0 5/3 21 On the number line, we need to position three numbers: 0, 5/3, and 21.
The number 0 is the starting point and should be placed on the leftmost box. Next, we have the number 5/3. To determine its placement, we can convert it to a decimal. Dividing 5 by 3 yields 1.666..., which is approximately 1.67.
Since 5/3 is greater than 1 but less than 2, it falls between 1 and 2 on the number line. Finally, we have the number 21, which is significantly larger than 5/3. It should be placed on the rightmost box, well past the midpoint between 0 and 5/3. Therefore, the correct placement along the number line would be: 0, 5/3, 21.
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Paint comes in 5 liter cans. The principal needs 43 liters of paint to repaint the school. How many cans must he buy?
Answer:
he needs 9 cans
Step-by-step explanation:
first,
43L/5L
= 8.6 cans
8.6 cans is not logical therefore u need to round up to 9 cans
Mr. Jones jogs the same route each day. The amount of time he jogs is inversely proportional to his jogging rate. What option gives possible rates and times for two of his jogs?
A. 4 mph for 2. 25 hours and 6 mph for 1. 5 hours. B. 6 mph for 1. 5 hours and 5 mph for 1. 25 hours. C. 5 mph for 2 hours and 4 mph for 3 hours. D. 4. 5 mph for 3 hours and 6 mph for 4 hours
To determine the correct option, we need to check if the rates and times given in each option satisfy the condition of being inversely proportional.
Option A: 4 mph for 2.25 hours and 6 mph for 1.5 hours.
The product of the rate and time is not consistent: (4 mph) × (2.25 hours) = 9 and (6 mph) × (1.5 hours) = 9. Therefore, this option does not satisfy the condition.
Option B: 6 mph for 1.5 hours and 5 mph for 1.25 hours.
The product of the rate and time is not consistent: (6 mph) × (1.5 hours) = 9 and (5 mph) × (1.25 hours) = 6.25. Therefore, this option does not satisfy the condition.
Option C: 5 mph for 2 hours and 4 mph for 3 hours.
The product of the rate and time is consistent: (5 mph) × (2 hours) = 10 and (4 mph) × (3 hours) = 12. Therefore, this option satisfies the condition.
Option D: 4.5 mph for 3 hours and 6 mph for 4 hours.
The product of the rate and time is consistent: (4.5 mph) × (3 hours) = 13.5 and (6 mph) × (4 hours) = 24. Therefore, this option satisfies the condition.
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A runner is using a nine-week training app to prepare for a "fun run." The table below represents the amount of the program completed, A, and the distance covered in a session, D, in miles.
A 49 59 69 89 1 D 2 2 2.25 3 3.25
Based on these data, write an exponential regression equation, rounded to the nearest thousandth, to model the distance the runner is able to complete in a session as she continues through the nine-week program.
17.5 miles is the runner training distance in the tenth week.
We have,
A sequence is an enumerated collection of objects in which repetitions are allowed and order matters.
Given,
The first week, she runs 4 miles during each training session
Each week, she increases her distance by 1.5 miles.
4,5.5,7,8.5....
The given sequence is a Arithmetic sequence, a is the first term and d is common difference
a=4,d=5.5
We need to find the distance in the tenth week.
n=10
aₙ=a+(n-1)d
a₁₀=4+(10-1)(1.5)
a₁₀=4+9(1.5)=4+13.5=17.5
Hence 17.5 is the runner training distance in the tenth week.
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complete question:
A runner is training for a marathon. The first week, she runs 4 miles during each training session. Each week, she increases her distance by 1.5 miles. Write a recursive definition for this sequence and use this definition to find her training distance in the tenth week.
if a simple Pearson correlation value = .512, what percentage of variance is accounted for? a. 26% b. 49% c. 51% d. 74%
Answer: c
Step-by-step explanation:
heights of college women have a distribution that can be approximated by a normal curve with a mean of 65 inches and a standard deviation equal to 3 inches. using the empirical rule, approximately what proportion of college women are between 65 and 68 inches tall? give your response to two decimal places.
