Isaac pay for the cost of racket is 39.9.
The cost of a pair of shoes is three times the cost of a bag, so we can write:
Cost of shoes = 3b
Similarly, the cost of the racket is twice the cost of the bag, so we can write:
Cost of racket = 2b
Now we can use the given information to set up an equation:
Cost of racket + Cost of bag + Cost of shoes = $119.70
Substituting the expressions we found above, we get:
2b + b + 3b = $119.70
Simplifying the equation:
6b = $119.70
Dividing both sides by 6:
b = $19.95
So the cost of the bag is $19.95.
We can use this to find the costs of the shoes and racket:
Cost of shoes = 3b = 3($19.95) = $59.85
Cost of racket = 2b = 2($19.95) = $39.90
Therefore, Isaac paid $39.90 for the racket.
A cost is an expenditure required to produce or sell a product or get an asset ready for normal use. In other words, it's the amount paid to manufacture a product, purchase inventory, sell merchandise, or get equipment ready to use in a business process.
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Scott y Mark fueron a escalar. Scott subió a la cima de un risco de 75 pies, y desde allí le arrojó una soga de 96 pies a Mark, que estaba debajo de él en tierra. Si la soga quedó tirante desde los pies de Mark hasta los pies de Scott, ja qué distancia de la base del acantilado (directamente debajo de Scott) se encuentra parado Mark? Dibuja un diagrama y coloca los datos. Luego calcula la longitud faltante. ¿Es irracional la longitud?
Alex has 70% of her weekly paycheck automatically deposited in her savings account. This week, $35 is deposited. Alex wants to know the total amount of her paycheck this week.
find the projection bp of the vector of the right hand sides to the column space of the coefficient matrix.
To find the projection bp of the vector of the right hand sides to the column space of the coefficient matrix, we can use the formula:
bp = A(A^T A)^-1 A^T b
where A is the coefficient matrix and b is the vector of the right hand sides.
To find the projection of the vector b onto the column space of the coefficient matrix A, you need to perform the following steps:
1. Calculate the orthogonal projection matrix P using the formula P = A(A^T * A)^-1 * A^T, where A^T is the transpose of A, and (A^T * A)^-1 is the inverse of the product of A^T and A.
2. Multiply the projection matrix P with the vector b to obtain the projection vector bp: bp = P * b.
In summary, to find the projection bp of the vector of the right-hand sides to the column space of the coefficient matrix, calculate the orthogonal projection matrix P and then multiply it with the vector b.
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The probability that X is at least 7 is:
a.) 5/36
b.) 6/36
c.) 15/36
d.) 21/36
Answer:
The correct answer is d.) 21/36.If we assume a fair six-sided number cube, there are a total of 36 possible outcomes. Out of these, the favorable outcomes for X being at least 7 are 21, 22, 23, 24, 25, and 26 (six outcomes). Therefore, the probability of X being at least 7 is 6/36, which can be simplified to 1/6, or approximately 0.1667.
Step-by-step explanation:
Estimate the product 306 × 673 by first rounding each number to the nearest hundred
After the estimation, the product of 306 and 673 by first rounding each number to the nearest hundred is 2100.
To round off the number to the nearest hundred, we have to check the first two digits of the number and if the number is below 50 then we round off it to the same hundred position. Similarly, if the number is above 50 then we round off it to the next hundred places.
Given the numbers are 306 and 673,
306 has 06 as the first two digits and it is below 50 then after rounding off it is rounded off to 300.
673 has 73 as the first two digits and it is after 50 then after rounding off it is rounded off to 700.
Thus, after the estimation, the product can be calculated as:
300 * 700 = 2100
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What is the volume of the cylinder
An investment of $10,000 earns interest at an annual rate of 6. 7% compounded continuously. Answer Part 1 and Part 2 with this information.
Part 1:
Find the instantaneous rate of change in the amount in the account after 2 years (in dollars per year). Round to the nearest cent.
$____per year.
Part 2
Find the instantaneous rate of change in the amount in the account at the time the amount is equal to $14,101. Round to the nearest cent.
