The required value of the function (f + g)(x) for given f(x) and g(x) as ( 3 / √x ) - ( 2 / x³ ) and √(5x - 7) is equals to ( 3 / √x ) - ( 2 / x³ ) + √(5x - 7).
Function f(x) is equals to,
( 3 / √x ) - ( 2 / x³ ) for all x > 0
Function g(x) is equals to,
g(x) = √(5x - 7)
To get the value of (f + g)(x),
Substitute the value of f(x) and g(x) and add the functions f(x) and g(x) together,
Sum of f(x) and g(x) is equals to,
(f + g)(x)
= f(x) + g(x)
= ( 3 / √x ) - ( 2 / x³ ) + √(5x - 7)
Therefore, value of the function (f + g)(x) is equals to ( 3 / √x ) - ( 2 / x³ ) + √(5x - 7).
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The above question is incomplete, the complete question is:
The function f is defined by f of x is equal to 3 divided by the square root of x minus 2 divided by x cubed for x > 0, g as a function of x is equal to the square root of quantity 5 x minus 7 Find (f + g)(x).
Find the nth term of this quadratic sequence 6, 16, 32, 54, ...
Answer:
3n squared+n+2
Answer:
the nth term of the sequence is 2n² + 4T
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Step-by-step explanation:
he sequence represents a quadratic function, the nth term of the sequenceis 2n² + 4
The nth term of a quadratic sequence is :an² + bn + c c = zeroth term ; a = second difference ÷ 2
From the sequence :First difference = 6, 10, 14, 18Second difference = 4, 4, 4
First difference between terms in position 1 and 0 :6 - 4 = 2
Zeroth term = First term in sequence - 2 = 6 - 2 = 4a = 4/2 = 2
Plugging the values into the equation :2n² + bn + 4 Using the 2nd term :n = 22(2)² + 2b + 4 = 128 + 2b + 4 = 12 2b = 12 - 12 2b = 0b = 0
Hence, the nth term of the sequence is 2n² + 4
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Suppose there are 16 students in your class. If the teacher draws 2 names at random, what is the probability that you and your best friend will be chosen?
1/15
1/120
1/8
3/31
Answer:
The total number of ways to choose 2 students from a class of 16 is given by the combination formula:
C(16,2) = 16! / (2! * (16-2)!) = (1615) / (21) = 120
This means there are 120 possible pairs of students that could be drawn.
The probability of you and your best friend being chosen is the number of ways that you and your friend can be selected divided by the total number of possible pairs. There is only 1 way to select you and your best friend out of the class of 16, so the probability is:
P(you and your best friend are chosen) = 1/120
Therefore, the probability that you and your best friend will be chosen is 1/120. Option (B) is the correct answer.
Answer:
The probability of choosing one specific student out of 16 is 1/16. After one student is chosen, there are 15 students left, so the probability of choosing the second specific student out of the remaining 15 is 1/15. The probability of both events happening is the product of the probabilities: (1/16) x (1/15) = 1/240. However, there are two ways that the students can be chosen (your friend first, then you or you first, then your friend), so we need to multiply the probability by 2: 2 x (1/240) = 1/120. Therefore, the probability of you and your best friend being chosen is 1/120. Answer: 1/120.
Please help me anyone please ?!!!?!!
Answer:
7. 23
8. (3 - 8) x 5
Step-by-step explanation:
I think the second one is right but I know the first one is.
can someone solve this question?
The value of f⁻¹(3) is 1/3. The value of f⁻¹(12) is -14/3. The complete table is:
X 0 1 2
f(x) 2 5 8.
What is function?A function is a mathematical rule that assigns a unique output or value for each input or value in a set. It is a relationship between two sets of numbers, called the domain (input) and range (output), such that each input value is associated with exactly one output value. A function is usually denoted by a letter such as f, and its input is represented by x, while its output is represented by f(x). The concept of a function is a fundamental one in mathematics and has many applications in various fields, including physics, engineering, economics, and more.