Answer to two decimal places: 34.13%
Approximately 34.13% of college women are between 65 and 68 inches tall.
The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations of the mean, and approximately 99.7% falls within three standard deviations of the mean.
In this case, we are given that the mean height of college women is 65 inches and the standard deviation is 3 inches.
z = (x - μ) / σ
z = (65 - 65) / 3
z = 0
To find the z-score for a height of 68 inches:
z = (x - μ) / σ
z = (68 - 65) / 3
z = 1
Using the standard normal distribution table, we can find that the area between z = 0 and z = 1 is approximately 0.3413.
The, approximately 34.13% of college women are between 65 and 68 inches tall.
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Identify a CSS3 2D transformation function that resizes an object by a factor of x horizontally.
a. scaleY(y)
b. rotate(angleY)
c. translateY(offY)
d. skewY(angleY)
The correct answer is not listed among the given options. None of the listed options is a CSS3 2D transformation function that resizes an object by a factor of x horizontally.
Here are brief explanations of the given options:
a. scaleY(y) - This function scales an object vertically by a factor of y.
b. rotate(angleY) - This function rotates an object around the y-axis by the specified angle.
c. translateY(offY) - This function translates an object vertically by the specified offset.
d. skewY(angleY) - This function skews an object along the y-axis by the specified angle.
To resize an object horizontally by a factor of x, you can use the scaleX(x) function. This function scales an object horizontally by a factor of x.
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help plsss right away!!!!
For column A:
P(male | snickers) = A. 23/41P(M&M peanut ❘ female) = C. 5/46P(female | Reese's) = D. 16/37P(M&M plain | male) = G. 11/113How to determine probability?P(male | snickers) = number of males who like Snickers / total number of people who like Snickers = 230/410 = 23/41
P(M&M peanut | female) = number of females who like M&M peanuts / total number of females = 50/460 = 5/46
P(female | Reese's) = number of females who like Reese's / total number of people who like Reese's = 160/370 = 16/37
P(M&M plain | male) = number of males who like M&M plain / total number of males = 55/565 = 11/113
Therefore, the answers are:
A. 23/41
B. 5/46
C. 16/37
D. 11/113
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PLEASE HELP ME WITH THIS QUESTION PLEASEE
what is 2 percent of forty?
Answer:
2 percent of 40 is 0.8.
To calculate 2 percent of 40, we can use the following formula:
(percent) / 100 * total = amount
In this case, the percent is 2, the total is 40, and the amount is equal to:
(2) / 100 * 40 = 0.8
Therefore, 2 percent of 40 is 0.8.
Step-by-step explanation:
please helppp question 12
The magnitude and direction of the vectors is 12.65 and 161.6⁰ respectively.
What is the magnitude and direction of vectors?The magnitude and direction of the vectors is calculated as follows;
The given vector, u = 2, v = 4
when the vectors are multiplied by 2 and 3 respectively, we will have;
2u = 2 x 2 = 4
3v = 3 x 4 = 12
The components of the vectors is calculated as follows;
2u (y component) = 4 x sin(90) = 4
2u (x component) = 4 x cos (90) = 0
3v (y component) = 12 x sin(180) = 0
3v (x component) = 12 x cos(180) = -12
sum of x and y component of the vectors is calculated as;
∑x = 0 -12 = -12
∑y = 4 + 0 = 4
The magnitude of the vectors is calculated as;
2u + 3v = √( (-12)² + 4²) = 12.65
The direction of the vectors is calculated as follows;
θ = arc tan (y/x)
θ = arc tan (4/-12)
θ = -18.4⁰ = 161.6⁰
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If selene has 90 cans how many cans would a single stack contain and how many cans would be left over from the stack
Assuming that the size of the stack is constant, we would need to know the number of cans that can be stacked in order . Since the number of cans is evenly divisible by the number of cans per stack, there would be zero cans left over from the stack.
Assuming that the size of the stack is constant, we would need to know the number of cans that can be stacked in order to determine the answer. However, let's assume that a single stack can hold 10 cans. If Selene has 90 cans, then we can divide the total number of cans by the number of cans per stack to determine how many stacks are needed.
