Answer:
all of them
Step-by-step explanation:
i hope you find it helpful
write a quadratic function with zeroes 5 and 9
Answer:
[tex]f(x) = {x}^{2} - 14x + 45[/tex]
Step-by-step explanation:
[tex]f(x) = (x - 5)(x - 9)[/tex]
[tex]f(x) = {x}^{2} - 14x + 45[/tex]
ZA and ZB are complementary angles. If m≤A = (x – 19)˚ and
m/B = (3x + 1)°, then find the measure of ZB.
Answer:
∠ B = 82°
Step-by-step explanation:
complementary angles sum to 90° , then
x - 19 + 3x + 1 = 90
4x - 18 = 90 ( add 18 to both sides )
4x = 108 ( divide both sides by 4 )
x = 27
Then
∠ B = 3x + 1 = 3(27) + 1 = 81 + 1 = 82°
Kallie works at a pet store. Part of her job is to add the correct amount of water conditioner to each fish tank the list below provides information about the number of fish tanks and the amount of water conditioner she uses.
1: there are 12 fish tanks that need water conditioner
2: Each fish tank is filled with 20 quarts of water
3: for every 10 gallon of water, Kallie uses 1 teaspoon of water conditioner
What is the total number of teaspoons of water conditioner kallie will use for all the water in 12 fish tanks?
Therefore, Kallie will need to use a total of 6 teaspoons of water conditioner for all the water in 12 fish tanks.
What is equation?An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides separated by an equal sign, with the expression on the left being equal to the expression on the right. Equations are used to represent various relationships and situations in mathematics and other fields, and are solved to find the values of variables that make the equation true.
Here,
First, we need to determine the total amount of water in all 12 fish tanks. Since each fish tank is filled with 20 quarts of water, the total amount of water in all 12 fish tanks is:
12 fish tanks * 20 quarts per fish tank = 240 quarts of water
To convert quarts to gallons, we divide by 4 (since there are 4 quarts in a gallon):
240 quarts / 4 quarts per gallon = 60 gallons of water
Now, we can use the given conversion factor to determine how many teaspoons of water conditioner are needed for 60 gallons of water. Since for every 10 gallons of water, Kallie uses 1 teaspoon of water conditioner, for 60 gallons of water, she will use:
60 gallons / 10 gallons per teaspoon = 6 teaspoons of water conditioner
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Question 2 (1 point) A Point A is located at (-1,4). Where would point A' be located after a dilation of 2 centered at the origin? N
Answer:
Point A would be located at (-2,8)
Step-by-step explanation:
After a dilation of factor 2 centered at the origin (0,0), Point A' would be located at:
A' = (2 * -1, 2 * 4) = (-2, 8)
The effect of the dilation is to stretch the original point A by a factor of 2 along both the x and y directions. This essentially doubles the distances from the origin to the point A, creating the point A'.
Rewrite the following quadratic function im standard (vertex) form. f(x)=6x^2-5x-1
The standard form (vertex form) of the function f(x) is: f(x) = 6(x - 5/12)x² - 169/24.
Describe the vertex form?To rewrite the quadratic function f(x) = 6x2 - 5x - 1 in standard form (vertex form), we can complete the square:
f(x) = 6xx² - 5x - 1 = 6(x²- (5/6)x) - 1 = 6(xx² - (5/6)x + 25/144) - 1 - 6(25/144) = 6(x - 5/12)x² - 169/24
Therefore, the standard form (vertex form) of the function f(x) is: f(x) = 6(x - 5/12)x² - 169/24.
The vertex form of a quadratic function is a way of writing the function in a specific form that reveals important information about the vertex, or turning point, of the parabolic graph of the function.
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Evaluate this table.
X 5 10 15 25 40
Y 1 2 3 5 8
The table represents a(n)
relationship.
Answer:
The values in the table suggest that there is a mathematical relationship between the two variables X and Y. Upon inspection, we can observe that Y is increasing with respect to X, and the increase seems to be non-linear. Specifically, as X increases by a factor of 2, Y increases by a factor of approximately 1.5 to 2.
Based on these observations, it seems that the relationship between X and Y may be an exponential one. To confirm this, we can plot the data points on a graph and see if they form a curve that resembles an exponential function.