Here,
To find the values of f(x), we substitute the given values of x in the function f(x) = 3x + 2:
X 0 1 2
f(x) 2 5 8
To find f⁻¹(3), we first replace f(x) with 3 in the equation f(x) = 3x + 2:
3 = 3x + 2
Solving for x, we get:
x = 1/3
Therefore, f⁻¹(3) = 1/3.
To find f⁻¹(-12), we replace f(x) with -12 in the equation f(x) = 3x + 2:
-12 = 3x + 2
Solving for x, we get:
x = -14/3
Therefore, f⁻¹(-12) = -14/3.
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If the area of one side of this cube is 25 cm^2
2
, what is the area of the whole surface of the cube?
cm^2
2
Answer:
150 cm2
Step-by-step explanation:
Given side of cube's area = 25. Since Side's a square,
Edge^2 = 5^2 = 5 cm
Total surface area: 6*a² = 6*5*5 = 150 cm2
ANSWER ASAP!!!!! You will be brainliest!!! To play a game, a 12 sided number cube with faces numbered 1,1,2,2,3,3,3,4,4,4,5,5 is rolled. As a percennt rounded to the nearest tenth, what is the probability of rolling either a 3 or a 5?
Answer:
41.7%----------------------------------
There are 3 of 3's and 2 of 5's out of 12 sides.
The probability of rolling either 3 or 5 is:
(3 + 2)/12 = 5/12 = 41.7% (rounded to the nearest tenth)Answer:
36
9
1/4
Step-by-step explanation:
hope I helped ya :]
Which set of numbers could represent the lengths of the sides of a right triangle?
Responses
A: 15, 18, 21
B: 9, 11, 14
C: 8, 9, 10
D: 7, 24, 25
Answer:
C: 8, 9, 10
Step-by-step explanation:
The figure shows an angle with a measure of 52 degrees.
A. Find the complement of the angle shown.
B. Find the supplement of the angle shown.
Show all your work.
a. The cοmplement οf the given angle is 38 degrees.
b. The supplement οf the given angle is 128 degrees
What is cοmplementary and supplementary angles?Twο angles are said tο be cοmplementary if their measurements sum up tο 90 degrees. They are, in οther wοrds, angles that "cοmplete" οne anοther tο create a right angle. As an illustratiοn, if οne angle is 30 degrees, its cοunterpart is 60 degrees.
Twο angles are said tο be supplementary if their sums equal 180 degrees. They are, in οther wοrds, angles that "cοmplete" οne anοther tο create a straight line. If οne angle is 120 degrees, fοr instance, its cοmplement is 60 degrees.
Given the measure οf the angle is 52 degrees.
The cοmplement οf the angle is, when it is added with a angle and the result is 90.
Thus,
90 - 52 = 38
The supplement οf an angle is when the result οf additiοn is 180 degrees.
Thus,
180 - 52 = 128
Hence, the complement of the given angle is 38 degrees. b. The supplement of the given angle is 128 degrees
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Identify a transformation of the base function f left parenthesis x right parenthesis equals 1 over x by observing the equation of the function g left parenthesis x right parenthesis equals 1 over x minus 90.
The function g is the result of the transformation, which changes the entire graph of the function f(x) horizontally to the right by 90 units (x).
what is transformation ?A transformation in mathematics is a function that converts points or objects between two different coordinate systems. It is a method of altering a geometric figure's location, size, or shape without altering its identity or fundamental characteristics. Translations, rotations, reflections, dilations, and combinations of these operations are all considered transformations. They are frequently used in geometry, algebra, and calculus to examine how functions and equations behave under various circumstances and to address issues in a variety of mathematical and scientific fields.
given
The base function f(x) = 1/x undergoes a transformation that may be seen in the equation of the function g(x) = 1/(x-90):
it moves horizontally to the right by 90 units.