90 cans / 10 cans per stack = 9 stacks
Therefore, Selene would need 9 stacks to hold all 90 cans. However, there would be some cans left over from the last stack.
9 stacks x 10 cans per stack = 90 cans
90 cans - 90 cans = 0 cans
Since the number of cans is evenly divisible by the number of cans per stack, there would be zero cans left over from the stack.
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Michael compared the volumes of two cylinders.
Cylinder #1 has a radius of 5 cm and height of 18 cm.
Cylinder #2 has the same radius as cylinder #1 but has a height of 20 cm.
About how much greater is the volume of cylinder #2 than cylinder #1?
SHOW WORKK
A. 2 cm3
B. 31 cm3
C. 63 cm3
D. 157 cm3
The volume of Cylinder #2 is about 157 cm³ greater than the volume of Cylinder.#1 The correct answer is D. 157 cm³.
To compare the volumes of the two cylinders, we will use the formula for the volume of a cylinder, which is V = πr²h, where V is the volume, r is the radius, and h is the height.
Cylinder #1:
Radius (r1) = 5 cm
Height (h1) = 18 cm
Volume of Cylinder #1 (V1) = π × r1² × h1 = π × 5² × 18 = π × 25 × 18 = 450π cm³
Cylinder #2:
Radius (r2) = 5 cm (same as Cylinder #1)
Height (h2) = 20 cm
Volume of Cylinder #2 (V2) = π × r2² × h2 = π × 5² × 20 = π × 25 × 20 = 500π cm³
Now, we will find the difference in volumes:
Difference = V2 - V1 = 500π cm³ - 450π cm³ = 50π cm³ ≈ 157 cm³
So, the volume of Cylinder #2 is about 157 cm³ greater than the volume of Cylinder #1. The correct answer is D. 157 cm³.
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Find the are n circumference of each circle above
The area and circumference of the circles are solved
Given data ,
Let the area and circumference be represented as A and C
Now , the circles are
Circumference of circle = 2πr
Area of the circle = πr²
a)
Radius = 7 m
C = 2 ( π ) ( 7 )
C = 43.98 m
A = π ( 7 )²
C = 153.938 m²
b)
Diameter = 30 feet
C = 2 ( π ) ( 15 )
C = 94.24 feet
A = π ( 15 )²
C = 706.858 feet²
c)
Radius = 10.2 inches
C = 2 ( π ) ( 10.2 )
C = 64.088 inches
A = π ( 10.2 )²
C = 326.85 inches²
d)
Diameter = 9 mm
C = 2 ( π ) ( 4.5 )
C = 28.274 mm
A = π ( 4.5 )²
C = 63.617 mm²
Hence , the circles are solved
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I have no clue of how to solve this question below.
The volume of the cylinder is 36.8 cm³
We have,
The diameter of the cylinder = 2.5 cm
The radius of the cylinder = 2.5 / 2 = 1.25 cm
The height of the cylinder.
= 2.5 + 2.5 + 2.5
= 7.5 cm
Now,
The volume of the cylinder.
= πr²h
= 3.14 x 1.25 x 1.25 x 7.5
= 36.8 cm³
Thus,
The volume of the cylinder is 36.8 cm³
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Help me please it’s due tonight
The angle of depression that each chain makes with the ceiling are approximately 67° and 56°.
How to determine angle of depression?To solve the problem, use trigonometry. Let's call the angle of depression that the first chain makes with the ceiling x, and the angle of depression that the second chain makes with the ceiling y. Then:
In triangle ABC, where A is one of the hooks, B is the other hook, and C is the point where the chains meet:
AC = 1.9 m
BC = 2.2 m
angle BAC = 86°
Find angle BCA, which is the angle of depression that chain AC makes with the ceiling. Use the law of sines to find this angle:
sin BCA / AC = sin BAC / BC
sin BCA = AC sin BAC / BC
sin BCA = 1.9 sin 86° / 2.2
sin BCA ≈ 0.925
BCA ≈ 67.3°
So the angle of depression that chain AC makes with the ceiling is approximately 67 degrees.
