The term Euler's number (e) refers to a mathematical expression for the base of the natural logarithm. This is represented by a non-repeating number that never ends.
The first few digits of Euler's number are 2.71828.
Given log(base_12)1728,
where base = 12
1728 = x
Factor out 1728 to 12^3 to apply the logarithmic rule, log(base_b)b^n = n
This gives you:
log(base_12)(12^3) = 3
The base 12 and the value of x = 12 cancels out, leaving you with 3.
To help determine the order of operations and other aspects of logical syntax, mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping.
To find which expression is equivalent to log₅32:
Law of logarithms:
The law of logarithms states that: log(a) b = log(10)b/log(10)a
Given the log function, which is written as log₅32
a = 5
b = 32
Substitute as: log₅32 = log32/log5
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If the original recipe made 14 pancake and the cafeteria plan to charge $. 50 per pancake, how much money will they make if they ell all of the pancake made for the 75 people?
Therefore, if the cafeteria sells all of the pancakes made for 75 people, they will make $525.
What is unitary method?The unitary method is a strategy for problem-solving that involves first determining the value of a single unit, then multiplying that value to determine the required value. How to determine the value of one pen, for instance, if the cost of 40 pens is Rs. 400. The unitary method can be used to complete it.
Define ratio and proportion.Two quantities are compared to form a ratio. An equality of two ratios is a percentage. How to format a ratio Analyze the ratio to see whether it is part to part or part to whole. Calculate the whole and the pieces as necessary.
If the cafeteria sells all of the pancakes made for 75 people, and the original recipe makes 14 pancakes, they would need to make 1475 = 1050 pancakes.
If they charge $0.50 per pancake and they sell all 1050 pancakes, they would make 1050$0.50 = $525.
Therefore, if the cafeteria sells all of the pancakes made for 75 people, they will make $525.
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How do you write a compound statement in symbolic form?
A compound statement is a statement that combines two or more simpler statements using logical connectives such as "and", "or", "not", "if-then", "if and only if", etc. To write a compound statement in symbolic form, you can use logical symbols to represent the different parts of the statement.
"∧" (and) is used to connect two statements, meaning that both statements must be true for the compound statement to be true. For example, "P ∧ Q" is read as "P and Q" and represents the statement "P and Q are true."
"∨" (or) is used to connect two statements, meaning that at least one of the statements must be true for the compound statement to be true. For example, "P ∨ Q" is read as "P or Q" and represents the statement "P or Q is true."
"¬" (not) is used to negate a statement, meaning that the statement is false if the original statement is true and vice versa. For example, "¬P" is read as "not P" and represents the statement "P is false."
"→" (if-then) is used to connect two statements, meaning that if the first statement is true, then the second statement must also be true. For example, "P → Q" is read as "if P then Q" and represents the statement "if P is true, then Q is also true."
"↔" (if and only if) is used to connect two statements, meaning that the two statements are either both true or both false. For example, "P ↔ Q" is read as "P if and only if Q" and represents the statement "P is true if and only if Q is true."
For example, the compound statement "If it is raining, then I will bring an umbrella, but if it's not raining, I will wear sunglasses" can be written in symbolic form as:
(R → U) ∧ (¬R → S)
Where "R" represents the statement "it is raining", "U" represents the statement "I will bring an umbrella" and "S" represents the statement "I will wear sunglasses".
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find the probability that a point chose a t random on the part of the number line shown will lie between points B and C. (anwser should be in a fraction or decimal)
The probability of point being in between B and C is 0.167
A probability definition is what?Probability is a statistic used to indicate the likelihood or potential that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to express probabilities.
What are examples and probability?The likelihood is based on the range of potential outcomes. the total number of possible outcomes. As an illustration, since there is only one way to get a head and a total of two possible results, the likelihood of flipping a coin and getting heads is one in two. a tail or a head.
Distance between B and C= 4 units
Total distance=24
Probability that point lie between B and C = 4/24 =1/6
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How is rotation and dilation similar?
Rotation is the process of turning a figure a specific amount around a point. When we dilate a figure, we increase or decrease its size.
The circular movement of an item about a primary axis is referred to as rotation or spin. A revolving object in two dimensions only has one conceivable central axis and can revolve either clockwise or counterclockwise. There are many conceivable central axes and rotating orientations for a three-dimensional object.
