Ansley's cousin is 12 years old, and Ansley's age can be found by plugging in 12 for Cousin's age.
How can we know that Ansley's age is 5 years less than 3 times her cousin's age?The problem tells us that Ansley's age is 5 years less than 3 times her cousin's age. We can write this as an equation:
Ansley's age = 3 × Cousin's age - 5
We also know that Ansley is 31 years old. So we can substitute 31 for Ansley's age in the equation:
31 = 3 × Cousin's age - 5
Now we solve for Cousin's age. First, we add 5 to both sides of the equation:
31 + 5 = 3 × Cousin's age
Simplifying:
36 = 3 × Cousin's age
Finally, we divide both sides by 3:
Cousin's age = 12
So Ansley's cousin is 12 years old, and Ansley's age can be found by plugging in 12 for Cousin's age in the expression we found earlier:
Ansley's age = 3 × Cousin's age - 5 = 3 × 12 - 5 = 31
So Ansley is indeed 31 years old.
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What is the solution of |x – 6| ≥ 1? 5 < x < 7 x ≤ –7 or x ≥ –5 x ≤ 5 or x ≥ 7 –7 < x < –5
Answer:
(c) x ≤ 5 or x ≥ 7
Step-by-step explanation:
You want the solution to |x -6| ≥ 1.
UnfoldThe absolute value relation represents two relations, one for the domain x < 6, and one for the domain x ≥ 6.
x < 6In this domain, the inequality becomes ...
-1 ≥ x -6
5 ≥ x . . . . . . add 6
x ≤ 5 . . . . . . . put x on the left
x ≥ 6In this domain, the inequality is ...
x -6 ≥ 1
x ≥ 7
The disjoint solution sets are x ≤ 5 or x ≥ 7.
__
Additional comment
For |x -a| ≤ b, we can "unfold" this to the compound inequality ...
-b ≤ (x -a) ≤ b
copying the inequality symbol to the left side, and writing the opposite of the constant there.
We can do the same thing with the inequality ...
|x -a| ≥ b
but it doesn't really make sense as a compound inequality.
Instead, we have to write it as ...
-b ≥ (x -a) or (x -a) ≥ b
in recognition of the fact that the solution spaces are disjoint.
Anne's Road Paving Company mixed 16 1/4 tons of cement. They used 6 3/4 tons of the cement to pave a street downtown. How much cement did they have left?
Answer is 9.5 tons of cement
Anne's Road Paving Company initially mixed 16 1/4 tons of cement. They used 6 3/4 tons for paving a street downtown. To find the remaining amount of cement, subtract the used amount from the initial amount:
16 1/4 - 6 3/4 = 15 1/4 - 5 3/4 = 9 1/2 tons.
So, they had 9 1/2 tons of cement left.
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In QRS, the measure of angle S=90°, the measure of angle Q=6°, and RS = 20 feet. Find the
length of SQ to the nearest tenth of a foot.
R
20
6°
s
Q
X
The length of SQ to the nearest tenth of a foot is approximately 2.1 feet.
To find the length of SQ, we can use trigonometry. First, we can find the measure of angle R by subtracting the measures of angles Q and S from 180°:
R = 180° - 90° - 6° = 84°
Then, we can use the sine function to find the length of SX (which is equal to SQ):
sin(Q) = SQ / RS
sin(6°) = SQ / 20
SQ = 20 * sin(6°)
SQ ≈ 2.07 feet (rounded to the nearest tenth of a foot)
Therefore, the length of SQ to the nearest tenth of a foot is approximately 2.1 feet.
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The probability that Max will have to stop for a passing train on his route to work is 0. 5. The probability that there will be construction on Max's route to work ,begin emphasis,and,end emphasis, that he will have to stop for a train is 0. 4. What is the probability that there was construction if Max had to stop for a passing train on his route to work?
The problem is asking for the probability of construction given that Max had to stop for a passing train on his route to work. This can be solved using Bayes' theorem, which states that the probability of A given B is equal to the probability of B given A multiplied by the probability of A, divided by the probability of B.