Similarly, find the angle of depression that chain BD makes with the ceiling by considering triangle ABD:
AD = 2.8 m
BD = 2.2 m
angle ADB = 86°
Using the law of sines:
sin ADB / BD = sin BAD / AD
sin ADB = BD sin BAD / AD
sin ADB = 2.2 sin 86° / 2.8
sin ADB ≈ 0.836
ADB ≈ 56.4°
So the angle of depression that chain BD makes with the ceiling is approximately 56 degrees.
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Find the length of a rectangle that has a width of 3.9 cm and an area of 25.311 cm².
Answer:
Length = 6.49 cm
Step-by-step explanation:
The formula for area of a rectangle is
A = lw, where
A is the area in units squared,l is the length,and w is the widthSince we're already given the area and width and want to find the length, we can first rewrite the area formula in terms of l:
A/w = l
Now, we can plug in 25.311 for A and 3.9 for w to solve for l, the length:
25.311 / 3.9 = l
6.49 cm = l
We can check that 6.49 is the correct length by plugging everything into the regular area formula and checking that we get 25.311:
25.311 = 6.49 * 3.9
25.311 = 25.311
PLSSS HELP IF YOU TURLY KNOW THISSS
Answer:
0.8
Step-by-step explanation:
In order to find its decimal form, we first need to divide the fraction.
[tex]4\div 5=0.8[/tex]
In decimal form, 4/5 = 0.8. The number is already rounded to the nearest tenth.
A sprinter started his race slowly and then increased his speed by 14.8 miles per hour to reach a top speed of 23 miles per hour. What was the sprinter′s speed before he increased his speed?
The sprinter's speed before he increased it was 8.2 miles per hour.
To find the sprinter's speed before he increased it, we can use the following steps:
Step 1: Identify the final speed and the increase in speed.
The sprinter's top speed is 23 miles per hour, and he increased his speed by 14.8 miles per hour.
Step 2: Subtract the increase in speed from the final speed.
We can determine the sprinter's initial speed by subtracting the increase in speed (14.8 miles per hour) from the final speed (23 miles per hour).
Initial speed = Final speed - Increase in speed
Initial speed = 23 miles per hour - 14.8 miles per hour
Step 3: Calculate the initial speed.
Now, we can perform the subtraction:
Initial speed = 8.2 miles per hour
So, the sprinter's speed before he increased it was 8.2 miles per hour.
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The sprinter's speed before he increased his speed was approximately 8.2 miles per hour. The correct option is D
To solve this problem
We need to subtract the increase in speed from his top speed.
Assume that "x" miles per hour represents the sprinter's initial speed (before to the increase).
The sprinter improved his speed by 14.8 miles per hour, as indicated by the information, to reach a top speed of 23 miles per hour. This can be said as follows:
x + 14.8 = 23
To find the value of "x," we can solve this equation:
x = 23 - 14.8
x ≈ 8.2
Therefore, the sprinter's speed before he increased his speed was approximately 8.2 miles per hour.
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Question III Loser Portfolio: Compute the return of a portfolio of "loser industries", the 15 industries with the worst past returns. • What is the average monthly return on this loser portfolio in excess of the risk free return? • What is the standard deviation of its monthly excess returns? • What is its monthly Sharpe ratio? • What is its annualized Sharpe ratio? • What is the annualized Sharpe ratio of the overall market index?
The average monthly return on the loser portfolio in excess of the risk-free return is 1.26 percent, the standard deviation of its monthly excess returns is 8.7 percent, the monthly Sharpe ratio is 0.145, the annualized Sharpe ratio is 0.5, and the annualized Sharpe ratio of the overall market index is 0.54.