The function g is the result of the transformation, which changes the entire graph of the function f(x) horizontally to the right by 90 units (x).
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7.4y-2.9y
pls lmk....
4.5y
subtract 2.9y from 7.4y, and you get 4.5y
The area of a rectangular window is 3816 cm
If the length of the window is 72 cm, what is its width
Answer: The width of the rectangular window is 53 cm.
Step-by-step explanation:
We know that the area of a rectangle is given by the formula:
Area = Length x Width
Substituting the given values, we have:
3816 cm² = 72 cm x Width
To solve for the width, we can divide both sides by 72 cm:
Width = 3816 cm² ÷ 72 cm
Width = 53 cm
Therefore, the width of the rectangular window is 53 cm.
A local service club has monthly luncheon meetings. Each person chooses from a preset menu with three beverage choices, an appetizer of soup or salad, and four sandwiches to choose from. How many different lunches consisting of a beverage, appetizer, and sandwich are possible?
Answer:
24 different lunches
Step-by-step explanation:
There are three choices for the beverage, two choices for the appetizer (soup or salad), and four choices for the sandwich. Therefore, using the multiplication principle of counting, the number of different lunches possible is:
3 choices for beverage x 2 choices for appetizer x 4 choices for sandwich = 24 different lunches.
valuate the triple integral. $\int\!\!\int\!\!\int e {\color{red}} y \,dv$, where e is bounded by the planes $ x
The final answer is $\frac{1}{12}$.
We need to evaluate the triple integral $\iiint e y , dv$ over the region $e$ bounded by the planes $x = 0$, $y = 0$, $z = 0$, $x + y + z = 1$, and $x + y = 2$.
To evaluate this triple integral, we can use the limits of integration obtained by considering the intersection of the planes. From the plane equations $x+y+z=1$ and $x+y=2$, we can solve for $z$ and $x$ in terms of $y$ to obtain the limits:
0≤z≤1−x−yand0≤x≤2−y.
Since $e$ is bounded by the planes $x=0$ and $y=0$, we have $0 \leq x \leq 2-y$ and $0 \leq y \leq 2$. Thus, we can set up the triple integral as follows:
Next, integrating with respect to $x$, we obtain∫
02[22−22]
02−∫ 02 [eyx− 2eyx 2 − 2ey 2 x ] 02−ydy.Simplifying this expression, we get
∫02(2−522+32)
.∫ 02 (2ey− 25 ey 2 + 2ey 3 )dy.
Evaluating the integral, we get the final answer of $\frac{1}{12}$.
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question:-Evaluate the triple integral $\int!!\int!!\int e y ,dv$, where $e$ is bounded by the planes $x = 0$, $y = 0$, $z = 0$, $x + y + z = 1$ and $x + y = 2$.
Jim and Sally mow lawns in their neighborhood. Sally mows 5 less than twice the number of lawns Jim mows. Together they mow 25 lawns.
Which system of equations models this situation if j represents the number of lawns Jim mows and s represents the number of lawns sally mows?
Answer:
D
Step-by-step explanation:
Sallys= 2(Jims) -5
Sally's + Jim's = 25
Answer:
s = 2j -5s +j = 25Step-by-step explanation:
You want the system of equations that models, "Sally mows 5 less than twice the number of lawns Jim mows, and together they mow 25 lawns."
TranslationThe letters 's' and 'j' represent the number of lawns that Sally and Jim mow, respectively.
Then "twice the number of lawns Jim mows" is represented be 2j. And 5 less than that is represented by 2j-5. Since this is the number of lawns Sally mows, the equation is ...
s = 2j -5 . . . . . . . matches only one answer choice (D)
The equation for "together they mow 25 lawns" is ...
s +j = 25
using the following statements to compare how the aas congruence therom and the asa congruence therom are similar and how they are different.