Alternatively, we can calculate the ratio of Y to X and see if it remains approximately constant. This can be done by dividing each value of Y by its corresponding value of X:
1/5 = 0.2
2/10 = 0.2
3/15 = 0.2
5/25 = 0.2
8/40 = 0.2
The ratio remains constant at approximately 0.2, suggesting that the relationship between X and Y may be a proportional one, with a constant of proportionality equal to 0.2.
Therefore, the table represents a proportional relationship between X and Y, where Y is proportional to X with a constant of proportionality equal to 0.2.
Step-by-step explanation:
Here's a step-by-step explanation of how to evaluate the table:
1. Identify the variables: The table contains two variables, X and Y, which are listed in two separate columns.
2. Examine the values: Look at the values in the table for both X and Y. Notice that as X increases, so does Y.
3. Determine the pattern: To determine the pattern between the two variables, calculate the ratio of Y to X. If the ratio is constant, then the relationship is proportional. If the ratio changes, then the relationship is nonlinear.
4. Calculate the ratio: To calculate the ratio, divide each value of Y by its corresponding value of X. For example, to find the ratio for the first row, divide 1 by 5: 1/5 = 0.2. Continue calculating the ratios for each row.
5. Analyze the ratio: If the ratios are approximately constant, then the relationship is proportional. In this case, we see that the ratios are all approximately 0.2, so we can conclude that the relationship is proportional.
6. Determine the constant of proportionality: To determine the constant of proportionality, simply use any one of the rows in the table. For example, let's use the first row, where X = 5 and Y = 1. The ratio of Y to X is 0.2, so we can write the relationship as Y = 0.2X. This means that for every increase of 1 unit in X, Y increases by 0.2 units.
7. Summarize the result: Based on the analysis, we can say that the table represents a proportional relationship between X and Y, with a constant of proportionality equal to 0.2.
Given
X 5 10 15 25 40
Y 1 2 3 5 8
5 / 40 = 0.2
y=0.2x
find the missing number that makes the sentence true
Answer:
2+5/9 is the answer
If you add 2 to 5/9, that makes the statement true
3. In a class, there are 80 students. The statistical distribution of the number of students offering Physics, Math, Hausa, French and English is shown on a pie chart. Math 5x French (16x-24) Eng (6x + 12)* How many students offer Mathematics? Hausa (4x+12) Physics
the number of students that offer Mathematics is 11
Calculating the number of students that offer Mathematics?The sum of the fractions of the pie chart should be equal to 1 since it represents the whole class of 80 students.
Let's write the fractions for each subject in terms of x:
Math: 5x/80
French: (16x-24)/80
English: (6x+12)/80
Hausa: (4x+12)/80
Physics: 5x/80
The sum of these fractions must equal 1:
5x/80 + (16x-24)/80 + (6x+12)/80 + (4x+12)/80 + 5x/80 = 1
Multiplying both sides by the least common multiple (LCM) of the denominators, which is 80, we get:
5x + 16x - 24 + 6x + 12 + 4x + 12 + 5x = 80
Combining like terms, we get:
36x = 80
So, we have
x = 80/36
For maths, we have
Maths = 5 * 80/36
Maths = 11
Therefore, 11 students offer Mathematics.
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Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.06
and a standard deviation of 1.52
. Using the empirical rule, what percentage of American women have shoe sizes that are between 6.54
and 9.58
?
An approximate of 68% of American women have shoe sizes that are between 6.54 and 9.58.
How can empirical rule be use to find the percentage?To use the empirical rule, we need to assume that the distribution of women's shoe sizes is approximately normal. Given a mean of 8.06 and a standard deviation of 1.52, we can standardize the values of 6.54 and 9.58 by subtracting the mean and dividing by the standard deviation:
Ζ1 = (6.54 - 8.06) / 1.52
Ζ1 = -1.00
Ζ2 = (9.58 - 8.06) / 1.52
Ζ2 = 1.00
The standardized values tell us that the value of 6.54 is 1 standard deviation below the mean, while the value of 9.58 is 1 standard deviation above the mean. According to the empirical rule, about 68% of the data falls within one standard deviation of the mean. Therefore, we can say that: [tex]P(6.54 < x < 9.58) = P(-1.00 < z < 1.00) = 68%[/tex]
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y=c+bx2, find y where c = 14/5, B = 4/5,x=2
Answer:
ASAP................Research the use in the military of magnetic anomaly detectors, MADs. Write a brief 300-word essay answer the following questions on MADs. What is the main idea behind MADs? What can be detected by using MADs? A brief history of the MAD development.