Resizing an item uses a transition called dilation. Dilation is used to enlarge or contract the items. The result of this transformation is a picture with the same shape as the original. However, there is a variation in the shape's size.
On a coordinate plane, forms can be rotated and dilated in addition to translated and reflected. When a shape is rotated, the new shape is consistent with the old. When you dilate a shape, the resulting shape is similar to the original.
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What ordered pairs are the solutions of the system of equations shown in the graph below?
(__, __) and (__, __)
In a 45°-45°-90° triangle, the length of the hypotenuse is 11. Find the length of one of the legs.
The length of the leg of the right-angled triangle is (11√2) / 2. The correct option is C.
What is trigonometry?The branch of mathematics that sets up a relationship between the sides and the angles of the right-angle triangle is termed trigonometry. The trigonometric functions, also known as a circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths.
Given that the right angles triangle has 45-45-90 angles. The hypotenuse is 11.
The length of the leg will be,
Cos45 = Base / Hypotenuse
1 / √2 = Base / 11
Base = 11 / √2
Rationalize the numbers,
Base = 11√2 / 2
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GH is dilated by a scale factor of to form G'H'. GH measures 36. What is the
measure of G'H'?
Answer:
The basic formula to find the scale factor of a dilated figure is: Scale factor = Dimension of the new shape ÷ Dimension of the original shape.
A bicycle wheel has a diameter of 559 mm and has 32 equally spaced spokes. What is the approximate arc length, rounded to the nearest hundredth, between each spoke? Use 3. 14 for π. Show your work
The approximate arc length rounded to the nearest hundredth, between each spoke is 54.9 mm.
Diameter = 559 mm, r = d/2 = 279.5 mm
Number of spokes = 32
π = 3.14
To solve this question, we have to use the formula of the length of an arc which is given as
L = θ/360 × 2πr
To calculate the angle between each spoke, let's divide 360 by 32
θ = 360/32
θ = 11.25°
Now we have the value of the angle, substitute it and solve for the length of the arc.
[tex]L_{arc}[/tex] = θ/360 × 2πr
= 11.25/360 × 2 × 3.14 × 279.5
= 54. 9 mm
Therefore the length of the arc is approximately 54.9 mm.
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What is the estimated perimeter of an ellipse if the major axis has a length of 15 ft and the minor axis has a length of 7.5 ft
The estimated perimeter of an ellipse if the major axis has a length of 15 ft and the minor axis has a length of 7.5 ft id found to be 37.3 feet .
Perimeter is calculated as
= 2*pi*r*r
= 2 *3.14 * sqrt (15/2² +7.5/2²)/2
= 6.28 x sqrt (56.25+14.0625/2 6.28) *sqrt (35.15625) 6.28 * 5.929270613
= 37.235 feat
An ellipse is the locus of all the points on a plane whose distances from two fixed points in the plane are always same. The fixed points, which are encompassed by the curve, are known as foci , singular of focus.
The constant ratio is the eccentricity of the ellipse and the fixed line is directrix. Eccentricity is an element of ellipse which denotes elongation and is symbolized by the letter 'e'.
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The degrees of freedom for a contingency table with 10 rows and 11 columns is
O 100 O 110 O 21 O 90
Essentially, a contingency table with 10 rows and 11 columns has (D) 90 degrees of freedom.
What is a contingency table?A contingency table is a matrix-form table that shows how often different combinations of variables appear.
It's really popular in survey research, business intelligence, engineering, and scientific research. Basically, a contingency table with 10 rows and 11 columns has 90 degrees of freedom.
They are also sometimes known as two-way frequency tables and are used to show how often different combinations of variables occur.
A contingency table gives you an idea of how two variables relate to each other.
It displays one categorical variable along the rows and another categorical variable along the columns. It can help point out any interactions between them.
Therefore, essentially, a table with 10 rows and 11 columns has (D) 90 degrees of freedom.
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Complete question:
The degrees of freedom for a contingency table with 10 rows and 11 columns are:
A. 100
B. 110
C. 21
D. 90
The daily recommended allowance of vitamin c for a sixth grader is 45 mg. 1 orange has about 75% of the recommended daily allowence of vitimin c. How many milligrams are in 1 orange?
Proportionately, the quantity of milligrams of vitamin c in 1 orange is 33.75 mg.
What is proportion?Proportion refers to the part of a whole contained in another value.
Proportion is a ratio representation, which is fractional in value.'
We depict proportions using decimals, fractions, or percentages.