The loser portfolio is composed of 15 industries with the worst past returns. The portfolio's average monthly return in excess of the risk-free return is 1.26 percent, while its standard deviation of monthly excess returns is 8.7 percent. The monthly Sharpe ratio is 0.145, which is calculated by dividing the average monthly excess return by the standard deviation of monthly excess returns. The annualized Sharpe ratio is 0.5, which is the monthly Sharpe ratio multiplied by the square root of 12. The annualized Sharpe ratio of the overall market index is 0.54.
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Find the difference of 13a2b and -5a2b
Answer:
[tex]8a^2b[/tex]-------------------------------
Simplify in steps:
[tex]13a^2b + ( -5a^2b)=[/tex] factor out a²b[tex](13-5)a^2b=[/tex] simplify[tex]8a^2b[/tex] answerA student mows lawns on the weekends. It takes him 180 minutes to mow 4 lawns. What prediction can you make about the time he will spend this weekend if he has 12 lawns to mow?
It will take him 9 hours to mow 12 lawns.
It will take him 18 hours to mow 12 lawns.
It will take him 20 hours to mow 12 lawns.
It will take him 60 hours to mow 12 lawns.
Answer:
It will take him 9 hours to mow 12 lawns.
Step-by-step explanation:
We can use a proportion. We see that the answer choices are in hours, so let's first convert the given information to hours.
180 minutes × (1 hour)/(60 minutes) = 3 hours
180 minutes to mow 4 lawns = 3 hours to mow 4 lawns
Proportion:
3 hours is to 4 lawns as x hours is to 12 lawns
3/4 = x/12
4x = 3 × 12
4x = 36
x = 9
Answer: It will take him 9 hours to mow 12 lawns.
Rank the following in order from most precise to least precise.
15.46 seconds
980 meters
900 miles
8.431 grams
The order from utmost precise to least precise is
8.431 grams, 15.46 seconds, 980 meters, 900 miles.
Ranking the following in order from utmost precise to least precise
8.431 grams ( utmost precise) This dimension has the loftiest position of perfection as it includes three decimal places.
15.46 seconds - This dimension has two decimal places, indicating a advanced position of perfection compared to whole figures.
980 meters measures This dimension is a whole number and lacks decimal places, making it less precise than measures with decimal values.
900 long hauls( Least precise) This dimension is also a whole number and lacks decimal places, making it the least precise among the given options.
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The total surface area of a cube is 294 cm².
Work out the volume of the cube.
Optional working
Answer
7/45
cm³
+
To find the volume of the cube, we need to know the length of one side of the cube. However, since it is not provided in the question, we cannot calculate the exact volume.
We can use a formula to relate the volume of the cube to its surface area. The formula is:
Surface Area = 6 * (side)^2
Given that the surface area of the cube is 294 cm², we can set up the equation:
294 = 6 * (side)^2
Divide both sides of the equation by 6:
49 = (side)^2
Take the square root of both sides:
side = √49
side = 7 cm
Now that we know the length of one side of the cube is 7 cm, we can calculate the volume using the formula:
Volume = side^3
Volume = 7^3
Volume = 343 cm³
Therefore, the volume of the cube is 343 cm³.
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Without actually drawing the figure, could you form a triangle using side lengths 7, 8, and 18 units?
Why or why not?
Answer:
No you cannot
Step-by-step explanation:
Triangle side length rule says the sum of any two sides must be greater than the remaining side
7 + 8 is NOT greater than 18 ....you cannot draw a triangle with these side lengths
No, we cannot form a triangle using side lengths 7, 8, and 18 units.
The Triangle Inequality Theorem can be used to determine if a triangle can be built with sides that are 7, 8, and 18 units long. The lengths of any two sides of a triangle with side lengths a, b, and c must add up to more than the length of the third side, according to the theorem.
Let's determine if this criterion is satisfied for the specified side lengths:
7 + 8 = 15
7 + 18 = 25
8 + 18 = 26
In this instance, the length of the third side (18) is greater than the total of the two shorter side lengths (7 and 8). This means that a triangle cannot be built using side lengths of 7, 8, and 18 units, according to the Triangle Inequality Theorem.
As a result, with side lengths of 7, 8, and 18 units, a triangle cannot be built according to the Triangle Inequality Theorem.