$_____per year
1. The instantaneous rate of change in the amount after 2 years is [tex]$1,605.64[/tex] per year
2. The instantaneous rate of change in the amount at the time the amount is equal to [tex]$14,101[/tex] is approximately $994.78 per year
[tex]A = P[/tex]× [tex]e^{rt}[/tex]
where P is the principal (initial investment), r is the annual interest rate as a decimal, and t is the time in years.
For this problem, we have P = $10,000, r = 0.067 (6.7% as a decimal), and we want to find the instantaneous rate of change in the amount after 2 years, so t = 2.
Part 1:
To find the instantaneous rate of change, we need to take the derivative of the function A(t) with respect to t:
[tex]dA/dt = Pre^{rt}[/tex]
At[tex]t = 2[/tex], we have:
[tex]A(2) = $10,000e^{0.0672}[/tex]
[tex]= $11,868.94[/tex]
[tex]dA/dt = $10,0000.067e^{0.067}[/tex]×[tex]2)[/tex]
[tex]= $1,605.64[/tex]
So the instantaneous rate of change in the amount after 2 years is $1,605.64 per year
Part 2:
To find the time at which the amount in the account is $14,101, we need to solve the equation A = $14,101 for t:
[tex]$14,101[/tex][tex]= $10,000[/tex] × [tex]e^{0.067t}[/tex]
Dividing both sides by $10,000:
[tex]1.4101 = e^{0.067t}[/tex]
Taking the natural logarithm of both sides:
[tex]ln(1.4101) = 0.067t[/tex]
Solving for t:
[tex]t = ln(1.4101)/0.067[/tex]
≈ [tex]3.5 years[/tex]
So the time at which the amount in the account is $14,101 is approximately 3.5 years.
To find the instantaneous rate of change at this time, we need to evaluate the derivative at t = 3.5:
[tex]dA/dt = $10,0000.067e^{0.067}[/tex]×[tex]3.5)[/tex]
≈ [tex]$994.78[/tex]
So the instantaneous rate of change in the amount at the time the amount is equal to $14,101 is approximately $994.78 per year
Compound interest is the interest you earn on interest. This can be illustrated by using basic math: if you have $100 and it earns 5% interest each year, you'll have $105 at the end of the first year. At the end of the second year, you'll have $110.25
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What is 6 1/2 + 3 1/2
Answer: 10
Step-by-step explanation:
6 1/2 + 3 1/2
13/2 + 7/2
20/2
= 10
Step 1: Add the whole numbers
6 + 3 = 9
Step 2: Add the fractions
We can simply add the numerators while keeping the same denominator because the denominators are the same.
1/2 + 1/2 = 2/2 = 1
Step 3: Combine them
The answer is 10 since 9 and 1 are whole numbers. So we simply add them.
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SummaryOverall, we add the whole numbers first, then add the fractions. If the denominators are identical, we add the numerators and keep the same denominator. The solution will be displayed as a mixed number/fraction or a whole number. Since there was no fraction at the end, in this case, we simply added the whole numbers.
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FAQWhat is a numerator?The top-written number in a fraction is the numerator. It shows how many parts of the whole you are talking about.
For example, the numerator of the fraction 3/5 is 3, which means there are 3 parts out of a total of 5 equal parts.
The number of parts taken out of the whole is therefore represented by the numerator.
What is a denominator?The number written at the bottom of a fraction works as the denominator. It gives the number of equally sized parts of the whole.
As an example, the denominator of the fraction 2/5 is 5, which shows that the entire is divided into four equal parts.
The total number of equal parts that make up the whole is represented by the denominator.
What is a mixed number/fraction?Mixing a full number and a fraction creates a mixed number. The whole number comes first, then a space, and then the fraction is written.
For example, the mixed number 2 1/2 is a whole number and a fraction, with 2 being the whole number. The word "and" between a mixed number's whole and fraction might be removed or included.
A proper fraction, or one that is less than one whole, must make up the fractional part of the mixed number. Quantities that are not whole numbers but instead consist of several whole numbers are represented by mixed numbers.
What is a whole number?A number that represents a finished item or thing is said to be a whole number. It is a number that is neither a decimal nor a fraction.
All natural numbers and zero are considered whole numbers. These are positive integers without decimals, fractions, or negative numbers. A few examples of entire numbers include 1, 2, 3, and so forth.
For counting items that cannot be divided into smaller portions, whole numbers are used.
