The probability of tossing "heads" on each of your first two tosses is 0.25
Probability = n(E)/n(S)
Where E is the possibility of an event occurring
and S is the sample space i.e., the total number of possible outcomes.
Probability is the measure of how likely an event can happen. The probability of any event lies between 0 and 1 only.
The sample space for when you toss one coin is - H, T
So the probability to get an H is= 1/2
=0.5
Similarly, The sample space for when you toss two coins is HH, TT, HT, TH
So, the probability to get an H on each of your first two tosses is = 1/4
=0.25
Therefore, The probability of tossing "heads" on each of your first two tosses is 0.25
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Bookwork code: N39
The diagram below shows a sketch of the curve
y = x² + 10x - 12.
P is the turning point of the curve.
Work out the coordinates of P.
Answer:
(-5,-37)
Step-by-step explanation:
Complete the square
(x+5)^2-37
3/4÷3/8 write a sentence to describe your answer
Answer:
We multiply 3/4 with the reciprocal of 3/8 which is 8/3. This leaves us with 3/4x8/3 which equals 24/12. This can be simplified into 2.
question 7 fill in the blank: while cleaning data, a data analyst can use a changelog to keep a chronological list of changes they make. they can refer to it during the___period if there are errors or questions.
A changelog is a record of changes made to a dataset during the data cleaning process.
It can serve as a helpful reference for data analysts and other stakeholders to understand the changes that were made and the reasons why they were made. It is important for data analysts to keep a detailed changelog in order to learn from their mistakes and to provide a detailed audit trail for any future questions about the data.
The formula for calculating the value of a changelog is as follows:
changelog= Number of Changes Made + Complexity of Changes + Accuracy of Changes
The number of changes made to the data set is an important factor in calculating the changelog value. The more changes that are made, the more valuable the changelog will be. The complexity of the changes is also important. If the changes are complex and require a lot of effort to make, the changelog will be more valuable. Finally, the accuracy of the changes is also a factor. If the changes are accurate and correct, then the changelog will be more valuable.
Overall, a changelog is a valuable tool for data analysts to keep a detailed record of the changes they make to a dataset. This record can be used to ensure accuracy and to provide a detailed audit trail for any questions that may arise in the future.
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solve for y2?
please
[tex]m = \frac {y2 - y1}{x2 - x1} [/tex]
Please refer to the questions in the image below
a) The sampling method used is non-probability sampling.
b) The sample is going to be representative.
What is sampling?
Sampling is a technique for choosing certain individuals or a small portion of the population in order to draw conclusions about the population as a whole and estimate its characteristics. Researchers frequently utilise various sampling techniques in market research so they do not have to study the full community in order to gather useful information.
In non-probability sampling, participants are chosen at random by the researcher. This type of sampling is not a set or predetermined selection procedure. This makes it challenging for every component of the population to have an equal chance of being included in a sample.
The sampling method used in non-probability sampling as the people selected are selected on a convenience basis.
A representative sample is a subset of a population that seeks to accurately reflect the characteristics of the larger group.
Since, the students are chosen randomly but from the same room of the same school they might represent the characteristics of all other students.
Therefore, the sample is representative.
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PART C Why might you want to represent the triangle's perimeter with one of the
expressions you wrote in Part B, rather than with 6x - 6.3?
In order yo calculate the perimeter of a triangle, one should add the length of its sides. For example, if the triangle has sides a, b, and c, then the perimeter of that triangle will be P = a + b + c
What is the perimeter?Any two-dimensional figure's perimeter is determined by the space surrounding it. By adding the lengths of the sides, we can determine the perimeter of any closed shape.
The perimeter of any polygon is equal to the sum of its side lengths. Considering a triangle, sum of the three sides equals the perimeter.
Units are always a part of the final response. The final answer should be given in centimeters if the triangle's sides are in that unit.
An overview was given as your information is incomplete and couldn't be found.
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Which of the following is a counterexample for the hypothesis " If a team plays at home, then they win the game?"
a. The team plays at home and wins the game.
b. The team plays away and wins the game.
c. The team plays at home and loses the game.
d. The team plays away and loses the game.
The graph above shows the cost of parking at the airport. Jay left on a business trip and had to leave his car at the airport for 5 days. Based on the graph, what is the total cost that he paid for the parking?