In this case, let A be the event that there is construction on Max's route, and let B be the event that Max has to stop for a passing train. We are looking for the probability of A given B.
Using Bayes' theorem, we have:
P(A|B) = P(B|A) * P(A) / P(B)
We know that P(B) = 0.5, the probability that Max has to stop for a passing train. We also know that P(B|A) = 0.4, the probability that there is construction and Max has to stop for a passing train.
To find P(A), the probability of construction on Max's route, we need to use the complement of the event A, which is the probability that there is no construction:
P(not A) = 1 - P(A) = 1 - 0.4 = 0.6
Finally, we can plug in the values and solve for P(A|B):
P(A|B) = 0.4 * 0.4 / 0.5 = 0.32
Therefore, the probability that there was construction if Max had to stop for a passing train on his route to work is 0.32 or 32%.
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list all the Factors. circle the GCF.
6:
9:
list 7 multiples.Circle the LCM.
5:
2:
Answer:
List all the factors
6: 3, 2, 1 (6)
9:3,(9),1
5:(5),1
2:(2),1
Step-by-step explanation:
Please asap!!! will give 100 brainlest!!! (there's more than one answer)
select all the correct measures of center and variation for the following data set.
10, 20, 31, 17, 18, 5, 22, 25, 14, 43
a. first quartile = 12
b. iqr = 11
c. median = 19
d. third quartile = 25
e. mad = 7
First quartile is 14, IQR is 14, median is 19, third quartile is 28 and MAD is 7.
a. First quartile = 12 and d. Third quartile = 25 are not necessarily correct measures of quartiles for this dataset. To calculate the quartiles, we need to first order the data set and then find the value(s) that divide it into four equal parts. In this case, the sorted dataset is:
5, 10, 14, 17, 18, 20, 22, 25, 31, 43
The first quartile is the median of the lower half of the data: (5, 10, 14, 17, 18) and is 14.
b. IQR = 11 is not correct. The IQR (Interquartile Range) is the difference between the third quartile and the first quartile, which is 28-14=14 for this dataset.
c. Median = 19 is a correct measure of center.
d. The third quartile is the median of the upper half of the data: (22, 25, 31, 43) and is 28.
e. MAD = 7 is a correct measure of variation.
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A bag of sweets contains only gobstoppers and sherbert lemons.
There are 3 gobstoppers for every 4 sherbert lemons.
There are 56 sweets in the bag. How many gobstoppers are there?
An oil tank is the shape of a right rectangular prism. The inside of the tank is 36. 5 cm long, 52 cm wide, and 29 cm
high. If 45 liters of oil have been removed from the tank since it was full, what is the current depth of oil left in the
tank?
The current depth of oil left in the tank is approximately 4.64 cm.
The volume of the oil tank can be found by multiplying its length, width, and height:
Volume of the oil tank = length x width x height
= 36.5 cm x 52 cm x 29 cm
= 53,854 cubic cm
If 45 liters of oil have been removed from the tank, the current volume of oil in the tank is:
Current volume of oil = Total volume of tank - Volume of oil removed
= 53,854 cubic cm - 45,000 cubic cm (1 liter = 1000 cubic cm)
= 8,854 cubic cm
Let's assume that the depth of oil left in the tank is x cm. Then the volume of oil left in the tank can be found by multiplying the length, width, and depth of oil:
Volume of oil left in tank = length x width x depth of oil
= 36.5 cm x 52 cm x x cm
= 1906x cubic cm
Now we can set up an equation to find the value of x:
1906x = 8,854
Dividing both sides by 1906, we get:
x = 4.64 cm
Therefore, the current depth of oil left in the tank is approximately 4.64 cm.
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Emma is making a scale drawing of her farm using the scale 1 centimeter to 2. 5 feet. In the drawing, she drew a well with a diameter of 0. 5 ccentimeter. Which is the closest to the actual circumference of the well?
The circumference of the well is 3.93 ft.
Given, Emma is making a scale drawing of her farm using the scale 1 cm=2.5 ft
Diameter of the well she drew = 0.5 cm
We need to convert the diameter of the well from centimeters to feet, using the given scale.
i.e. 0.5cm = 2.5/2 = 1.25 ft
We know the radius is half of the diameter.