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Help me solve this please and thanks! :’)
Answer:
385 in ^3
Step-by-step explanation:
11 x 7 x 5
Let S2(r) = {x ∈ R3 : |x| = r} for r >
0 and let f : S2 → S2(r) by f(x) = rx. Prove
that f is one-to-one and onto but not an isometry if r /= 1
The function f(x) = rx mapping S2 to S2(r) is one-to-one, onto, but not an isometry if r ≠ 1.
To prove that the function f: S2 → S2(r) defined by f(x) = rx is one-to-one, onto, and not an isometry if r ≠ 1, we'll consider the following:
1. One-to-one: For f to be one-to-one, for every distinct pair of points x, y ∈ S2, we must have f(x) ≠ f(y). Suppose x ≠ y, then rx ≠ ry since r > 0. This shows that f is one-to-one.
2. Onto: To show that f is onto, we must show that for every point y ∈ S2(r), there exists a point x ∈ S2 such that f(x) = y. For y ∈ S2(r), we can find x = (1/r)y, which satisfies |x| = 1, so x ∈ S2. Then f(x) = r(1/r)y = y, proving that f is onto.
3. Not an isometry if r ≠ 1: An isometry is a function that preserves distances between points. If f were an isometry, we'd have |f(x) - f(y)| = |x - y| for all x, y ∈ S2. Consider x, y ∈ S2 with |x - y| = d. Then, |f(x) - f(y)| = |rx - ry| = r|x - y| = rd. If r ≠ 1, rd ≠ d, so f does not preserve distances, and therefore f is not an isometry.
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Q2. (a) An ordinary fair die is rolled and the number on the uppermost face is observed. If the die is rolled for 4 consecutive times, find the probability that the numbers observed from the rolls are all different. (3 marks)
Hi there! To answer your question, we'll consider the terms: ordinary fair die, rolled 4 consecutive times, and probability.
An ordinary fair die has 6 sides, each with an equal probability of 1/6. Since you're rolling the die 4 consecutive times and want all different numbers, we can calculate the probability as follows:
For the first roll, any of the 6 numbers can appear, so the probability is 6/6.
For the second roll, you have 5 remaining numbers, so the probability is 5/6.
For the third roll, there are 4 remaining numbers, so the probability is 4/6.
Finally, for the fourth roll, there are 3 remaining numbers, so the probability is 3/6.
Now, multiply the probabilities together to find the overall probability of observing all different numbers:
(6/6) × (5/6) × (4/6) × (3/6) = 1 × 5/6 × 2/3 × 1/2 = 5/36
So, the probability of observing all different numbers in 4 consecutive rolls of an ordinary fair die is 5/36.
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The histograms display the frequency of temperatures in two different locations in a 30-day period.
A graph with the x-axis labeled Temperature in Degrees, with intervals 60 to 69, 70 to 79, 80 to 89, 90 to 99, 100 to 109, 110 to 119. The y-axis is labeled Frequency and begins at 0 with tick marks every one unit up to 14. A shaded bar stops at 10 above 60 to 69, at 9 above 70 to 79, at 5 above 80 to 89, at 4 above 90 to 99, and at 2 above 100 to 109. There is no shaded bar above 110 to 119. The graph is titled Temps in Sunny Town.
A graph with the x-axis labeled Temperature in Degrees, with intervals 60 to 69, 70 to 79, 80 to 89, 90 to 99, 100 to 109, 110 to 119. The y-axis is labeled Frequency and begins at 0 with tick marks every one unit up to 16. A shaded bar stops at 2 above 60 to 69, at 4 above 70 to 79, at 12 above 80 to 89, at 6 above 90 to 99, at 4 above 100 to 109, and at 2 above 110 to 119. The graph is titled Temps in Desert Landing.
When comparing the data, which measure of center should be used to determine which location typically has the cooler temperature?
Range of center should be used to determine which location typically has the cooler temperature.
Now, We know that;
Range is measures the difference between the highest and lowest values in a dataset, providing a clear measure of variability for both sets of data. It is not affected by skewness or symmetry, which makes it a useful measure of variability for comparing the temperature consistency between Desert Landing and Flower Town.
A histogram is a graphical representation of the distribution of a dataset. It is a way to display the frequency of different values or ranges of values in a dataset.