A.
$33.50
B.
$31.00
C.
$26.00
D.
$20.00
The total cost that he paid for the parking for 5 days is $26.00.
What is a linear equation?When an algebraic equation is graphed, it always results in a straight line since each term in a linear equation has an exponent of 1. For this reason, it is referred to as a "linear equation."
There are both one-variable and two-variable linear equations.
The graph above shows the cost of parking at the airport,
since the graph is a straight line so it is the condition for a linear equation,
from the graph, the y-axis shows the price in dollars and the x-axis shows the number of days,
line of the graph passes from coordinates (0, 6) and (6, 30)
equation of line is y = mx + c
where c is initial cost when x is 0, y is 6
c = 6
and m = (y₂- y₁)/(x₂ - x₁)
m = (30 - 6)/(6 - 0)
m = 4
equation of line is,
y = 4x + 6
cost for 5 days,
x = 5
y = 4(5) + 6
y = 26
cost is $26.00
Hence cost for 5 days is $26.00, option C is correct.
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give an example of a nonempty subset u of r2 such that u is closed under scalar multiplication, but u is not a subspace of r2.
One example of a nonempty subset u of R^2 such that u is closed under scalar multiplication, but u is not a subspace of R^2, is the set u = {(x, y) | x, y are rational numbers}.
This set is closed under scalar multiplication as if a and (x,y) are in u, a*(x,y) will also be in u because rational numbers are closed under multiplication.
A subset can be described with different properties, one of them is closure, a subset U is closed under an operation if the operation is applied to any two elements of U, and the result is also an element of U.
In mathematics, a subset is a set that is completely contained within another set. It is defined as a set U that is a part of a set S, such that every element of U is also an element of S. The notation used to indicate that U is a subset of S is U ⊆ S.
However, this set u is not a subspace of R^2 because it doesn't contain the zero vector (0,0) and it doesn't close under addition, as the sum of two rational numbers may not be a rational number.
Therefore, One example of a nonempty subset u of R^2 such that u is closed under scalar multiplication, but u is not a subspace of R^2, is the set u = {(x, y) | x, y are rational numbers}.
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given a and b in exercises 11 and 12, write the augmented matrix for the linear system that corresponds to the matrix equation Ax = b. Then solve the system and write the solution as a vector. A= |1 2 4| | -2 |
|0 1 4| b= | 2 |
|-2 -4 -3| | 9 |
|1 2 1| | 0 |
A= |-3 -1 2| b= | 1 |
|0 5 3| | -1 |
The augmented matrix for the linear system corresponding to the matrix equation Ax = b is |1 2 4 0| |-2 2 9| |0 1 4 1| |2 -1| |-2 -4 -3 -1| |9 -1|. Solving the system yields the solution vector (x, y, z) = (5, -1, 4).
The augmented matrix for the linear system Ax = b is a way of combining the two matrices into one matrix. In this case, it is the combination of the coefficient matrix A and the constant matrix b. This new matrix contains all of the information needed to solve the linear system, as it contains both the coefficients of the variables and the solutions of the equations. We can solve the linear system by performing row operations to the augmented matrix, which is equivalent to solving the equations in the original matrix equation. After performing the row operations, we will be left with the solution vector (x, y, z) = (5, -1, 4). This solution vector represents the values of the variables that satisfy the original equation.
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The table below shows some values of a function g(x)
Which function represents the relationship shown in the table?
Answer:
g(x) = 4x - (1/2)
Second choice
Step-by-step explanation:
Using points: (-2, -8.5) and (-1, -4.5)
y2 - y1 -4.5 - (-8.5) -4.5 + 8.5 4
slope = ----------- = --------------- = --------------- = --- = 4
x2 - x1 -1 - (-2) -1 + 2 1
Answer:
the answer should be g(x) = 4x - (1/2)
Step-by-step explanation:
Which graph shows the new position of the rectangle after a translation?
The graph that shows the new position of the rectangle after a translation is; Option C
How to interpret Translation in Transformation?When it comes to transformation in mathematics, there are different types such as reflection, dilation, rotation, translation.
Now, we are dealing with translation which is defined as a transformation of a shape in a plane that preserves the length, which tells us that the object is transformed without getting its dimensions affected.