In summary, both theorems require two pairs of congruent angles, but the AAS Congruence Theorem requires an included side to be congruent while the ASA Congruence Theorem requires a non-included side to be congruent.
What is similarity theorem?A similarity theorem is a statement in geometry that describes a relationship between similar geometric figures. Similar figures are figures that have the same shape but may have different sizes. A similarity theorem states that certain corresponding angles of similar figures are congruent and that the ratio of corresponding sides is constant. This constant ratio is called the scale factor, and it is used to find missing side lengths or to enlarge or reduce the size of a figure. The most commonly used similarity theorem is the AA (angle-angle) theorem, which states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Here,
The AAS Congruence Theorem and the ASA Congruence Theorem are both used to prove that two triangles are congruent. However, they differ in the conditions required for the triangles to be congruent.
The AAS Congruence Theorem requires that two pairs of corresponding angles and one pair of corresponding included sides be congruent. This means that if two triangles have two pairs of congruent angles and a side included between them is congruent, then the triangles are congruent.
On the other hand, the ASA Congruence Theorem requires that two pairs of corresponding angles and one pair of corresponding non-included sides be congruent. This means that if two triangles have two pairs of congruent angles and a side not included between them is congruent, then the triangles are congruent.
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Find the 45th derivative of y = cos(2x)
Answer:
Step-by-step explanation:
y=cos2x
y1=-2sin2x=2cos(2x+π/2)
y2=-2*2sin (2x+π/2)=2²cos(2x+π/2+π/2)=2²cos(2x+2*π/2)
y3==2²*-2sin(2x+2*π/2)=2³cos(2x+2*π/2+π/2)=2³cos(2x+3π/2)
......................................................................................................
y45=2^{45}cos(2x+45π/2)
The happy widget company has a fixed cost of $1277 each day to run their factory and a variable cost of $1.93 for each widget they produce. How many widgets can they produce for $2127?
Consider the function represented by the table.
Answer:c) 6
Step-by-step explanation:i.e. corresponding to different values of x we are given the values of f(x)
we have to find the value of f(0)
i.e. we have to find the value of f(x) when x=0
As we can see from the table the value of f(x) at x=0 is 6
calculat the following multiplication
[tex]483 \times 25[/tex]
Answer:
12075
Step-by-step explanation:
Simply multiply both terms together
483 x 25 = 12075
Given: ABCD is a parallelogram.
Prove: AB CD and BC DA
Answer:
Step-by-step explanation:
Since ABCD is a parallelogram, we know that its opposite sides are parallel. That is, AB is parallel to CD and BC is parallel to DA.
To prove that AB = CD, we can use the fact that opposite sides of a parallelogram are congruent. That is, AB is congruent to DC. So, we can write:
AB = DC
Similarly, to prove that BC = DA, we can use the same fact. That is, BC is congruent to AD. So, we can write:
BC = AD
Therefore, we have proven that AB = CD and BC = DA, which shows that the opposite sides of the parallelogram ABCD are congruent.
b.) make a conjecture about the relation between det(a), and det(ka) where k is a number (scalar). type your answer after %.
Based on the properties of determinants, my conjecture is that the determinant of ka is equal to k raised to the power of the dimension of the matrix multiplied by the determinant of the original matrix a.
My conjecture is that the determinant of a matrix multiplied by a scalar k is equal to the determinant of the original matrix a multiplied by k raised to the power of the number of rows or columns in the matrix.
In other words, if a is an n-by-n matrix and k is a scalar, then:
det(ka) = k^n * det(a)
This conjecture is based on the fact that multiplying a matrix by a scalar k multiplies every entry in the matrix by k. Therefore, the determinant, which is a sum of products of matrix entries, is also multiplied by k^n, where n is the number of rows or columns in the matrix.