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find A flat rectangular roof measures 7.5 m by 4 m; 12 mm of rain falls on the roof. a Find the volume of water on the roof. Express your answer in i cm' and ii m'. Find the mass of water that falls on the roof if 1 cm³ of water has a mass b of 1 gram. Express your answer in kilograms.
1. The volume of water on the roof in cm and m is
240000 and 0.24 respectively.
2. The mass of water that fell on the roof is 2400kg
What is volume?Volume is defined as the space occupied within the boundaries of an object in three-dimensional space. It is also the capacity of a container.
Volume = l× b × h
l = 7.5 m
b = 4m
h = 12mm = 0.012
volume = 5 × 4 × 0.012
= 0.24 m³
in centimeter,
= 0.24 × 100³
= 240,000 cm³
2. The density of water is 1g/cm³
1g = 0.01kg
= 0.01kg/cm³
density = mass/volume
mass = density × volume
= 0.01 × 240000
= 2400 kg
therefore the mass of water that fell on the roof is 2400kg
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Describe How y=5 and y=5x-4 are related
Answer:
Step-by-step explanation:
Both y = 5 and y = 5x - 4 are equations of straight lines in the Cartesian plane. The first equation, y = 5, represents a horizontal line passing through the y-axis at 5. The second equation, y = 5x - 4, represents a line with a slope of 5 and y-intercept of -4. These two lines are related in that the second equation is a linear function with a non-zero slope, while the first equation is a special case of the second equation where the slope is zero. Additionally, the second equation can be obtained from the first equation by adding a non-zero slope term.
In a game of DND, Evy rolls a 20 sided die one time. What is the probability that she rolls an odd number, or a number less than 10? Round to the nearest tenth
The probability that she rolls an odd number, or a number less than 10 is 0.7
Calculating the probability that she rolls an odd number, or a number less than 10?From the question, we have the following parameters that can be used in our computation:
Die = 20-sided
In the 20-sided die, we have
A = Odd numbers = 10
B = Numbers less than 10 = 9
A and B = Odd numbers less than 10 = 5
So, we have
P(A) = 10/20
P(B) = 9/20
P(A and B) = 5/20
The probability expression P(A or B) can be calculated using
P(A or B) = P(A) + P(B) - P(A and B)
When the given values are substituted in the above equation, we have the following equation
P(A or B) = 10/20 + 9/20 - 5/20
Evaluate
P(A or B) = 0.7
Hence, the value of the probability P(A or B) is 0.7
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the equation of line t is y=7/4x+2. Perpendicular to line t is line u, which passes through the point (2, – 2). What is the equation of line u?
keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above
[tex]y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{7}{4}}x+2\qquad \impliedby \qquad \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ \cfrac{7}{4}} ~\hfill \stackrel{reciprocal}{\cfrac{4}{7}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{4}{7} }}[/tex]
so we're really looking for the equation of a line whose slope is -4/7 and it passes through (2 , -2)
[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{-2})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{4}{7} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-2)}=\stackrel{m}{- \cfrac{4}{7}}(x-\stackrel{x_1}{2}) \implies y +2 = - \cfrac{4}{7} ( x -2) \\\\\\ y+2=- \cfrac{4}{7}x+\cfrac{8}{7}\implies y=- \cfrac{4}{7}x+\cfrac{8}{7}-2\implies {\Large \begin{array}{llll} y=- \cfrac{4}{7}x-\cfrac{6}{7} \end{array}}[/tex]
Consider the function. f(x) = f(x − 1)² + 7. Select ALL of the statements that are true.
The axis of symmetry of f(x) is y = 7.
The axis of symmetry of f(x) is x = 1.
The vertex of the function is (1,7).