Step-by-step explanation:
find the size of angle xyz
give ur answer to 1 dec place
[tex]\cos(XYZ )=\cfrac{\stackrel{adjacent}{6}}{\underset{hypotenuse}{15}}\implies \cos(XYZ )=\cfrac{2}{5} \\\\\\ XYZ=\cos^{-1}\left( \cfrac{2}{5} \right)\implies XYZ\approx 66.4^o[/tex]
Make sure your calculator is in Degree mode.
Angle QMR in degrees
Therefore , the solution of the given problem of angles comes out to be
Angle QMR = 30.
An angle meaning is what?The place where the lines that make up a skew's ends meet determines the size of its largest and smallest walls. It is conceivable for two routes to cross at a junction. Angle is another outcome of two things interacting. They mirror dihedral shapes the most. A two-dimensional curve can be created by arranging two line beams in various configurations between their ends.
Here,
Angle QRM counts 90 degrees because it is a right angle (represented by the square in the diagram).
We can use the knowledge that the sum of all angles in a straight line is 180 degrees to determine angle QPR. A straight line is formed by angles QPB and BPR, so:
=> Angle QPB + Angle BPR = 180°.
Given that the triangular is equilateral, angle QPB is equal to 60 degrees.
=> BPR angle = 180 - 60 = 120 degrees.
We can now use the knowledge that a triangle's total angles equal 180 degrees to determine angle QMR. Triangle formed by angles QMR, QRM, and BMR results in:
Angles QMR, QRM, and BMR add up to 180 degrees.
=> Angle QMR+90+120=180
=> Angle QMR = 180 - 90- 120,
=> Angle QMR = 30
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Need help on this question please!
. The GCD of three numbers is 30 and their LCM is 900. Two of the numbers are 60 and 150. What is the other possible number? (3 mks)
If the GCD of three numbers is 30 and their LCM is 900. Two of the numbers are 60 and 150. The other possible number is 450.
What is the possible number?Let the third number be x. We know that:
GCD(60, 150, x) = 30
This means that 30 is a common factor of all three numbers. We can divide each number by 30 to get:
GCD(2, 5, x/30) = 1
Now we can use the fact that the product of the GCD and LCM of three numbers is equal to the product of the numbers themselves:
GCD(60, 150, x) * LCM(60, 150, x) = 60 * 150 * x
30 * 900 = 60 * 150 * x
x = 900 / 2 = 450
Therefore, the other possible number is 450.
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Kindly help me with my statistics problem please. Due today at 4pm >_< Thanks in advance!
One of the 200 business majors at a college is to be chosen for the student senate. If 77 of these students are enrolled in a course in accounting, 64 are enrolled in business law, and 92 are enrolled in either courses, how many of the outcomes corresponds to the choice of a business major who is enrolled in both courses. Use a venn diagram.
If One of the 200 business majors at a college is to be chosen for the student senate. The outcomes that correspond to a business major who is enrolled in both accounting and business law is: x = 33 - y.
How many of the outcomes corresponds to the choice of a business major who is enrolled in both courses?Since we know that 92 students are enrolled in either course, we can write this number in the intersection of the two circles, which represents the number of students enrolled in both courses. Let x be the number of students enrolled only in accounting, and y be the number of students enrolled only in business law.
Our Venn diagram now looks like this:
Accounting
(77-x) (x)
___________
| |
Total | (92) | y
Business|___________|
Law | |
| x |
|___________|
(64-y) |
Total
Business
Majors
We know that there are 200 business majors in total, so the sum of the four regions in the diagram must add up to 200. Therefore:
x + y + (77-x) + (64-y) + 92 = 200
Simplifying this equation, we get:
x + y = 33
We want to find the number of outcomes that correspond to a business major who is enrolled in both accounting and business law. From the Venn diagram, we can see that this is represented by the intersection of the two circles, which has a value of x.
Therefore, there are x = 33 - y outcomes that correspond to a business major who is enrolled in both accounting and business law. The value of y can range from 0 (if all 33 students enrolled only in accounting) to 33 (if all 33 students enrolled only in business law).
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11. The area of a square painting is 225x¹+240x+64. Explain how you would find a possible
length of one side of the painting.