The x-axis of a histogram typically represents the values or ranges of values, and the y-axis represents the frequency or count of those values.
The data is divided into bins, and each bin is represented by a bar whose height corresponds to the number of observations in that bin. Histograms are used to visualize the distribution of data, detect outliers, and identify patterns or trends in the data.
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A cylinder has a radius of 2.5 meters. It’s volume is 37.5 pi cubic meters. What is the height of the cylinder
[tex]\textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=2.5\\ V=37.5\pi \end{cases}\implies 37.5\pi =\pi (2.5)^2 h \\\\\\ \cfrac{37.5\pi }{2.5^2 \pi }=h\implies \cfrac{37.5}{6.25}=h\implies 6=h[/tex]
find the zeroes of 4(3x−2) ^2 −3(3x−2)(x+5)−7(x+5) ^2
The zeroes of polynomial are 43/5 and -3/4.
We have,
4 (3x-2)² -3 (3x-2) (x+5) - 7(x+5)²
simplifying the above expression we get
4 (9x² + 4 -12x ) -3 (3x² + 15x - 2x - 10) - 7(x² + 25 + 10x)
= 36x² - 48x + 16 - 9x² -39x + 30 - 7x² - 175 - 70x
= 20x² -157x -129
Now, solving the quadratic equation we get
x = 43/5 and x= -3/4
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A sample of 275 students, 26 that they are vegetarians. Of the vegetarians, 9 eat both fish and eggs, three eggs but not fish, 7 eat neither. Choose one of the vegetarians at random. What is the probability play the chosen student eats fish or eggs?
The probability that the chosen student eats fish or eggs is 12/26 = 0.4615 or approximately 46.15%
To answer your question, let's first organize the information given:
Total vegetarians: 26
Eat both fish and eggs: 9
Eat eggs but not fish: 3
Eat neither fish nor eggs: 7
We want to find the probability that the chosen vegetarian student eats fish or eggs. To do this, we need to find the total number of vegetarians who eat fish or eggs. Since 9 eat both fish and eggs, and 3 eat eggs but not fish, we can deduce that 9 + 3 = 12 vegetarians eat fish or eggs.
Now, to find the probability, we'll divide the number of vegetarians who eat fish or eggs (12) by the total number of vegetarians (26).
Probability = (Number of vegetarians who eat fish or eggs) / (Total number of vegetarians)
Probability = 12 / 26
Probability ≈ 0.4615
So, the probability that the chosen vegetarian student eats fish or eggs is approximately 0.4615 or 46.15%.
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Can someone help me with this question
The value of the unknown angle is 40⁰
What is circle theorem?A chord of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is a secant line. More generally, a chord is a line segment joining two points on any curve
In geometry, a circular segment (symbol: also known as a disk segment, is a region of a disk which is "cut off" from the rest of the disk by a secant or a chord.
Circle theorems are properties that show relationships between angles within the geometry of a circle. We can use these theorems along with prior knowledge of other angle properties to calculate missing angles, without the use of a protractor. This has very useful applications within design and engineering.
Angles in the same segment are equal
The two angles marked are seen to be in the same segment and as such they are equal angles
The value of each of the angles is 40⁰
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How can you tell if three positive numbers form a Pythagorean triple.
Answer: Here i will explain it to you and give an example
Here's an example: let's say you have three positive integers, 5, 12, and 13. To check if they form a Pythagorean triple, you can compute 5^2 + 12^2 = 25 + 144 = 169, which is equal to 13^2. Since the equation holds, the three numbers 5, 12, and 13 form a Pythagorean triple.
In fact, this is a well-known Pythagorean triple, because it is one of the smallest triples, and it is frequently used in geometry and mathematics. The triple (5, 12, 13) satisfies the Pythagorean theorem and represents the lengths of the sides of a right triangle.
Step-by-step explanation: Three positive numbers form a Pythagorean triple if they satisfy the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In other words, if a, b, and c are the lengths of the sides of a triangle such that c is the length of the hypotenuse (the longest side) and a and b are the lengths of the other two sides, then the Pythagorean theorem states that a^2 + b^2 = c^2.