In this case, we see the rectangle in the graph and we can conclude that the translation of the rectangle shows the correct translation would be the second option because it shows that the length is preserved without getting its dimensions or shape altered.
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using properties of logarithms and trigonometric identities in exercises 87, 88, 89, and 90, show that the two formulas are equivalent.
We need to use the substitution method of integration and logarithm properties to provide the statements.
We know that
cotx = cosx/sinx
∫cotx dx = ∫[cosx/sinx]dx
Let sinx be z
differentiating with respect to x we get
cosx dx = dz
Using this we get
∫[1/z]dz
We know that ∫1/x dx = logx + C
Hence we get
logz + C
Using the value of z here we get
log|sinx| + C
Here we know that we cannot have log of a negative value hence the mod sign has to be used.
Now,
∫cotx dx = ∫cosx.cosecx dx
let cosecx be z
diff with respect to x we get
-cot x.cosec x dx = dz
or, -cosx.cosec²x dx = dz
or, dx = dz/[-z²cosx]
Hence we get
∫[cosx.z]/[-z²cosx]dz
or, -∫1/z dz
or, -logz + C
or, -log|cosecx| + C
Hence Proved
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Complete Question
Using Properties of Logarithms and Trigonometric Identities In Exercises 88 - 92 show that the two formulas are equivalent.
∫cotx dx = ln|sinx| + C
∫cotx dx = -ln|cosecx| + C
10) in the game of scrabble, each player begins by drawing 7 tiles from a bag containing 100 tiles. there are 42 vowels, 56 consonants, and 2 blank tiles in the bag. emma chooses her 7 tiles and is surprised to discover that all of them are vowels. we can use a simulation to see if this result is likely to happen by chance. Part A: State the question of interest using the language of probability. (2 points)
Part B: How would you use random digits to imitate one repetition of the process? What variable would you measure? (3 points)
Part C: Use digits from a table of random digits or use a calculator to perform one repetition. Submit the list of random digits and indicate those that represent vowels. (3 points)
Part D: In 500 repetitions of the simulation, there was one time when all seven tiles were vowels. What conclusion can you draw from this result? (2 points) (10 points
The probability of randomly drawing seven vowels from a bag of 100 tiles consisting of 42 vowels, 56 consonants and 2 blank tiles is low. One time throughout the simulation's 500 iterations, all seven tiles were vowels.
Part A: What is the probability of randomly drawing 7 vowels from a bag of 100 tiles consisting of 42 vowels, 56 consonants and 2 blank tiles?
Part B: To imitate one repetition of the process, you would use a table of random digits or a calculator to randomly generate seven digits, each between 0 and 9. The variable you would measure is the number of vowels drawn.
Part C:
Random Digits: 4, 2, 5, 7, 8, 8, 4
Vowels: 2, 8, 8, 4
Part D: This result suggests that it is possible to randomly draw all vowels in one draw, although it is unlikely.
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10) This shaded region is bounded by the curve with equation y = 3x² + 1
and the lines y = 4 and y = 2.
Calculate the exact area of the region.
Area under the curve y = 3x² + 1 which lies between y = 4 and y = 2 is
4 - [tex]\frac{8}{3}\sqrt{\frac{1}{3} }[/tex] sq, units.
What is Definite integral ?Using microscopic slivers or stripes of the region, a definite integral is a formal estimate of the area beneath a function. Integrals can reflect a region's (signed) area, the total value of a function that changes over time, or the amount of a particular thing given its density.
By performing a definite integral between the two locations, one may determine the area under a curve between two points. Integrate y = f(x) between the limits of a and b to determine the area under the curve y = f(x) between x = a and x = b. Areas below the x-axis will have a negative result, while those above the x-axis will have a positive result.
The given function is y = 3x² + 1
when y = 2 value of x in the function is ± (√(1/3))
and, when y = 4 value of x in the function is ± (1)
Area under the curve y = 2 is
∫3x² + 1 . dx from x ± (√(1/3)).
= (x³ + x) from x ± (√(1/3)).= [tex]\frac{8}{3}\sqrt{\frac{1}{3} }[/tex]
Similarly for the curve y = 4
∫3x² + 1 . dx from x ±1
= (x³ + x) from x ± 1.= 4
Area under the curve y = 3x² + 1 which lies between y = 4 and y = 2 is
4 - [tex]\frac{8}{3}\sqrt{\frac{1}{3} }[/tex] sq, units.