So, r = 1.25/2 = 0.625
We know that the formula for the circumference of a circle is C = 2πr
C = 2*3.14*0.625
= 3.93 ft
Hence, the circumference of the well is 3.93 ft.
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There are four spaces each so you can put either parentasis or brakets
In the given function the domain is [-1, ∞]
Range is [-3, ∞]
The interval when function is positive [0, ∞]
The domain of a function is the set of values that we are allowed to plug into our function.
This set is the x values in a function such as f(x).
The range of a function is the set of values that the function assumes
In the given function the domain is [-1, ∞]
Range is [-3, ∞]
The interval when function is positive [0, ∞]
The interval when function is negative [-∞, -1]
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Question #6
Samantha is fishing at the pier, holding her rod with two hands. At the same time she also
has her gun equipped, so she is able to defend herself when robbers approach her. What is
Samantha doing wrong?
Your answer is 150 characters short!
Enter your answer here.
Samantha is doing two things wrong: she is breaking the law and it is unsafe to handle a firearm while fishing.
Samantha is breaking the law by possessing a firearm while fishing at the pier. Most fishing piers are considered public places and carrying a firearm in public places is usually prohibited unless the person has a valid permit or is a law enforcement officer. It could cause panic among other people around her. It is important to follow local laws and regulations regarding firearms and to prioritize safety when in public spaces.
Additionally, it is unsafe and irresponsible to handle a firearm while fishing as it can cause accidents or injuries to oneself or others. She should not hold her fishing rod with both hands if she needs to be prepared to use her gun for self-defense. It would be difficult to access and use the gun effectively while holding the rod with both hands.
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The image shows three sets of stuffed bears. Each set represents a term of the sequence (1, 4, 7,. . . ). An arrangement of stuffed toy bears in groups of 1, 4, and 7
What is the next term in the sequence?
Describe the domain of the sequence. Describe the range of the sequence
The next term in the sequence of the series which have groups of 1, 4, and 7 is 10.
The fundamental concepts in mathematics are series and sequence. A series is the total of all components, but a sequence is an ordered group of items in which repeats of any kind are permitted. One of the typical examples of a series or a sequence is a mathematical progression.
We have the series as 1, 4, 7, ....
First term = a = 1
Common difference = d = 3
Using the formula for the Term is
T = a + (n-1)d
T = 1 + (n-1)3
= 1 + 3n - 3
T = 3n - 2
To find the next term in the series we need to find the 4th term so
T₄ = 3(4) - 2
= 12 - 2
T₄ = 10.
The domain of the sequence T = 3n - 2 is all Real numbers n ∈ Real numbers.
The range is given as
R ∈ (-∞, ∞).
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Answer:
1, 4, and 7 is 10
Step-by-step explanation:
The pattern sequence follows the add 3 rule so, the next term in the sequence will be 10.
The index of the terms of represents the domain of a function, which is { 1, 2, 3, . . .}.
The range includes the terms of the sequence {1, 4, 7, . . .}.
A cell phone leans against a wall. The bottom of the phone is 4 inches from the base of the wall, and the top of the phone makes an angle of 52 degrees with the wall. Find the length, x, of the phone so you can buy a new case. Round to the nearest hundreths place
The length of the phone is approximately 6.08 inches, so you can buy a case that fits this size.
To find the length, x, of the phone, we can use trigonometry. We know that the bottom of the phone is 4 inches from the base of the wall, so we can use the tangent function to find the length of the phone.
tangent(52 degrees) = opposite/adjacent
The opposite side is x (the length of the phone) and the adjacent side is 4 inches.
So,
tangent(52 degrees) = x/4
Multiplying both sides by 4, we get:
4 * tangent(52 degrees) = x
Using a calculator, we find that:
x ≈ 6.08 inches
Therefore, the length of the phone is approximately 6.08 inches, so you can buy a case that fits this size.