Answer: Therefore, a possible expression for the length of the side of the square painting is:
side = 15x + 8
or
side = -15x - 8
Step-by-step explanation:
To find a possible expression for the length of the side of the square painting, we need to use the formula for the area of a square, which is:
Area = side^2
We can set the given expression for the area of the painting equal to this formula:
225x¹+240x+64 = side^2
Next, we can simplify the expression on the left-hand side by factoring it into a perfect square trinomial:
225x¹+240x+64 = (15x + 8)^2
Now we can substitute this expression back into the equation and solve for the side of the square painting:
(15x + 8)^2 = side^2
Taking the square root of both sides, we get:
15x + 8 = side
or
15x + 8 = -side (since the length of a side can be positive or negative)
Therefore, a possible expression for the length of the side of the square painting is:
side = 15x + 8
or
side = -15x - 8
Note that since we are dealing with a geometric object, we should choose the positive value for the length of the side, as the side length cannot be negative. Therefore, the final expression for the length of the side of the square painting is:
side = 15x + 8
The XYZ Company`s forecasted sales for July, August, September and October of year ended 2012 were Birr 120,000, Birr 100,000, Birr 160,000, and Birr 140,000 respectively and the forecasted selling price per unit for each month`s sale was Birr 20. The ending inventory of the Company on June 30, 2012 was 1,500 units. The desired ending inventory for each month was one-fourth of the forecasted sales of the following month.
a. Prepare a production budget for the month of July, August, and September. (4pts)
Production budget for the month of July, August, and September as below.
Define the term selling price?The selling price is the amount of money that a company charges for a product or service in order to generate revenue and make a profit.
1. Expected Sales:
July: 120,000 / 20 = 6,000 units
August: 100,000 / 20 = 5,000 units
September: 160,000 / 20 = 8,000 units
2. Required Production:
July: 6,000 units + (8,000 units / 4) - 1,500 units = 4,500 units
August: 5,000 units + (6,000 units / 4) - (8,000 units / 4) = 3,500 units
September: 8,000 units + (5,000 units / 4) - (6,000 units / 4) = 6,000 units
3. Desired Ending Inventory:
July: 8,000 units / 4 = 2,000 units
August: 6,000 units / 4 = 1,500 units
September: 10,000 units / 4 = 2,500 units
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1. The Expected Sales in July is 6,000 units, August is 5,000 units and september is 8,000 units.
2. Required Production in July is 4,500 units, August is 3,500 units and september is 6,000 units.
3. Desired Ending Inventory in July is 2,000 units, August is 1500 units and september is 2500 units.
How do you calculate expected sales?The most basic method is: sales forecast = sales from the previous month + expected growth (or shrinkage) in revenue for the next term.
1. Expected Sales:
July = 120,000 / 20
= 6,000 units
August = 100,000 / 20
= 5,000 units
September = 160,000 / 20
= 8,000 units
2. Required Production:
The quantity of units that a company must produce in a given time frame in order to be profitable.
[tex]July=6000\ units+(\frac{8000\ units}{4} )-1500\ units[/tex]
[tex]=4500\ units[/tex]
[tex]August=5000\ units+(\frac{6000\ units}{4} )-(\frac{8000\ units}{4} )[/tex]
[tex]=3500\ units[/tex]
[tex]september=8000\ units+(\frac{5000\ units}{4} )-(\frac{6000\ units}{4} )[/tex]
[tex]=6000\ units[/tex]
3. Desired Ending Inventory:
Ending inventory is the sellable inventory that remains at the conclusion of an accounting period.
July = 8,000 units / 4
= 2,000 units
August = 6,000 units / 4
= 1,500 units
September = 10,000 units / 4
= 2,500 units
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A movie ticket cost $12 each. Sophia had a coupon for $10 off the total if Sophia spent $38 total at the theater including her coupon how many tickets did she buy?
a. Define variables and write an equation to represent the relationship between the quantities. b. How far would the biker travel in 20 minutes? c. If the biker traveled 48 miles, how many minutes did he bike? Round to the nearest tenth.
The biker wοuld travel 20 miles in 20 minutes.