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compare the quantity in column a with the quantity in column b. choose the best answer. column a column b the volume of a cone with base radius
Comparing column a with column b we get that column b is greater than column a. Hence, option B is the correct answer to this question.
Column a: We can find the surface area of a cone by using the formula -
Surface area = π * r * (r + l)
where r is the radius of the base and l is the slant height of the cone.
Substituting r = 3 cm and l = 8 cm, we get,
Surface area = π * 3 * (3 + 8) = π * 3 * 11 = 33π
Hence, Surface area = 33π [tex]cm^2[/tex]
Column b: We can find the surface area of a cone by using the formula -
Surface area = π * r * (r + l)
where r is the radius of the base and l is the slant height of the cone.
Substituting r = 4 cm and l = 6 cm, we get,
Surface area = π * 4 * (4 + 6) = π * 4 * 10 = 40π
Hence, Surface area = 40π [tex]cm^2[/tex]
Comparing we get that,
The surface area of the cone with a base radius of 3 cm and slant height of 8 cm < the surface area of the cone with a base radius of 4 cm and a slant height of 6 cm.
Hence, column a < column b.
Thus, the correct option is B. The quantity in column B is greater.
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The complete question is -
Compare the quantity in column a with the quantity in column b. choose the best answer.
column a: The surface area of the cone with a base radius of 3 cm and a slant height of 8 cm.
column b: The surface area of the cone with a base radius of 4 cm and a slant height of 6 cm.
A. The quantity in column A is greater.
B. The quantity in column B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
The scatter plot below shows the population of a village (P) over time (t). Describe the relationship between the population of the village and time.
A. The population is decreasing over time.
B. The population is increasing over time.
C. The population remains roughly the same.
D. None of these
The correct option regarding the relationship between the population of the village and time is given as follows:
A. The population is decreasing over time.
How to describe the relationship?The variables of the scatter plot are given as follows:
Input variable: time t.Output variable: Population p.From the points on the scatter plot, we have that when the value of t increases, the value of P decreases.
This means that the scatter plot shows a decreasing relationship, and thus the correct option is given by option A.
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Which answer is it bc I’m in direee need
Correct option is C, The height of triangular prism is not 6 units, it is 15 units.
What number of sides does a triangle have?A triangle base, a translated copy, and three faces that join the appropriate sides make up a triangular prism, a three-sided prism. There are no rectangular sides in an oblique right triangle; it is only a right triangle. A right triangle is a uniform triangular prism, which has square sides and equilateral bases.
In essence, it is a polyhedron with three parallel sides, three surface normals in the same plane, and three surface normals (which is not necessarily parallel to the base planes). These three faces make it a parallelogram. Every cross-section that follows the base faces yields the same triangle.
For the given triangular prism,
The perimeter of base = Side + Side + Side = 13+11+8 = 32 units
The area of base = 1/2 × Base × Height = 1/2 × 11 × 6 = 33 units²
The height of prism = 15 units
The lateral surface area = (side + side + side) × height = 15 × (13 + 11 + 8) = 480 units²
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Mr. Abad's class is composed of 28 boys and 35 girls. If he is going to make groups of boys and groups of girls for the activities, what is the biggest number of children in the group if they are of the same number?
what is asked?
what is given?
word clue?
complete answers?
The biggest number of children in the group would be 14 boys and 14 girls.
What is mean by basic division?Basic division is an arithmetic operation that involves two numbers. It is used to find out how many times one number can be divided into another. It is sometimes referred to as division or division by a divisor.
Division is the inverse operation of multiplication. This means that division is the opposite of multiplication. When a number is multiplied by another number, the result is the product of the two numbers. When a number is divided by another number, the result is the quotient of the two numbers.
The process of division involves finding the number of times that one number can be divided into another. Division is performed by dividing the dividend (the number being divided) by the divisor (the number that divides the dividend). The result of the division is the quotient.
Division can be done using a variety of methods. Long division is a traditional method of division, often taught to students in elementary school. It involves breaking down each number into individual digits and then dividing them one at a time.