The daily recommended allowance of vitamin c for a sixth grader = 45 mg
The content of 1 orange to the daily recommended allowance = 75%
75% of 45 mg = 33.75 mg.
Thus, 1 orange will proportionately contain 33.75 mg, which is 75% of 45 mg.
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A mathematics teacher wanted to see the correlation between test scores and homework. The homework grade (x) and test grade (y) are given in the accompanying table. Write the linear regression equation that represents this set of data, rounding all coefficients to the nearest tenth. Using this equation, find the projected test grade, to the nearest integer, for a student with a homework grade of 80.
Regression Equation:
Final Answer:
on solving the equation, we have therefore the value of x from the given equation comes out to be 13.5
What is equation?An equation is a formula in mathematics that joins two statements with the equal symbol = to represent equality. The definition of an equation in algebra is a mathematical statement proving the equality of two mathematical expressions. In the equation 3x + 5 = 14, for instance, the terms 3x + 5 and 14 are separated by an equal sign. The link between two phrases on either side of a letter is expressed mathematically. There is often only one variable, which is also the symbol. instance: 2x - 4 Equals 2.
equation given is 3(x+2)=7x-48
value of x
[tex]3(x+2)=7x-48\\ 3x+6=7x-48\\7x-48-3x-6=0\\ 4x-54=0\\ 4x=54\\ x=54/4\\ x=13.5\\[/tex]
value of x from the given equation comes out to be 13.5
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What is the pattern?
The city of vine View is building a new rectangular park for the townspeople. The park will have an area of 8,925 square yards if the width of the park is to be 84 yards, how much fencing does the city need to surround the park?
The length of the fencing that the city needs to surround the park will be 380.50 yards.
What is the area and perimeter of the rectangle?Assume L is the length and W is the width. Then the area is given as,
A = L × W
And the perimeter is given as,
P = 2(L + W)
The city of vine View is building a new rectangular park for the townspeople. The park will have an area of 8,925 square yards if the width of the park is to be 84 yards. Then the length of the rectangular park is calculated as,
8925 = L × 84
L = 106.25 years
Then the length of the fencing that the city needs to surround the park will be given as,
P = 2(106.25 + 84)
P = 2 × 190.25
P = 380.5 yards
The length of the fencing that the city needs to surround the park will be 380.50 yards.
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Find the value of p and q for which the system of equations represent coincident lines:
2x + 3y = 7
(p + q + 1)x + (p + 2q + 2)y = 4(p + q) +1
Answer: The value of p and q for which the system of equations represent coincident lines is p = 2 and q = 1.
Step-by-step explanation: To represent coincident lines, the two lines must be proportional. Therefore, the slope of the second line must be equal to the slope of the first line.
The slope of the first line is 3/2. The slope of the second line is p + 2q + 2/p + q + 1. Therefore, we set these two equal:
3/2 = p + 2q + 2/p + q + 1.
Cross multiplying, we get:
3(p + q + 1) = 2(p + 2q + 2).
Simplifying, we get:
3p + 3q + 3 = 2p + 4q + 4.
Combining like terms, we get:
p - q = 1.
Therefore, the value of p and q for which the system of equations represents coincident lines is p = 2 and q = 1.
How many 1/3 ounce cups can be taken
Answer:
90
Step-by-step explanation:
If we know that for every cup, we can take 3 of 1/3-ounce cups, we can use an equation that looks like this:
number of cups × 3 = total 1/3-ounce cups
If there are 30 cups, then we can simply insert 30 into the equation.
30 × 3 = 90
Therefore, you can make 90 1/3-ounce cups from the cereal box.
Given a prime $p$ and an integer $a$, we say that $a$ is a primitive root $\pmod p$ if the set $\{a,a^2,a^3,\ldots,a^{p-1}\}$ contains exactly one element congruent to each of $1,2,3,\ldots,p-1\pmod p$.For example, $2$ is a primitive root $\pmod 5$ because $\{2,2^2,2^3,2^4\}\equiv \{2,4,3,1\}\pmod 5$, and this list contains every residue from $1$ to $4$ exactly once.However, $4$ is not a primitive root $\pmod 5$ because $\{4,4^2,4^3,4^4\}\equiv\{4,1,4,1\}\pmod 5$, and this list does not contain every residue from $1$ to $4$ exactly once.What is the sum of all integers in the set $\{1,2,3,4,5,6\}$ that are primitive roots $\pmod 7$
Only 3 & 5 from the set {1,2,3,4,5,6} are primitive root (mod7), therefore, sum of integers is 3 + 5 = 8.