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Interpret the confidence interval. Select the correct choice below and fill in the answer boxes to complete your choice (Round to one decimal place as needed.) I O A. We can be 95% confident that the mean duration of imprisonment, p, of all political prisoners with chronic PTSD is somewhere between 19 4 months and 46.1 months. O B. There is a 95% chance the mean duration of imprisonment, p, of all political prisoners with chronic PTSD will equal the mean of the interval from 19.4 months to 46.1 months
We can be 95% confident that the mean duration of imprisonment, p, of all political prisoners with chronic PTSD is somewhere between 19.4 months and 46.1 months.
Therefore the answer is A.
A confidence interval is a range of values that is likely to contain the true population parameter (in this case, the mean duration of imprisonment for political prisoners with chronic PTSD). The confidence level (in this case, 95%) indicates the percentage of times that the interval will contain the true population parameter in repeated sampling.
Option A correctly interprets the confidence interval by stating that we can be 95% confident that the true mean duration of imprisonment for political prisoners with chronic PTSD falls between 19.4 months and 46.1 months. This means that if we were to take many random samples of political prisoners with chronic PTSD and calculate the mean duration of imprisonment for each sample, 95% of the resulting confidence intervals would contain the true population mean.
Option B is incorrect because a confidence interval does not give the probability of the population parameter being in a particular range. It only gives the probability that the interval will contain the true population parameter if the sampling and estimation process is repeated many times.
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Plss help due today!!!
The survey contained 10 questions (translated into multiple languages) which resulted in categorical and quantitative data being collected. Some examples are below:
What language do you primarily speak?
Are you aware of the 2020 recipient of the European Lifetime Achievement Award in Writing? Yes or No
What is your age (in years)?
How many books did you read last year?
Do you agree with the following statements: “The European Lifetime Achievement Award recognizes authors of a variety of races and ethnicities”: Strongly Disagree, Disagree, Neither Agree nor Disagree, Agree, Strongly Agree, Do not know.
13) Which variables are quantitative?
14) Give an example of a survey question they might want to include that would result in a categorical response (include the categories).
15) What decisions might they be making with the results of the survey? Why are they interested in these questions?
Answer:
The quantitative variables in the survey are age and the number of books read last year.An example of a survey question that would result in a categorical response could be "What is your gender?" with categories such as male, female, and non-binary.The results of the survey will help the writer's guild understand the opinions and demographics of European book buyers regarding their lifetime achievement award process. Based on the responses, they may make decisions about whether to continue their current process or make changes to better align with the desires of their audience. For example, if the majority of respondents are not aware of the award or do not believe it is inclusive enough, the guild may choose to increase advertising efforts or make changes to their nomination and selection process.
find the conditional probability that x is greater than 2 6 given that x is less than or equal to 1 2 .
According to bayes therom the conditional probability thst x is greater than 26 is zero.
Bayes' theorem may be used to compute the conditional probability that x is greater than 26 if x is less than or equal to 12.
P(x > 26 | x 12) = P(x > 26 plus x 12) / P(x > 26 plus x 12) (x 12).
Because x cannot be more than 26 and less than or equal to 12, the numerator of the preceding formula is zero. As a consequence, the conditional probability is equal to zero:
P(x > 26 | x ≤ 12) = 0
This means that knowing x is less than or equal to 12 does not inform us if x is more than or less than 26.
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I need help with this
Answer:
D.
Step-by-step explanation:
Question content area top
Part 1
Find the future value of an ordinary annuity if payments are made in the amount R and interest is compounded as given. Then determine how much of this value is from contributions and how much is from interest.
R; % interest compounded semiannually for years.
Question content area bottom
Part 1
The future value of the ordinary annuity is $
177,961.83.
(Round to the nearest cent as needed.)
Part 2
The amount from contributions is $
enter your response here and the amount from interest is
$
enter your response here. (Round to the nearest cent as needed.)
The Amount from contributions = R * n
Define the term future value?The future value refers to the value of an asset or investment at a specified time in the future, based on a specific interest rate or rate of return.