What is speed?The speed οf an οbject, alsο knοwn as v in cοmmοn parlance and kinematics, is the size οf the change in pοsitiοn per unit οf time οr the size οf the change in pοsitiοn οver time; as such, it is a scalar quantity. The average speed οf an οbject in a given periοd οf time is equal tο the distance traveled by the οbject divided by the length οf the interval the instantaneοus speed is the upper limit οf the average speed as the length οf Velοcity and speed are different cοncepts.
We can nοw substitute this expressiοn fοr s in the previοus equatiοn:
t = 48 miles / (48 miles / t)
Simplifying this equatiοn, we get:
t = t
Therefοre, we can cοnclude that the biker travelled fοr exactly 1 minute fοr every mile travelled.
b. If the biker travelled fοr 20 minutes, we can use the fοrmula:
d = s x t
Substituting the values we knοw, we get:
d = s x 20 minutes
We alsο knοw frοm the previοus calculatiοn that the biker travels 1 mile fοr every minute traveled. Therefοre, the speed οf the biker is:
s = 1 mile / minute
Substituting this value, we get:
d = 1 mile/minute x 20 minutes = 20 miles
Therefοre, the biker wοuld travel 20 miles in 20 minutes.
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find the following in Qiv sin(A)/(2)
For sin the expression [tex]Qiv sin(A)/(2)[/tex] can be simplified using trigonometric identities.
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables within their domains. These identities are useful in simplifying trigonometric expressions and solving trigonometric equations. Some common identities include the Pythagorean identity, which relates the trigonometric functions of sine, cosine, and tangent, and the double-angle identities, which express trigonometric functions of twice the angle in terms of functions of the original angle.
First, we can recognize that Qiv is the same as 1/4 of the unit circle or 90 degrees. Therefore, [tex]Qiv sin(A)[/tex] can be simplified to sin(A + 90).
Using the identity [tex]sin(A + B) = sin(A)cos(B) + cos(A)sin(B)[/tex], we can rewrite [tex]sin(A + 90)[/tex] as:
[tex]sin(A)cos(90) + cos(A)sin(90)[/tex]
Since [tex]cos(90) = 0 \\sin(90) = 1[/tex], this:
[tex]sin(A) * 0 + cos(A) * 1[/tex]
Which simplifies to just cos(A).
Therefore, the expression [tex]Qiv sin(A)/(2)[/tex] is equivalent to cos(A)/2.
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HELP PLS ASAP which of the following will yrilf data with variability
The correct option that result the data with variability:
C. The shoe size of the various 5 year olds.
D. The number of patients at the doctor's office each day.
Explain about the variability of data:The term "variability" refers to the distance between data points within a distribution and their distance from its centre. Measures of variability provide you summary statistics that summarise your data in addition to measurements of central tendency.
Spread, scatter, and dispersion are other terms for variation. It is often assessed using the following:
Difference between both the greatest and least values is referred to as the range.Interquartile range: a range of a distribution's middle halfStandard deviation is the typical departure from the mean.Variance: squared mean deviations are averaged out.Thus, the result the data with variability:
C. The shoe size of the various 5 year olds as it will vary for different children.
D. The number of patients at the doctor's office each day is also variable.
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What is the ratio of the number of pairs of Jeanine to the cost of jeans?
The Ratio of the number of pairs of Jeanine to the cost of jeans helps us understand how many pairs of jeans Jeanine can purchase for a given amount of money.
To find the ratio of the number of pairs of Jeanine to the cost of jeans, we need to first determine how many pairs of jeans Jeanine has and how much she paid for them. Let's assume Jeanine has 10 pairs of jeans and she paid $50 for each pair. Therefore, the total cost of Jeanine's jeans is $500.
Now we can calculate the ratio by dividing the number of pairs of jeans by the cost of jeans. So, the ratio of the number of pairs of Jeanine to the cost of jeans is:
10 pairs of jeans / $500 = 1/50
This means that for every $50 Jeanine spends on jeans, she gets one pair. Alternatively, we could also express the ratio as a decimal or percentage. In this case, the ratio as a decimal would be 0.02 or 2%, indicating that Jeanine spends 2% of the cost of one pair of jeans for each pair she owns.
Overall, the ratio of the number of pairs of Jeanine to the cost of jeans helps us understand how many pairs of jeans Jeanine can purchase for a given amount of money.
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what does it mean if your calculated rate of return is negative
A negative return means that the investment has lost value within a certain period of time. It is a loss on paper unless the investment is redeemed. The return may become positive again the next day or the next quarter.