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Luke has scored a goal in 15 of his 26 soccer games this season and has a hit in 12 of his 16 baseball games this season. Based on the results in his season so far, Luke wants to figure out the probability that he will score a goal in his next soccer game and get a hit in his next baseball game. Enter the probability as a fraction in reduced form
The probability of Luke scoring a goal in his next soccer game and getting a hit in his next baseball game is 45/104 in reduced form.
The probability of Luke scoring a goal in his next soccer game is the ratio of the number of games he scored a goal to the total number of soccer games he played so far. Thus, the probability of scoring a goal in his next game is 15/26.Similarly, the probability of Luke getting a hit in his next baseball game is the ratio of the number of games he had a hit to the total number of baseball games he played so far.
Thus, the probability of getting a hit in his next game is 12/16.Since the events are independent, we can use the product rule to find the probability of both events happening together. Thus, the probability of scoring a goal in his next soccer game and getting a hit in his next baseball game is (15/26) x (12/16) = 45/104 in reduced form.
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what the size of angle g 82,104,76
Answer:
angle g is 98 degrees.
Step-by-step explanation:
assuming the figure is a quadrilateral,
angle g + 82 + 104 + 76 = 360 ( Property of Quadrilateral)
262 + angle g = 360
angle g = 98 degrees
11. The function f(t) = 40 sin (π/45 t) +48 models the height in feet of one car od a Ferris wheel called the Colossus, where t is the time in seconds. Each of the functions below models the motion of a different Ferris wheel. Which Ferris wheel has the same diameter as The Colossus?
a. g(t) = 40 cos (π/45 t) +50
b. h(t) = 39 cos (π/60 t) + 49
c. j(t) = 39 sin (π/45 t) + 48
d. k(t) = 39 sin (π/45 t) + 49
Ferris wheel has the same diameter as The Colossus is g(t) = 40 cos (π/45 t) +50. So, correct option is A.
To determine which of the given functions represents a Ferris wheel with the same diameter as The Colossus, we need to use the fact that the diameter of a Ferris wheel is equal to the amplitude of the sinusoidal function that models its height.
The amplitude of the function f(t) = 40 sin (π/45 t) +48 is 40, so the diameter of The Colossus is 40 feet. We need to find the function that also has an amplitude of 40.
Looking at the given answer choices, we see that function g(t) has an amplitude of 40 cos (π/45 t) +50, which is equal to 40. This means that the Ferris wheel represented by function g(t) has a diameter of 40 feet, the same as The Colossus.
Functions h(t), j(t), and k(t) all have amplitudes that are less than 40, so they represent Ferris wheels with smaller diameters than The Colossus.
Therefore, the answer is A.
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Type the correct answer in the box.
The given equation, V = 1/3 πr²h, solved for h is:
h = 3V / πr²
Subject of formulae: Solving the equation for hFrom the question, we are to solve the give equation for h
From the given information,
The given equation is
V = 1/3 πr²h
To solve the equation for h, we will isolate h
Solving the equation for h
V = 1/3 πr²h
Multiply both sides of the equation by 3
3 × V = 3 × 1/3 πr²h
3V = πr²h
Divide both sides of the equation by πr²
3V / πr² = πr²h / πr²
3V / πr² = h
This can be written as
h = 3V / πr²
Hence, the equation solved for h is:
h = 3V / πr²
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A baker has small and large bags of sugar for making cakes. The large bag contains 30 cups of sugar and it's 2. 5 times larger than the small bag. The small bag contains enough sugar to make nine cakes and have. 75 cups of sugar remaining
How many cakes can be made with a large bag of sugar?
The number of cakes that can be made with a large bag of sugar, we first need to determine the amount of sugar in a small bag and then calculate the amount of sugar needed for one cake.