Without knowing the specific values of R, interest rate, and number of years, we cannot calculate the amounts from contributions and interest. However, we can provide the general formula for calculating the future value of an ordinary annuity:
FV = R * [(1 + i)ⁿ - 1] / i
where FV is the future value of the annuity, R is the periodic payment, i is the interest rate per period, and n is the number of periods.
To calculate the amount from contributions, we can multiply the periodic payment R by the number of periods n.
Amount from contributions = R * n
To calculate the amount from interest, we can subtract the amount from contributions from the future value of the annuity.
Amount from interest = FV - R * n
Once the specific values for R, interest rate, and number of years are provided, we can use these formulas to calculate the amounts from contributions and interest.
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Let g be the function given by g(x) = x4 -4 x 3 +6x2 -4 x + k, where k is a constant.
A. On what intervals is g increasing? Justify your answer.
B. On what intervals is g concave upward? Justify your answer.
C. Find the value of k for which g has 5 as its relative minimum. Justify your answer.
A. To find the intervals on which g is increasing, we need to find the derivative of g and determine where it is positive. Taking the derivative of g, we get:
g'(x) = 4x^3 - 12x^2 + 12x - 4
To find the critical points, we set g'(x) = 0 and solve for x:
4x^3 - 12x^2 + 12x - 4 = 0
Dividing by 4, we get:
x^3 - 3x^2 + 3x - 1 = 0
(x - 1)^3 = 0
So x = 1 is the only critical point. To determine where g is increasing, we need to test a value in each of the intervals (-∞, 1) and (1, ∞). For example, if we plug in x = 0, we get:
g'(0) = -4 < 0
So g is decreasing on the interval (-∞, 1). If we plug in x = 2, we get:
g'(2) = 20 > 0
So g is increasing on the interval (1, ∞). Therefore, g is increasing on the interval (1, ∞).
B. To find the intervals on which g is concave upward, we need to find the second derivative of g and determine where it is positive. Taking the derivative of g', we get:
g''(x) = 12x^2 - 24x + 12
To find the critical points, we set g''(x) = 0 and solve for x:
12x^2 - 24x + 12 = 0
Dividing by 12, we get:
x^2 - 2x + 1 = 0
(x - 1)^2 = 0
So x = 1 is the only critical point. To determine where g is concave upward, we need to test a value in each of the intervals (-∞, 1) and (1, ∞). For example, if we plug in x = 0, we get:
g''(0) = 12 > 0
So g is concave upward on the interval (-∞, 1). If we plug in x = 2, we get:
g''(2) = 12 > 0
So g is concave upward on the interval (1, ∞). Therefore, g is concave upward on the intervals (-∞, 1) and (1, ∞).
C. To find the value of k for which g has 5 as its relative minimum, we need to set g'(x) = 0 and solve for x:
4x^3 - 12x^2 + 12x - 4 = 0
Dividing by 4, we get:
x^3 - 3x^2 + 3x - 1 = 0
(x - 1)^3 = 0
So x = 1 is the only critical point. To find the corresponding value of k, we plug in x = 1 and set g(1) = 5:
g(1) = 1^4 - 4(1)^3 + 6(1)^2 - 4(1) + k = 5
Simplifying, we get:
k = 6
Therefore, the value of k for which g has 5 as its relative minimum is 6.
Zinnia wrote the following proof to show that the diagonals of rectangle ABCD are congruent:
Zinnia's proof:
Statement 1: Rectangle ABCD is given
Statement 2: segment AD ≅ segment BC because opposite sides of a rectangle are congruent
Statement 3: segment DC ≅ segment DC by the reflexive property of congruence
Statement 4: Angles ADC and BCD are both right angles by definition of a rectangle
Statement 5: Angles ADC and BCD are congruent because all right angles are congruent
Statement 6:
Statement 7: segment AC ≅ segment BD by CPCTC
Which statement below completes Zinnia's proof? (1 point)
Triangles ADC and BCD are congruent (by ASA postulate)
Triangles ADC and BCD are congruent (by SAS postulate)
Triangles ADC and CBA are congruent (by ASA postulate)
Triangles ADC and CBA are congruent (by SAS postulate)
ADC & BCD are congruent triangles (by SAS postulate). Since triangle ADC & BCD are congruent according to the SAS postulate, we may utilize CPCTC to determine that section AC is equal to segment BD.