Another type of division is short division, which is a simpler and faster method of division that can be done in one step. Short division can be used when the divisor is a single-digit number.
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evaluate the expression when x=20 and y=4.
Answer: where is the expression
Step-by-step explanation:
Word problem example but want to see step by step.
A thief steals a number of rare plants from a nursery. On the way out, the thief meets three security guards, one after another. To each security guard, the thief is forced to give one-half the plants that he still has, plus 3 more. Finally, the thief leaves the nursery with 4 plants. How many plants were originally stolen?
There were originally 36 plants in the nursery.
What is the number of plants?From the question, we have that we can be able to show the total number of the plants as x.
Hence;
When he passes the 1st guard:
x - (0.5x+2)
x -0.5x - 2
0.5x-2 plants left
When he passes the 2nd guard
(0.5x-2) - (0.5(.5x-2)+2)
0.5x - 2 - (0.25x-1+2)
0.5x - 2 - (0.25x+1)
0.5x - 2 - 0.25x - 1
When he passes the 3rd guard
(0.25x-3) - (0.5(0.25x-3)+2) = 1
0.25x - 3 - (0.125x-1.5+2) = 1
0.25x - 3 -(0.125x+.5) = 1
0.25x - 3 - 0.125x - 0.5 = 1
0.125x - 3.5 = 1
0.125x = 1 + 3.5
Then;
x = 4.5/0.125
x = 36 plants
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Marisol spent 34.5 hours on her science project. She worked 1.5 hours each day on her project. How many days did Marisol work on her science project?
Answer:
23
Step-by-step explanation:
just divide 34.5 and 1.5 that gives you the ammount of days she worked on her project
find tan(a) in the triangle
Answer:
35/12
Step-by-step explanation:
In a right angled triangle tangent is the ratio of perpendicular and base . In the given triangle
p = 35b = 12So ,
[tex]\longrightarrow \tan\alpha =\dfrac{p}{b} \\[/tex]
[tex]\longrightarrow \underline{\underline{ \tan\alpha =\dfrac{35}{12} }} \\[/tex]
And we are done!
Please help me
Use the converse of the Pythagorean theorem to decide whether ΔABC is a right triangle by answering the questions.
1. Use the Pythagorean theorem to find AD. Show your work. (3 points)
2. Find AC. Show your work. (3 points)
3. Is ΔABC a right triangle? Explain. (4 points)
Using Pythagorean theorem, AD = 9 units and AC = 25 units.
ΔABC is a right triangle.
How to use the Pythagorean theorem to find AD?The Pythagorean theorem 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.
Using Pythagorean theorem:
AB² = AD² + BD²
AD = √(AB² - BD²)
AD = √(15² - 12²)
AD = 9 units
AC = AD + CD
AC = 9 + 16 = 25 units
For ΔABC:
The hypotenuse is AC = 25.
Using Pythagorean theorem to confirm if ΔABC a right triangle. That is:
AC² = BC² + AB²
25² = 20² + 15²
625 = 625
Since the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore, ΔABC is a right triangle.
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The number of flaws per square yard in a type of carpet material varies with mean flaws per square yard and standard deviation flaws per square yard.
a. Without doing any calculations, explain which event is more likely:
randomly selecting a
square yard of material and finding 2 or more flaws
randomly selecting square yards of material and finding an average of or more flaws
b. Explain why you cannot use a Normal distribution to calculate the probability of the first event in part (a).
c. Calculate the probability of the second event in part (a)
a. It is more likely to randomly select square yards of material and finding an average of or more flaws.
b. Normal distribution assumes that the data is continuous and symmetric, but in this case, the number of flaws per square yard is a count data, and it is not symmetric. Therefore, a normal distribution cannot be used to calculate the probability of the first event in part (a).
c. To calculate the probability of the second event in part (a), we would need the mean and standard deviation of the number of flaws per square yard, as well as the specific threshold of "or more flaws" to calculate the area under the curve that is greater than or equal to that threshold. Without that information, I cannot provide an accurate probability calculation.
Carpet Flaw Probability CalculationI determined that it was not possible to calculate the probability of the second event in part (a) without the specific mean and standard deviation of the number of flaws per square yard and the specific threshold of "or more flaws." This is because in order to calculate the probability of an event, you need to know the underlying distribution of the data and the specific cut-off point for the event in question.