1. Find the amount of sugar in a small bag:
Since the large bag contains 30 cups of sugar and is 2.5 times larger than the small bag, we can write the equation:
Small bag = Large bag / 2.5
Small bag = 30 cups / 2.5
Small bag = 12 cups of sugar
2. Determine the amount of sugar needed for one cake:
The small bag contains enough sugar to make 9 cakes and have 0.75 cups of sugar remaining. So, we can subtract the remaining sugar from the total amount in the small bag:
Sugar used for 9 cakes = 12 cups - 0.75 cups
Sugar used for 9 cakes = 11.25 cups
Now, we can find the amount of sugar needed for one cake:
Sugar per cake = Sugar used for 9 cakes / 9
Sugar per cake = 11.25 cups / 9
Sugar per cake = 1.25 cups
3. Calculate the number of cakes that can be made with a large bag of sugar:
Cakes from large bag = Large bag sugar / Sugar per cake
Cakes from large bag = 30 cups / 1.25 cups
Cakes from large bag = 24
Therefore, a baker can make 24 cakes with a large bag of sugar.
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Find the value of a2+bc√−d, when
a = –3, b = 2, c = 100, and d = –2
To find the value of a² + bc√(-d) when a = -3, b = 2, c = 100, and d = -2, follow these steps:
Step 1: Substitute the values into the expression.
a² + bc√(-d) = (-3)² + (2)(100)√(-(-2))
Step 2: Simplify the expression.
(-3)² + (2)(100)√(2) = 9 + 200√2
So, the value of a² + bc√(-d) when
a = -3,
b = 2,
c = 100,
d = -2 is 9 + 200√2.
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A favorite activity at LNHS is throwing paper
balls into the trashcan while the teacher isn't
looking. Suppose a paper ball is shot from 5 feet
off the ground, and the paper ball reaches a
height of 10 feet after 3 seconds.
*Write the equation that models the height (h)
of the paper ball at any given second (t).
Help me!!
The equation that models the height (h) of the paper ball at any given second (t) is: [tex]h = -16t^2 + 49.67t + 5.[/tex]
To write the equation that models the height (h) of the paper ball at any given second (t), we can use the formula:
[tex]h = -16t^2 + vt + s[/tex]
where v is the initial velocity (in feet per second), s is the initial height (in feet), and t is the time (in seconds).
In this case, we know that the paper ball was shot from 5 feet off the ground, so s = 5. We also know that the paper ball reached a height of 10 feet after 3 seconds, so we can use this information to find the initial velocity:
[tex]h = -16t^2 + vt + s[/tex]
[tex]10 = -16(3)^2 + v(3) + 5[/tex]
10 = -144 + 3v + 5
149 = 3v
v = 49.67 (rounded to two decimal places)
Now we can substitute the values for v and s into the equation:
[tex]h = -16t^2 + vt + s\\h = -16t^2 + 49.67t + 5[/tex]
Therefore, the equation that models the height (h) of the paper ball at any given second (t) is:
[tex]h = -16t^2 + 49.67t + 5.[/tex]
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Jessica records the number of winners at the dunk-a-teacher booth at the town fair as shown in the table. if there are 750 contestants on monday, how many should jessica expect to dunk a teacher? enter your answer in the box.
If there are 750 contestants on Monday, Jessica should expect around 75 of them to dunk a teacher.
Based on the table provided, we can see that the percentage of winners at the dunk-a-teacher booth varies from day to day.
On Monday, 10% of contestants were able to dunk a teacher.
Therefore, if there are 750 contestants on Monday, Jessica should expect around 75 of them to dunk a teacher.
This is calculated by multiplying 750 by 0.10, which gives us 75. It's important to note that this is just an estimation, as the actual number of winners may be slightly higher or lower than 75.
However, this gives Jessica a rough idea of what to expect at the booth on Monday.
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The slant height if the cone is 13 cm. What is the volume of a cone having a radius of 5 cm and a slant height of 13 cm.
The formula for the volume of a cone is:
V = (1/3)πr^2h
where r is the radius of the base of the cone and h is the height of the cone.
We are given that the radius of the cone is 5 cm and the slant height is 13 cm. We can use the Pythagorean theorem to find the height of the cone:
h^2 = l^2 - r^2
where l is the slant height of the cone. Substituting the given values, we get:
h^2 = 13^2 - 5^2
h^2 = 144
h = 12
Now we can substitute the values of r and h into the formula for the volume of the cone:
V = (1/3)πr^2h
V = (1/3)π(5^2)(12)
V = (1/3)π(25)(12)
V = (1/3)π(300)
V = 100π
Therefore, the volume of the cone is 100π cubic centimeters.