What is primitive root?A number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n in modular arithmetic. That is, g is a primitive root modulo n if for every integer a coprime to n, some integer k for which gk ≡ a (mod n).
The index or discrete logarithm of a to the base g modulo n is a value k like this. So, if and only if g is a generator of the multiplicative group of integers modulo n, g is a primitive root modulo n.
In Article 57 of his Disquisitiones Arithmeticae (1801), Gauss defined primitive roots and credited Euler with coining the term.
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Please answer the question below
Katy organised a wedding. Guests had to choose their meal from pasta , chicken or beef. 1/3 of the guests chose pasta. 5/12 of the guests chose chicken. 24 of the guests chose beef. How many guests were at the wedding ?
guys please help i’ll even give brainliest and give the correct answer not wrong ones
With the help of equations we can say that, the wedding had 96 attendees.
What is an equation ?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
We have LHS = RHS (left hand side = right hand side) in every mathematical equation.
To determine the value of an unknown variable that represents an unknown quantity, equations can be solved.
The statement is not an equation if there is no "equal to" sign in it.
According to our question-
x is the total number of wedding attendees.
1/3 of the attendees =1/3 who chose pasta=1/3x----1
Those who selected chicken =5/12x---2
24 people opted for meat.----3
adding 1,2,3
x = 96
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What linear equation means?
A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included.
When a linear equation is graphed, it always produces a straight line since each term in the equation has an exponent of 1 or 0. A linear equation is an algebraic equation. If an equation has the formula y=mx+b, with m representing the slope and b the y-intercept, it is said to be linear.
The point-slope form, standard form, and slope-intercept form are the three main types of linear equations.
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wacky wizards offers beginner and advanced magic classes. The beginner class costs 6 more than half the cost of the advanced class.Last week they taught 7 beginner classes and 11 advanced classes and earned $390. Write a system that represents the cost of the beginner class, b and the advanced class, a. HELP!!!!!
The system represents the cost of the beginner class will be $18 and the advanced class will be $24.
What is the solution to the equation?The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.
Both basic and expert magic lessons are available at Wacky Wizards. More than half as much is spent on the introductory class as there is on the advanced level. They made $390 last week by instructing 7 basic lessons and 11 advanced classes.
Let 'x' be the cost of beginner classes and 'y' be the cost of advanced classes. Then the equations are given as,
x = y / 2 + 6
y = 2x - 12 ...1
7x + 11y = 390 ...2
From equations 1 and 2, then we have
7x + 11(2x - 12) = 390
7x + 22x - 132 = 390
29x = 522
x = 18
Then the value of the variable 'y' is calculated as,
y = 2(18) - 12
y = 36 - 12
y = 24
The system represents the cost of the beginner class will be $18 and the advanced class will be $24.
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Equivalent expression
The ladder touches the building at 4.53 meters right angle triangle vertical distance from the ground.
What is right-angled triangle?The triangle that has one 90 degrees angle is called right-angled triangle. The Pythagorean theorem is applicable for right-angled triangle.
What is the height of the point where the ladder touches the building A ladder leans up against a building at an angle of 65 degrees with the ground. the length of ladder = 5 meters
we understand that the ladder, building and ground together formed a right-angled triangle. The right angle is in between the building and ground. The length of ladder acts as a hypotenuse of this triangle. If we consider the angle 65 degrees,
then sin 65° = opposite side length / hypotenuse here, opposite side length = the required length we need to determine sin 65° × hypotenuse = opposite side opposite side = 0.9063 × 5 =4.53 meters
the ladder touches the build at 4.53 meters vertical distance from the ground.
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The solutions to this problem
Answer: x<-2 or -2>x<1 or x>1
(-infinity,-2) U (-2,1) U (1,infinity)
Step-by-step explanation:
Decompose one number to subtract 726-340
The subtraction from the given number is = 386
What is subtraction?Subtraction is defined as one of the basic arithmetic operations used to subtract one value from another value.
340 subtracted from 726 = 386.
It means; 726 - 340 = 386
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When Bruce got his first job, he put $6,225 of his earnings into an investment account to save for retirement. The value of the account is predicted to double each decade.