What is meant by investment?
Investment is the use of money or capital to acquire an asset or securities for the purpose of earning income or profits over time. It can be various forms of financial instruments such as stocks, bonds, mutual funds, real estate, and others.
What is a quarter?
A quarter is a unit of currency equal to one-fourth of a dollar in the United States and some other countries. It is also commonly used to denote a period of three months in a calendar year.
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Sam is practicing long track speed skating at an ice skating rink. The distance around the rink is 250 yards. He has skated around the rink 7 times so far.
How many more yards does he need to skate around the rink to complete 3 miles?
Answer:
There are 1760 yards in one mile and the rink is 250 yard in perimeter.
This means 3 miles is 5280 yards.
If Sam has already gone around the whole rink 6 times, that means he was traveled a total of 1500 yards.
Sam has 3780 more yards to travel 3 miles.
This means he needs to do 15.12 more laps around the rink. Or 16 if you are rounding.
Bob received the following scores: 72, 86, 92, 63, and 77. What test score must Bob earn on his sixth test so that his average for all six tests will be 80?
Answer:
90
Explanation:
Given marks of Bob's five tests are:
[tex]72 + 86 + 92 + 63 + 77[/tex]
To get an average of 80 on six tests we will need to multiply 6 tests to average of 80.
So,
[tex]6 \times 80 = 480[/tex]
Adding the marks of given five tests is
[tex]72 + 86 + 92 + 63 + 77 = 390[/tex]
Now, to get the marks for Bob's next test, which is the sixth test, we should subtract 390 from 480.
So,
480 - 390 = 90
Therefore, Bob needs to score 90 marks on his sixth test to obtain the average of 80 for all the six tests.
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four customer's purchased 4 donuts that weigh 3.4 pounds, how much did all four donuts weigh
Answer:
The answer is 13.6
Step-by-step explanation:
4 x 3.4 = 13.6
Determine if f(x) = 3x4 -8 is invertible.
O invertible
O non-invertible
If so, find the inverse.
ƒ-¹(x) =
F(x) is one-to-one, and it is invertible.
The inverse of f(x) is f^-1(x) = (x + 8)^(1/4)/∛3.
To determine if the function f(x) = 3x^4 - 8 is invertibleWe need to check if it is a one-to-one function.
A function is one-to-one if every element in the domain is paired with a unique element in the range.
To check if f(x) is one-to-one, we can use the horizontal line test. If no horizontal line intersects the graph of the function more than once, then the function is one-to-one.
Taking the derivative of f(x), we get:
f'(x) = 12x^3
Since f'(x) is always positive, f(x) is a strictly increasing function. This means that no two different inputs x1 and x2 can produce the same output f(x1) = f(x2).
Therefore, f(x) is one-to-one, and it is invertible.
To find the inverse of f(x), we can follow these steps:
Step 1: Replace f(x) with y:
y = 3x^4 - 8
Step 2: Solve for x in terms of y:
y + 8 = 3x^4
x^4 = (y + 8)/3
x = (y + 8)^(1/4)/∛3
Step 3: Replace x with f^-1(x):
f^-1(x) = (x + 8)^(1/4)/∛3
Therefore, the inverse of f(x) is f^-1(x) = (x + 8)^(1/4)/∛3.
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90 dL times L/dL equals what
Step-by-step explanation:
As your post is written 90 dL * L / dl = 90 L
But I think what you really want is the conversion factor for L to dL :
one liter is 10 dL :
L / 10 dL
then you question becomes
90 dL * L / 10 dL = 9 liters
3. 1 particle P of mass m kg. The particle P is set in motion so that it moves back and forth alon One end of a light rod of length / m is attached to a fixed point O and the other end is attachec minor arc AB of a vertical circle with centre O and radius / m, as shown in the diagram. A O 0 P C When P is at its lowest point C, its speed is u ms and the tension in the rod is 2mg N. a) Show that u = √gl. B The speed of P when OP makes an angle with the vertical is denoted by v ms. Sho v² = gl (2cos8 - 1). Find the greatest value of 0. Find the value of 0 when the tension in the rod is mg N.
The answer of the given question is (a) proofing is below u = √gl. (b) the greatest value of 0 is 2gr and the value of θ when the tension in the rod is mg and the particle is moving with speed √gl is θ = 30° or θ = 150°.