Therefore, to determine if three positive numbers form a Pythagorean triple, you need to check if the sum of the squares of the two smaller numbers is equal to the square of the largest number. For example, if you have three numbers 3, 4, and 5, you can check if they form a Pythagorean triple by computing 3^2 + 4^2 = 9 + 16 = 25, which is equal to 5^2. Since the equation holds, the numbers 3, 4, and 5 form a Pythagorean triple.
Hope this helped. Have a great day.
You have
$
5.00
$5.00 and you need to make copies of a flyer at a store that charges
$
0.15
$0.15 per copy. Find the inequality that represents the number of copies you can make. Use
�
x as the variable.
-
What is the maximum number of copies you can afford to make?
The inequality is 0.15x ≤ 5.00. If copies of a flyer at a store that charges $0.15 per copy, the maximum number of copies you can afford to make is 33.
The inequality that represents the number of copies you can make is:
0.15x ≤ 5.00
Here, x represents the number of copies you can make, and 0.15 is the cost per copy in dollars. The inequality states that the total cost of copies must be less than or equal to the amount of money you have.
To find the maximum number of copies you can afford to make, we need to solve for x:
0.15x ≤ 5.00
x ≤ 5.00/0.15
x ≤ 33.33
Since you cannot make a fraction of a copy, the actual number of copies you can make is 33 or less.
In conclusion, the inequality that represents the number of copies you can make is 0.15x ≤ 5.00, and the maximum number of copies you can afford to make is 33.
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You have $5.00 and you need to make copies of a flyer at a store that charges $0.15 per copy. Find the inequality that represents the number of copies you can make. Use x as the variable. What is the maximum number of copies you can afford to make?
Question 9:6 + 3 + 7 Marks Let O = (0,0), and a = (2,-1) be points in R2. SetG = Bd? (0,1) = {v = (x, y) € R2: d2(0,v) < 1} H = Bd: (a, 1) = {v = (x,y) € R2: d1(a, v) <1}(a) Describe G and H in terms of (x, y)-curves alone, and where applicable without making use of any absolute value symbol. (b) Give the set S of all possible values of y if v = (13,y) € H. (c) Sketch G and H in separate Cartesian coordinates systems (x,y), indicating only O, a and all possible x-intercepts and y-intercepts.
G and H in terms of x and y is given by H = [tex]B^d[/tex](a, 1) and G = [tex]B^{d_2}(0, 1)[/tex] , the set S of all possible values of y is x+y≥0 the Cartesian coordinates systems is S = [-7/5, -3/5].
Choosing a point O of the line (the origin), a unit of length, and an orientation for the line are all steps in choosing a Cartesian coordinate system for a one-dimensional space, or for a straight line. The line "is oriented" (or "points") from the negative half towards the positive half when an orientation determines which of the two half-lines given by O is the positive half and which is the negative half. Then, depending on which half-line contains P, the distance between each point P on the line and O can be specified.
a) O = (0, 0) a = (2, -1)∈R²
G = [tex]B^{d_2}(0, 1)[/tex]
D = [tex]\sqrt{x^2+y^2}[/tex] < 1
So this is a circle until center at (0, 0) and no point on
[tex]x^2+y^2[/tex] and every point inside it
H = [tex]B^d[/tex](a, 1) = {v=(x,y)∈R²: d(a, v)≤1}
b) x-2 + y=1 ≤ 1
x-y ≤4
For, x-2≤0, y+1≥0 we get,
2-x+y+1≤1 y-x ≤-2
For, x-2≤0, y+1≤0, 2-x-y-1≤1
x+y≥0
c) Therefore,
d(a, (13/5, y) ≤ 1
(13/5 -2) + (y +1) ≤ 1
S = [-7/5, -3/5].
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can you help i'm stuck
The value of the output is independent of the value of the input.
How to determine what the graph indicate about the relationship between input and output?
In the graph, the input is the x value (x-axis) and the output is the y value (y-axis).
Looking at the graph, you notice the y values are constant (the same) while the x values changes.
What this means is that whatever the value of the input (x value), the value of the output (y value) will remain the same. That is the value of the output is independent of the value of the input.
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A. Write true or false after each sentence. If the sentence is
false, change the Capitalization word or words to make it true.
1. In the expression 7x + 15, 15 is a COEFFICIENT .
2. 3x + 7 means (3x + 7) DIVIDED BY 2
3. You can rewrite 2(4 + 8) as (2)(4) + (2)(8) using the DISTRIBUTIVE PROPERTY.
In the expression 7x + 15, 15 is a COEFFICIENT: False.