The president of the student council wants to survey the student population about parking. She decides to use a
random number table to take a random sample of 100 of the 1,020 students at the school. What is the smallest number
of digits that should be used to label the population?
The smallest number of digits that should be used to label the population is 4, since there are 1,020 students in the school.
To determine the smallest number of digits needed to label the population, we will follow these steps:
1. Determine the total number of students in the population.
2. Identify the number of digits needed to represent the largest student number.
3. Apply this number of digits to all student labels.
Let's apply these steps to your question:
1. The total number of students is 1,020.
2. The largest student number is 1,020, which requires 4 digits (since it has four numbers: 1, 0, 2, and 0).
3. To label the entire student population consistently, use 4 digits for all student labels.
So, the smallest number of digits that should be used to label the population is 4.
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For how many different integers $k$ are there rational solutions to the quadratic equation
[tex]\[x^2 + kx + 4k = 0?\][/tex]
For k = 0 and k = 16, there are rational solutions to the quadratic equation [tex]x^{2} + kx + 4k = 0[/tex]
We are given a quadratic equation [tex]x^{2} + kx + 4k = 0[/tex]
An algebraic equation in x with a degree of 2 is known as a quadratic equation. It is written in the format [tex]a[/tex][tex]x^{2}[/tex] [tex]+ bx + c[/tex] = 0. To find out whether there exists two solutions, one solution, or no solution for a quadratic equation, we use the discriminant of the quadratic equation.
We will find the solutions to this quadratic equation with the help of discriminant formula
As we know from the equation that b = k, a = 1, and c = 4k.
[tex]b^2 - 4ac = 0[/tex]
[tex]k^2 - 4(4k) = 0[/tex]
[tex]k^2 - 16k = 0[/tex]
k (k-16) = 0
k = 0 or k - 16 = 0
k = 0 or k = 16
So, for k = 0 or k = 16 the equation [tex]x^{2} + kx + 4k = 0[/tex] has only one solution.
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Use the summation formulas to rewrite the expression without the summation notation.
∑ 8i+7/n^2
The expression without the summation notation for ∑ 8i+7/n²2 using the summation formulas is (4n + 3)/2n.
To rewrite the expression without the summation notation, we need to use the summation formulas. We can start by expanding the given summation:
∑ 8i+7/n²2 = 8(1)/n²2 + 8(2)/n²2 + 8(3)/n²2 + ... + 8(n)/n²2 + 7/n²2
Next, we can simplify each term by factoring out 8/n²2:
= (8/n²2)(1 + 2 + 3 + ... + n) + 7/n²2
Using the formula for the sum of the first n positive integers, we have:
= (8/n²2)(n(n+1)/2) + 7/n²2
= (4n² + 4n)/2n² + 7/n²2
= (4n + 3)/2n
Therefore, the expression is (4n + 3)/2n.
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Mrs. mueller writes an inequality on the board. the table shows the responses of four students for possible values of x.
x>6
student
jacob
kendra
luke
maya
response
6
8
10
12
which student has a correct response to mrs. mueller's inequality?
o jacob
o kendra
o luke
o maya
The inequality given by Mrs. Mueller is x>6, which means that x is greater than 6. To check which student has given the correct response, we need to check if their values of x satisfy the given inequality.
Looking at the table, we see that all four students have given values of x that are greater than 6. However, we need to choose the student who has given the correct response to the inequality.
Jacob has given the response 8, which satisfies the inequality x>6. Kendra has given the response 10, which also satisfies the inequality. Luke has given the response 12, which is also greater than 6 and satisfies the inequality. Maya has given the response 10, which is the same as Kendra's response and also satisfies the inequality.
Therefore, we can say that all four students have given correct responses to Mrs. Mueller's inequality.
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What’s the answer I need help pls?