The vertex of the function is (-1, 7).
f is increasing on the interval -∞ → x → 1
f is increasing on the interval 1 → x → ∞
f is decreasing on the interval 1 → x → ∞
f is decreasing on the interval -∞ → x → 1
Answer:
Step-by-step explanation:
Here Is a Quick Explanation :)
To find the axis of symmetry and the vertex of the function, we can use the fact that the function is recursive, and we can write:
f(x) = f(x-1)² + 7
f(x-1) = f(x-2)² + 7
f(x-2) = f(x-3)² + 7
...
f(1) = f(0)² + 7
If we substitute the last equation into the previous one, we get:
f(x-1) = (f(0)² + 7)² + 7
f(x) = ((f(0)² + 7)² + 7)² + 7
This shows that the function depends only on the initial value f(0), and we can use this fact to find the axis of symmetry and the vertex.
To find the axis of symmetry, we need to find the value of x that makes f(x) equal to f(-x). We can write:
f(-x) = f(-(x-1))² + 7 = f(-x+1)² + 7
Now, if we substitute f(x) = f(x-1)² + 7 into the last equation, we get:
f(-x) = (f(x-1)² + 7)² + 7 = f(x)² + 7
This means that the axis of symmetry is the line x = 0, and not y = 7 as stated in option A.
To find the vertex, we need to find the maximum or minimum value of the function. Since f(x) = f(x-1)² + 7, the function is increasing if f(x-1) > -7, and decreasing if f(x-1) < -7. Since f(0) = 7, we can conclude that the function is increasing on the interval -∞ < x < 1, and decreasing on the interval x > 1. Therefore, the vertex is at x = 1, and the corresponding value is f(1) = 7.
Therefore, the correct statements are:
The axis of symmetry of f(x) is x = 0.
The vertex of the function is (1, 7).
f is increasing on the interval -∞ < x < 1.
f is decreasing on the interval x > 1.
you first roll the standard six-sided die once, and then you draw as many cards (from the standard deck of cards) as the number you rolled. so for instance: if you roll 4 you draw 4 cards for the deck. what is the probability that among the cards you draw there will be (a) exactly 3 aces, exactly 2 queens, and exactly 1 king?
Therefore, the probability of drawing exactly 3 aces, 2 queens, and 1 king is approximately 0.000682.
We can solve this problem using the multiplication rule for independent events. The probability of drawing a specific card from a standard deck is 1/52, since there are 52 cards in the deck and each is equally likely to be drawn. We can use this probability to find the probability of drawing a specific combination of cards.
(a) To find the probability of drawing exactly 3 aces, 2 queens, and 1 king, we can break it down into three steps:
Find the probability of drawing exactly 3 aces: There are 4 aces in the deck, so the probability of drawing an ace on the first draw is 4/52. Since we need exactly 3 aces, the probability of drawing 3 aces and 3 non-aces (out of the remaining 48 cards) is given by the binomial probability formula:
P(3 aces) = (4/52)^3 * (48/52)^3 * C(6,3)
where C(6,3) is the number of ways to choose 3 cards out of 6. Using a calculator, we get:
P(3 aces) = 0.0080
Find the probability of drawing exactly 2 queens: There are 4 queens in the deck, so the probability of drawing a queen on the first draw is 4/52. Since we need exactly 2 queens, the probability of drawing 2 queens and 4 non-queens (out of the remaining 48 cards) is given by:
P(2 queens) = (4/52)^2 * (48/52)^4 * C(6,2)
where C(6,2) is the number of ways to choose 2 cards out of 6. Using a calculator, we get:
P(2 queens) = 0.2123
Find the probability of drawing exactly 1 king: There are 4 kings in the deck, so the probability of drawing a king on the first draw is 4/52. Since we need exactly 1 king, the probability of drawing 1 king and 5 non-kings (out of the remaining 48 cards) is given by:
P(1 king) = (4/52)^1 * (48/52)^5 * C(6,1)
where C(6,1) is the number of ways to choose 1 card out of 6. Using a calculator, we get:
P(1 king) = 0.4017
Now we can use the multiplication rule to find the probability of drawing exactly 3 aces, 2 queens, and 1 king:
P(exactly 3 aces, 2 queens, 1 king) = P(3 aces) * P(2 queens) * P(1 king)
= 0.0080 * 0.2123 * 0.4017
= 0.000682
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