In the expression 7x + 15, 15 is a constant.
3x + 7 means (3x + 7) DIVIDED BY 2: False.
3x + 7 means 3x plus 7.
You can rewrite 2(4 + 8) as (2)(4) + (2)(8) using the DISTRIBUTIVE PROPERTY: True.
What is the distributive property of multiplication?In Mathematics, the distributive property of multiplication states that when the sum of two or more addends are multiplied by a particular numerical value, the same result and output would be obtained as when each addend is multiplied respectively by the same numerical value, and the products are added together.
By applying the distributive property of multiplication to left side of the equation, we have the following:
2(4 + 8) = (2)(4) + (2)(8)
2(12) = 8 + 16
24 = 24
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Answer all boxes and read the questions
The surface area of the paper towel is determined as 212.1 in².
What is the surface area of the cylinder?The surface area of the cylinder is calculated as follows;
S.A = 2πr (r + h )
where;
r is the radius of the cylinderh is the curved height of the cylinderThe radius of the cylinder = 5 in /2 = 2.5 in
The total surface area of the cylinder is calculated as follows;
S.A = 2πr (r + h )
S.A = 2π x 2.5 in ( 2.5 in + 11 in )
S.A = 212.1 in²
Thus, the surface area of the cylinder is equal to surface area of the paper towel.
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A triangle has 2 sides measuring 12 on each side. What is the base?
Answer:
To determine the base of a triangle, we need more information. The base of a triangle is one of its sides, typically denoted as the side opposite to the triangle's vertex or apex. In order to find the base, we need to know either the length of the other side and the angle between them, or the height of the triangle along with the length of one of its sides.
If you have additional information about the triangle, such as the length of another side or the height, please provide that information so that we can help you find the base.
Step-by-step explanation:
A card is picked from a standard deck of 52 cards. Determine the odds against and the odds in favor of selecting a black card
Picking a black card has a 1/2 chance of probability both ways.
The probability of an occurrence is a figure that represents how likely it is that the event will take place. In terms of percentage notation, it is expressed as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the more likely it is that the event will take place.
Hearts, clubs, spades, and diamonds make up the four suites of a normal deck of cards. 13 cards total—the ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king—make up each suit. There are two jokers, bringing the total number of cards in the deck to 54.
In a deck, there are 26 black cards. Picking any of these 26 cards had a probability of p=26/52 = 1/2 since picking any card has the same probability (1/52).
1-p=1/2 goes against probability.
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Claire has 6 large envelopes and 11 small envelopes. what is the ratio of large envelopes to the total number of evelopes?
Choices:
A 5 : 11
B 6 : 11
C 6 : 17
D 11 : 17
Solve: 4(-x-3) -4>-2
Answer:
x < -3.5 or x < -7/2
Step-by-step explanation:
We can solve the inequality by solving for and isolating x:
[tex]4(-x-3)-4 > -2\\-4x-12-4 > -2\\-4x-16 > -2\\-4x > 14\\\\x < -7/2\\or\\x < -3.5[/tex]
We can check that our solution is correct by plugging in a number less than -3.5 for x like -4:
[tex]4(-(-4)-3)-4 > -2\\4(4-3)-4 > -2\\4(1)-4 > -2\\4-4 > -2\\0 > -2[/tex]
The inequality is true for any value of x less than -3.5 so the answer is correct
Exercise 0.5. Calculate the Fourier series of the function f:(-1,1] →R, f(x) = 1-x^2 Use this series to prove that phi^2/6 = [infinity]Σn=1 1/n^2 (3 + 2 Marks)
π^2/6 = Σn=1^∞ 1/n^2
This completes the proof.
To calculate the Fourier series of the function f(x) = 1-x^2, we first extend it to a periodic function on (-∞, ∞) with period 2 by defining it as follows:
f(x) = 1 - x^2, -1 < x ≤ 1
f(x+2) = f(x), for all x in R
Since f is an even function, its Fourier series only contains cosine terms:
f(x) = a0/2 + Σn=1^∞ an cos(nπx/2), -∞ < x < ∞
where an = (2/π) ∫[-1,1] f(x) cos(nπx/2) dx.