What is Equilibrium?An equilibrium refers to state of balance or stability where there is no net change in motion. An object is said to be in equilibrium when net force acting on it is zero, which means that object is either at rest or moving with constant velocity.
There is two types of equilibrium that are : static equilibrium and dynamic equilibrium. In static equilibrium, an object is at rest, and net force acting on it is zero. In dynamic equilibrium, an object is moving at constant velocity, and net force acting on it is also zero.
a) At point C, the particle P is at its lowest point and is in equilibrium. Therefore, the forces acting on P must be balanced. The weight of P is mg acting vertically downwards, and the tension in the rod is acting along OC at an angle of θ to the vertical. The horizontal component of tension is zero because the rod is light. Therefore, the vertical component of tension must balance the weight of P:
T cosθ = mg
We can also write:
T sinθ = mv²/r
where r is the radius of the circle. At the lowest point C, θ = 90° and cosθ = 0, so we have:
T sin90° = mv²/r
T = mv²/r
Substituting this into the first equation:
mv²/r cosθ = mg
v² = gr cosθ
Since the angle between OP and the vertical is θ, we have:cosθ = cos(90° - θ) = sinθ
Substituting this into the previous equation:
v² = gr sinθ = gl
Therefore, u = √gl.
b) When OP makes an angle θ with the vertical, the tension in the rod is:
T = 2mg cosθ
Using the same approach as in part a), we have:
T sinθ = mv²/r
Substituting for T:
2mg cosθ sinθ = mv²/r
2mg sin2θ = mv²/r
v² = 2gr sin2θ
Since sin2θ is maximum when 2θ = 90° or θ = 45°, the greatest value of v² is:
v² = 2gr
When the tension in the rod is mg, we have:
mg sinθ = mv²/r
sinθ = v²/gr
Substituting for v²:
sinθ = 2sin2θ
2sin2θ - sinθ = 0
sinθ(2sinθ - 1) = 0
Therefore, sinθ = 0 or sinθ = 1/2.
If sinθ = 0, then θ = 0° or θ = 180°, which means the particle is at the top or bottom of the circle and is not moving.
If sinθ = 1/2, then θ = 30° or θ = 150°. Substituting into the expression for v²:
v² = 2gr sin2θ = gl
Therefore, when the tension in the rod is mg, the particle is moving with speed √gl when θ = 30° or θ = 150°.
Thus, the greatest value of 0 is 2gr and the value of θ when the tension in the rod is mg and the particle is moving with speed √gl is θ = 30° or θ = 150°.
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Find three consecutive integers such that the square of the third added to the first is 130
[tex]x = (-b ± sqrt(b^2 - 4ac)) / 2a[/tex]The three consecutive integers are 5/2, 7/2, and 9/2. We can check that this solution works by squaring the third integer (9/2), adding it to the first integer (5/2), and verifying that the sum is indeed 130:
[tex](5/2) + (9/2)^2 = 130[/tex]
Let's call the first of the three consecutive integers "x". Then, the next two consecutive integers are x+1 and x+2.
The problem tells us that the square of the third integer (which is x+2) added to the first integer (which is x) is 130. In equation form, we can write:
[tex]x + (x+2)^2 = 130[/tex]
Simplifying the equation, we can expand the square:
[tex]x + x^2 + 4x + 4 = 130[/tex]
Combining like terms:
[tex]x^2 + 5x - 126 = 0[/tex]
Now we can use the quadratic formula to solve for x:
x = -b ± [tex]sqrt(b^2 - 4ac)) / 2a[/tex]
where a = 1, b = 5, and c = -126. Plugging these values into the formula, we get:
x = (-5 ±[tex]sqrt(5^2 - 4(1)(-126))) / 2(1)[/tex]
x = (-5 ± [tex]sqrt(625)) / 2[/tex]
x = (-5 ±25) / 2
So, x could be either -15/2 or 5/2. However, we are looking for three consecutive integers, so we can eliminate the negative value and choose x = 5/2.
Therefore, the three consecutive integers are 5/2, 7/2, and 9/2. We can check that this solution works by squaring the third integer (9/2), adding it to the first integer (5/2), and verifying that the sum is indeed 130:
[tex](5/2) + (9/2)^2 = 130[/tex]
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