In the other part of the question, I stated that it is more likely to randomly select square yards of material and finding an average of or more flaws than it is to randomly select a square yard of material and finding 2 or more flaws.
I came to this conclusion without any calculations because in general, when you are looking for a specific number of flaws (in this case, 2 or more flaws) in a single square yard, it is typically less likely than finding an average of that number of flaws when you look at multiple square yards. This is because the number of flaws per square yard can vary, so even if the average is 2 or more flaws per square yard, it is still possible to find square yards with fewer than 2 flaws.
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anyone know this? 8th grade math
The amount of metal does artist need to make the sculpture is 990,845.54 cm²
What is total surface area and volume of rectangular prism? The total surface area and volume of a rectangular prism can be calculated using the following formulas: Total Surface Area = 2(lw + wh + lh)Volume = lwhWhere l = length, w = width, and h = height.For example, if the length of a rectangular prism is 3, the width is 4, and the height is 5, then the total surface area of the prism is 2(3*4 + 4*5 + 3*5) = 94, and the volume of the prism is 3*4*5 = 60.Therefore, the total surface area and volume of a rectangular prism depends on the length, width, and height of the prism. To calculate the total surface area and volume of a rectangular prism, simply use the formulas provided above.Step 1: Calculate the total surface area of the rectangular prism.
Surface area = 2(wl + hl + hw)
Surface Area = 2((21.8 x 24.4) + (21.8 x 39.6) + (24.4 x 39.6)) = 4722.88
Step 2: Calculate the total volume of the rectangular prism.
V = l * w * h *
Volume = 21.8 x 24.4 x 39.6 = 21064.03 cm³
Step 3: Multiply the total surface area by the total volume to get the total amount of metal needed.
Total Metal Needed = 4722.88 cm² x 21064.03 cm³ = 990,845.54cm²
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At the movie theatre, child admission is $6.00 and adult admission is $9.50. On Tuesday, 143 tickets were sold for a total sales of $1089.00. How many adult
tickets were sold that day?
Answer:
Step-by-step explanation:
Let x=child number
Let y=adult number
so, x+y=143
and 6x+9.5y=1089
Form x+y=143, we got x=143-y
so, 6x+9.5y=1089
will be 6(143-y) +9.5y=1089
then we got y=66
We are looking for adult ticket were sold, so it is 66 tickets.
Kareem makes muffins using a ratio of 3.5 cups of flour for every 1.75 cups of sugar.
If he has 3.5 cups of sugar, how much flour does he need to make the muffins?
Answer:
3.5 cups of sugar / 1.75 cups of sugar = 2
3.5 x 2 = 7 cups of flour
Kareem needs 7 cups of flour to make muffins.
In this diagram, circle A has radius = 5.8 and DC = 8.1. BC is tangent line, calculate the distance of BC.
The measure of tangent BC from the given figure is 12.6 units.
What is the Pythagoras theorem?The Pythagoras theorem which is also referred to as the Pythagorean theorem explains the relationship between the three sides of a right-angled triangle. According to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a triangle.
Given that, Circle A has radius =5.8 and DC=8.1. BC is tangent line.
Here, AB=AD= 5.8 units and AC=AD+DC =5.8+8.1
= 13.9
Using Pythagoras theorem, we have
AC²=AB²+BC²
13.9²=5.8²+BC²
BC²= 193.21-33.64
BC²= 159.57
BC=12.6 units
Therefore, the measure of tangent BC is 12.6 units.
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for each of the indefinite integrals below, choose which of the following substitutions would be most helpful in evaluating the integral. enter the appropriate letter (a,b, or c) in each blank.
The reduced integral form of the equation x = 4 tan θ is 4ln|sec x| + C
The term integral is referred as the process to make use of different simplification methods depending upon the type of the integrand.
Here we have the equation x = 4 tan θ
Here in this problem, we have to make use of the substitution method to simplify the integrals.
Here we have to apply these value on the integral form then we get,
=> ∫x dx
=> ∫4 tan θ dθ
As we all know that when we integrate we get the resulting value as,
Here the number 4 is a constant, so it can be written as,
=> 4 ∫tan θ dθ
Then the resulting value is written as,
=> 4ln|sec x| + C
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