To find the Fourier coefficients an, we first calculate a0:
a0 = (2/π) ∫[-1,1] f(x) dx
= (2/π) ∫[-1,1] (1 - x^2) dx
= 4/π
Next, we calculate an for n > 0:
an = (2/π) ∫[-1,1] f(x) cos(nπx/2) dx
= (2/π) ∫[-1,1] (1 - x^2) cos(nπx/2) dx
= 8/[n^3π^3 (1 - (-1)^n)] for n > 0
Therefore, the Fourier series of f is:
f(x) = 2/π - (8/π) Σn=1^∞ [1/((nπ)^2 (1 - (-1)^n))] cos(nπx/2), -∞ < x < ∞
Now, we can use this series to prove that:
Σn=1^∞ 1/n^2 = π^2/6
To do this, we start with the identity:
f(x) = (2/π) Σn=1^∞ [1/((nπ)^2 (1 - (-1)^n))] cos(nπx/2)
Integrating both sides over [-1,1], we get:
2/π ∫[-1,1] f(x) dx = (2/π) Σn=1^∞ [1/((nπ)^2 (1 - (-1)^n))] ∫[-1,1] cos(nπx/2) dx
The integral on the right-hand side is equal to 0 for odd values of n and 2 for even values of n. Therefore, we can simplify the equation as:
1 = (4/π) Σn=1^∞ [1/((nπ)^2)]
Multiplying both sides by (π^2/6), we get:
π^2/6 = Σn=1^∞ 1/n^2
This completes the proof.
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a manufacturer claims that their batteries, type a, exceeds their competitor, type b. a consumer organization collected data on the life of two types of automobile batteries. the summary statistics for 12 observations of each type are:
The manufacturer claims that their batteries, Type A, exceed their competitor, Type B. However, based on the summary statistics collected by the consumer organization, no definitive conclusion can be drawn as to which battery type lasts longer.
The summary statistics for Type A and Type B batteries must be compared to determine which one lasts longer. The statistics to compare include the mean, median, and range of each battery type.
If the mean and median lifespans of Type A are higher than those of Type B, and if the range of Type A is smaller than that of Type B, then it can be concluded that Type A lasts longer.
However, if the statistics show the opposite, or if there is overlap between the ranges of the two types, then no definitive conclusion can be made as to which battery type lasts longer.
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Jessica found an icicle 20 inches long. How long is it in feet?
Write your answer as a whole number or a mixed number in simplest form.
The length 20 inches of the icicle in feet is 1 2/3 feet
How long is the length in feet?From the question, we have the following parameters that can be used in our computation:
Jessica found an icicle 20 inches long.
This means that
Length = 20 inches
To convert inches to feet, we divide the length value by 12
So, we have
Length = 20/12 feet
Evaluate
Length = 1 2/3 feet
Hence, the length is 1 2/3 feet
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A thief is spotted by a policeman at a distance of 500 m. If the speed of the thief be 8 km/hr and that of the policeman be 10 km/hr, at what distance after he spots him will the policeman catch the thief? A. 4 km B. 3.4 km C. 2 km D 2.4 km
The answer is C. 2 km.
To find the distance after the policeman spots the thief where he will catch him, we'll need to use the terms distance, speed, and time.
Given:
- Distance between thief and policeman: 500 m
- Speed of thief: 8 km/hr
- Speed of policeman: 10 km/hr
Step 1: Convert the distance to kilometers (since the speeds are in km/hr).
500 m = 0.5 km
Step 2: Calculate the relative speed of the policeman to the thief.
Relative speed = Speed of policeman - Speed of thief = 10 km/hr - 8 km/hr = 2 km/hr
Step 3: Calculate the time it will take for the policeman to catch the thief.
Time = Distance / Relative speed = 0.5 km / 2 km/hr = 0.25 hr
Step 4: Calculate the distance traveled by the policeman in that time.
Distance traveled = Speed of policeman × Time = 10 km/hr × 0.25 hr = 2.5 km
Step 5: Subtract the initial distance between them to find the distance after the policeman spots the thief.
Distance after spotting = Distance traveled - Initial distance = 2.5 km - 0.5 km = 2 km
So, the policeman will catch the thief at a distance of 2 km after he spots him. The answer is C. 2 km.
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