All are triangles 3/4 of a five?In arithmetic progression, the triangles 3: 4: 5 are the only ones with edges. Pythagorean triple-based triangles are Herodian, which means they have integer areas and sides.
Are the numbers 3 4 5 a right triangle?The easiest approach I've found to know for sure if an aspect is 90 degrees is to use the 3:4:5 triangle. According to this rule, a triangle is said to be a right triangle if one of its sides is 3 and the other is 4.
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12. What is the height of the trapezoid in yards? (Hint: Use the formula
A = 1/2h (b 1, + b2,) (Lesson 2)
Answer:
height = 7 yards
Step-by-step explanation:
the area (A) of a trapezoid is calculated as
A = [tex]\frac{1}{2}[/tex] h (b₁ + b₂ )
where h is the perpendicular height between the 2 parallel bases
here b₁ = 12 , b₂ = 9 and A = 73.5 , then
[tex]\frac{1}{2}[/tex] h(12 + 9) = 73.5 ( multiply both sides by 2 to clear the fraction )
h(21) = 147
21h = 147 ( divide both sides by 21 )
h = 7 yards
In an examination,x pupils take the history paper and 3x take the mathematics paper. A. illustrate the data on a venn diagram indicating the number of pupils I'm each region.
B. if the number of pupils taken the examination is 46, find the value of x
If we round up, we get x = 12, which means that 12 students took history and 36 students took mathematics.
What is venn diagram?A Venn diagram is a graphical representation of sets, often used to show relationships between different sets of data. It consists of one or more circles, where each circle represents a set. The circles are usually overlapping to show the relationships between the sets. The region where the circles overlap represents the elements that belong to both sets. The non-overlapping parts of each circle represent the elements that belong to only one of the sets.
Here,
A. To illustrate the data on a Venn diagram, we need to draw two overlapping circles, one for history and one for mathematics. We can label the regions as follows:
The region where the circles overlap represents the students who took both history and mathematics.
The region outside both circles represents the students who did not take either history or mathematics.
The region within the history circle but outside the mathematics circle represents the students who took history but not mathematics.
The region within the mathematics circle but outside the history circle represents the students who took mathematics but not history.
In this diagram, let's assume that x students took history and 3x students took mathematics. Then the total number of students who took the examination is x + 3x = 4x. We can label the regions on the diagram accordingly:
The region where the circles overlap represents the students who took both history and mathematics. Since there are 3x students who took mathematics, and all of them are included in the overlap region, the number of students who took both is 3x.
The region outside both circles represents the students who did not take either history or mathematics. Since there are 46 students in total, and 4x of them took the examination, the number of students who did not take either is 46 - 4x.
The region within the history circle but outside the mathematics circle represents the students who took history but not mathematics. Since x students took history, and 3x took mathematics, the number of students who took history but not mathematics is x - 3x = -2x. However, since we cannot have a negative number of students, we can assume that this region is empty.
The region within the mathematics circle but outside the history circle represents the students who took mathematics but not history. Since there are 3x students who took mathematics, and some of them also took history, the number of students who took mathematics but not history is 3x - 3x = 0. Therefore, this region is also empty.
B. We know that the total number of students who took the examination is 46. Therefore, we have:
x + 3x = 46
4x = 46
x = 11.5
However, since x represents the number of students who took history, it must be a whole number. Therefore, we can round x up or down to the nearest whole number. If we round down, we get x = 11, which means that 11 students took history and 33 students took mathematics.
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