]
[tex]y=3(-1)^{5}-5(-1)^3+2[/tex]
y=4
(-1,4) Maximum point
--
x=0 [tex]y=3x^{5}-5x^3+2[/tex]
[tex]y=3(0)^{5}-5(0)^3+2[/tex] y=2
(0,2) Saddle Point
--
x=1 [tex]y=3x^{5}-5x^3+2[/tex]
[tex]y=3(1)^{5}-5(1)^3+2[/tex]
(1,0) Minimum point
f(x) increases when any value of x ∈ (-∞, -1] and x ∈ [1, ∞+) is plugged in the function. Or we can rewrite as x[tex]x\leq -1\\ \\x\geq 1[/tex]
-------------------------------------
b)On what intervals is the graph of f concave upward?
To find out which intervals are these, we need to calculate the 2nd derivative.
[tex]f(x)=3x^{5}-5x^3+2\\ f(x)'=15x^4-15x^{2} \\f(x)''=60x^3-30x\\[/tex]
Since we want the concave upward
f(x)''>0
Then:
[tex]60x^3-30x>0\\ 30x(2x^2-1)>0\\[/tex]
Rewriting
[tex]30x(\sqrt{2x}-1)[/tex]
Then
x=0, x=[tex]\frac{\sqrt{2}}{2} \\-\frac{\sqrt{2}}{2}[/tex]
Studying the sign
The intervals >0, when the graph is concave upwards.
are (√2/2,∞) and (0,-√2/2)
c) Write the equation of each horizontal tangent line to the graph of f.
Generally, every time somebody ask for a tangent line of any given function, they provide an information. A specific point.
But when they do not? Like in this case?
We have to look for where the 1st function's derivative is zero.
As we 've done previously in letter a. x=-1, x=0, x=1
Let's plug them back to the original function
[tex]y=3x^{5}-5x^{3}+2\\ y=3(-1)^{5}-5(-1)^{3}+2\\ y=-3-5+2\\ y=-8+2\\ y=4\\ \\y=3x^{5}-5x^{3}+2\\ y=3(0)^{5}-5(0)^{3}+2\\ y=2\\\\ y=3x^{5}-5x^{3}+2\\ y=3(1)^{5}-5(1)^{3}+2\\ y=0[/tex]
So each horizontal tangent line, tangents the maximum, saddle and the minimum point.
The cake store is having a 30 % off sale on all of its cakes. if the cake you want regularly costs $10 how much would you save with the discount?
#1: How many meters are in 0.05 kilometers?
A. 50
B. 500
C. 5,000
D. 50,000
Answer:
50 meters
A is correct
Step-by-step explanation:
Given: 0.05 kilometers
As we know, In 1 km = 1000 m
So, multiply 0.05 km to 1000 to change into meter
[tex]\Rightarrow 0.05\ km\times \dfrac{1000\ m}{1\ km}[/tex]
[tex]\Rightarrow \dfrac{5}{100}\times 1000[/tex]
[tex]\Rightarrow 50\ km[/tex]
Hence, In 0.05 km is 50 meters.
The meter equivalent of 0.05 km is 50 .
Given,
0.05 kilometers
As we know, In 1 km = 1000 m
So, multiply 0.05 km to 1000 to change into meter
0.05 * 1000
= 50 m
Hence, In 0.05 km is 50 meters.
Thus option A is correct .
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H(n) = −2n 2 4; find h(4)
If ef = 6 and eg = 21 what is the value of fg
If EF = 6 and EG = 21 what is the value of FG?
The answer is 15.
What is the mean number of hours that high school students spend on homework per week? It is known that the standard deviation of the population is 2.2 hours. In a random sample of 40 students, it is found that they average 13 hours per week on homework.
a) What is the standard deviation for this sample?
b) What is the critical value for a 90% confidence interval?
c) What is the critical value for a 95% confidence interval?
The standard deviation for the sample is approximately 0.348 hours.
The critical value for a 90% confidence interval is approximately 1.645, and for a 95% confidence interval, it is approximately 1.96.
To calculate the standard deviation for the sample, we use the formula:
[tex]\[ \text{Standard deviation for sample} = \frac{\text{Population standard deviation}}{\sqrt{\text{Sample size}}} \][/tex]
Given the population standard deviation of 2.2 hours and a sample size of 40, we plug these values into the formula:
[tex]\[ \text{Standard deviation for sample} = \frac{2.2}{\sqrt{40}} \][/tex]
[tex]\[ \text{Standard deviation for sample} \approx \frac{2.2}{6.324} \approx 0.348 \text{ hours} \][/tex]
So, the standard deviation for this sample is approximately 0.348 hours.
For the critical values of confidence intervals, we refer to the z-table or use a calculator.
For a 90% confidence interval, the critical value is approximately 1.645, and for a 95% confidence interval, it is approximately 1.96.
These values represent the number of standard deviations from the mean that encompass the specified confidence level.
A wave traveling in water has a frequency of 250 Hz and a wavelength of 6.0 m. What is the speed of the wave?
what is 4 divided by 2
Find the rate of change for the situation.
A chef cooks 9 lbs of chicken for 36 people and 17 lbs of chicken for 68 people.
A. 9/17lb per person
B. 4 lb per person
C. 1/4 lb per person
D. 36 people ...?
Final answer:
The rate of change for the chef's cooking scenario is calculated by dividing the lbs of chicken by the number of people for each case. In both situations, the rate of change is 1/4 lb per person.
Explanation:
The question asks to find the rate of change for a situation where a chef cooks a certain amount of chicken for a specific number of people. To calculate the rate of change, we need to divide the amount of chicken by the number of people for each situation, and then compare.
For 9 lbs of chicken for 36 people, the rate is 9 lbs ÷ 36 people = 0.25 lbs/person, which is equal to 1/4 lb per person. For 17 lbs of chicken for 68 people, the rate is 17 lbs ÷ 68 people = 0.25 lbs/person, which is also 1/4 lb per person.
Since the rate of change is the same in both cases, 1/4 lb per person, the correct answer is:
C. 1/4 lb per person
How do we find x-intercept of rational function
An X-intercept in a rational function is the point where the graph intersects the x-axis. It is derived by setting the function equal to zero and then solving for the variable x. This solution, x, represents the x-intercept of the rational function.
Explanation:In Mathematics, the x-intercept of a rational function is the point where the function intersects the x-axis. The x-intercept(s) can be found by setting the function equal to zero and solving for x. When graphing, this point determines where the line crosses the x-axis.
The same principle is applied to the calculation of the x-intercept as other functions. The given rational function is set equal to zero, and then algebraic methods are used to solve for the value of x. For example, if given the rational function y = 2x/3x + 1, setting this function to zero would read as 0 = 2x/(3x + 1). Cross-multiplication can then be applied to find the solution for x, which represents the x-intercept.
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