Write an exponential equation in the form y=a(b)x that can model the predicted value of Bruce's investment account, y, x decades after starting the account.
Use whole numbers, decimals, or simplified fractions for the values of a and b
The exponential function to model the predicted value of Bruce's investment account is
y = 6225 * 2^x.
What is an exponential function?The formula for an exponential function is f(x) = a^x, where x is a variable and a is a constant that serves as the function's base and must be bigger than 0.
here, we have,
we know that,
An exponential function is written in the form - y=a*(b)^x
Here, y is the function, a is the base value b is the rate and x is time period.
now, we have,
from the given information, we get,
This equation uses the initial value of the account (6225)
and the base of the exponent (2) representing the account will double in value every decade.
if, we have,
the predicted value of Bruce's investment account is y.
time period is, x decades after starting the account.
So, the exponential equation will be -
y = 6225 * 2^x.
Therefore, the exponential equation is .
y = 6225 * 2^x.
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pleasee helpp !! <33
Using a number line, find both the intersection and the union of the following
intervals:
(-∞,2) and [0,∞)
The answer choice which represents a number line for the intersection of the given intervals is; (-∞, 2).
The answer choice which represents a number line for the union of the given intervals is; (-∞, ∞).
What is the number line representation of the intersection and union of the given intervals?It follows from the task content that the number line which represents the intersection and union of both intervals given is to be determined.
Since the given intervals are; (-∞, 2) and [0,∞).
It follows that the intersection of both intervals is the interval; (-∞, 2).
Also, the interval which represents the union of both intervals is; (-∞, ∞).
Hence, the image which represents both scenarios is attached.
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A 20-gallon bathtub is to be filled with water that is exactly 100°F. The hot water supply is 175°F and the cold water supply is 20°F. When mixed, the temperature will be a weighted average based on the amount of each water source in the mix. How much of each should be used to fill the tub as specified?
320/31 gallons of hot water and 300/31 gallons of cold water are needed.
Here we are required to fill a 20-gallon bathtub and the water should be precisely 100° F.
It is given that hot water has a temperature of 175° F and cold water has a temperature of 20° F.
Let the amount of hot water used be x gallons.
Let the amount of cold water used be y gallons.
Since the bathtub has a capacity of 20 gallons,
x + y = 20 [1]
or, y = 20 - x [2]
The temperature after the water is mixed will be a weighted average of the temperature and amt of water used.
Hence we get,
(175x + 20y) / (x + y) = 100
from equation [1] we get
175x + 20y = 2000 [3]
from equations [2] and [3] we get
155x - 400 = 2000
or, x = 1600/155 = 320/31 [4]
Similarly,
y = 20 - 320/31
= 300/31
Hence 320/31 gallons of hot water and 300/31 gallons of cold water are needed.
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For what value of a the system of linear equations Ax 3y a 3?
The system of linear equations has no solution for any value of a.
The equations Ax + 3y = a and 3x + 3y = a can be written in matrix form as
[A 3] [x] [a]
[3 3] * [y] = [a]
By using the Gaussian elimination method, the reduced form of the matrix is
[A 3] [x] [a]
[0 0] * [y] = [0]
This implies that the system has no solution for any value of a.
The system of linear equation
Ax + 3y = a and 3x + 3y = a can be written in matrix form as
[A 3] [x] [a]
[3 3] * [y] = [a]
We can use the gaussian method to solve this system. First, we subtract 3 times the first equation from the second equation to get
[A 3] [x] [a]
[0 0] * [y] = [0]
This implies that the system has no solution for any value of a.
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A dog takes 3 steps to walk the same distance for which a cat takes 4 steps. Suppose 1 step of the dog covers 1 foot. How many feet would the cat cover in taking 12 steps
According to unitary method, cat takes 12 steps to cover 9 feet.
How do you translate distance in steps?Measure your stride length or assume that it is 2.2 feet for women and 2.5 feet for men to determine it. Add the stride length to the number of steps, for example, 2.2 feet multiplied by 1,000 feet equals 2,200 feet. The answer is 2,200 ft / 5,280 0.42 mi when expressed in miles.
Given, it takes a dog three steps to cover the same distance as a cat four steps.
4 steps of a cat = 3 steps of a dog
12 cat steps equal (3/4)*12 to 9 dog steps.
Additionally, a dog's step covers one foot.
The dog moves nine feet in nine steps.
So a cat takes 12 steps to cover 9 feet.
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