Answer:
(E). y = 2cos(3x)
Step-by-step explanation:
First, amplitude of cos(x) is 1 , then 2cos(x) has amplitude 2
Second, period of cos(x) is 2[tex]\pi[/tex] , then 3 × [tex]\frac{2\pi }{3}[/tex] = 2[tex]\pi[/tex]
So, the answer is y = 2cos(3x)
Mary creates a stack of 10 of piece am and a stack of 8 of piece N. Both stacks have equal volumes. Create an equation relating h and k
Let's assume that the pieces am and N have heights of h_am and h_N respectively, and let k be the number of times the height of piece N fits into the height of piece am (i.e., k is the ratio of the height of piece am to the height of piece N).
We know that the volume of each stack is equal. Let's use the following variables:
- h for the height of each piece of A
- k for the height of each piece of N
- 10 for the number of pieces in stack A
- 8 for the number of pieces in stack N
The equation for the volume of each stack is:
Volume of stack A = h x 10
Volume of stack N = k x 8
Since we know the volumes are equal, we can set the two equations equal to each other:
h x 10 = k x 8
To create an equation relating h and k, we can solve for one variable in terms of the other:
h = (8/10)k
or
k = (10/8)h
Either equation shows how h and k are related to each other. For example, if we know the value of h, we can use the first equation to find k.
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Qn in attachment . ..
The variance of first n even natural numbers is n²-1/12. So, the correct option is (a) .
The first n even natural numbers can be represented as 2, 4, 6, ..., 2n. The mean or expected value of this sequence is given by:
mean = (2 + 4 + 6 + ... + 2n) / n
= 2(1 + 2 + 3 + ... + n) / n
= 2n(n+1)/2n
= n+1
The variance of a sequence is the average of the squared differences from the mean, so we need to calculate:
Var = [(2- (n+1))² + (4- (n+1))² + (6- (n+1))² + ... + (2n- (n+1))²] / n
Simplifying the expression inside the brackets and using the formula for the sum of the first n integers, we get:
Var = [4(1² + 2² + 3² + ... + n²) - 4(n+1)(1 + 2 + 3 + ... + n) + n(n+1)²] / n
Substituting the formula for the sum of the first n squares and the sum of the first n integers, we get:
Var = [4n(n+1)(2n+1)/6 - 4(n+1)n(n+1)/2 + n(n+1)²] / n
Simplifying and factoring out (n+1), we get:
Var = (n+1)(n² - 1) / 3
Thus, the correct option is (a) n²-1/12.
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Use even and odd functions to evaluate the following integral. ſ(cosa + 3x4) dx -T
The integral of ſ(cosa + 3x^4) dx simplifies to ∫cos(x) dx, which can be evaluated as sin(x) + C, where C is the constant of integration.To evaluate the integral of ſ(cosa + 3x^4) dx using even and odd functions, we can decompose the integrand into even and odd parts.
Let's first identify the even and odd parts of the integrand. The function cos(x) is an even function because it is symmetric with respect to the y-axis, i.e., cos(-x) = cos(x). On the other hand, the function 3x^4 is an odd function because it is symmetric with respect to the origin, i.e., (-x)^4 = x^4.
We can rewrite the integrand as a sum of even and odd functions:
cos(x) + 3x^4 = (1/2) * (cos(x) + cos(-x)) + (1/2) * (3x^4 - 3(-x)^4)
Now, we can use the properties of even and odd functions to simplify the integral. The integral of an even function over a symmetric interval is equal to twice the integral of the function over half of the interval. Similarly, the integral of an odd function over a symmetric interval is equal to zero.
So, the integral of (1/2) * (cos(x) + cos(-x)) dx is equal to (1/2) * 2 * ∫cos(x) dx, since cos(x) is an even function.
And the integral of (1/2) * (3x^4 - 3(-x)^4) dx is equal to (1/2) * 0, since 3x^4 - 3(-x)^4 is an odd function and the interval of integration is symmetric.
Therefore, the integral of ſ(cosa + 3x^4) dx simplifies to ∫cos(x) dx, which can be evaluated as sin(x) + C, where C is the constant of integration.
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