1. The number of students who study all three subjects is 10.
2. The number of students who study only mathematics and chemistry is 8
What are set operations?Set operations are the operations that are applied on two or more sets to develop a relationship between them.
Let mathematics = M, physics = P, chemistry = C.
n(M∪P∪C) = 80
n(M) = 48
n(P) = 40
n(C) = 44
n(M∩P) = 20
n(M∩C) = ?
n(C∩P) = 24
n(M∩P) + n(M∩C) + n(C∩P) - 3n(C∩P∩M) = 32
20 + n(M∩C) + 24 - 3n(C∩P∩M) = 32
n(M∩C) = 3n(C∩P∩M) - 12 - *
We know,
n(M∪P∪C) = n(M) + n(P) + n(C) - n(M∩P) - n(M∩C) - n(C∩P) + n(C∩P∩M)
80 = 48 + 40 + 44 - 20 - n(M∩C) - 24 + n(C∩P∩M)
n(M∩C) = n(C∩P∩M) + 8 -
Equating
n(C∩P∩M) + 8 = 3n(C∩P∩M) - 12
2n(C∩P∩M) = 20
n(C∩P∩M) = 10
Putting the value of n(C∩P∩M) in *
n(M∩C) = 18
n(M∩C) - n(C∩P∩M) = 8
Therefore, eight students study mathematics and chemistry.
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please solve 2y+x=6 and 2y-3x=-2
Linear Equations
2y+x=6 ... (i)
2y-3x=-2 (ii)
x=2
y=2
To solve this problem, we will use a substitution method.
We would first transform equation (i) to get the formula for x-value.
2y+x=6
x=6-2y... (iii)
Next, we will substitute the equation (iii) into (ii) to find the value of y.
2y-3x=-2
2y-3(6-2y)=-2
2y-18+6y=-2
8y-18=-2
8y=16
y=2... (iv)
Then we would substitute the equation (iii) with the value y on the equation (iv)
x=6-2y
x=6-2(2)
x=6-4
×=2
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Answer:
(2, 2 )
Step-by-step explanation:
2y + x = 6 → (1)
2y - 3x = - 2 → (2)
subtract (2) from (1) term by term to eliminate y
(2y - 2y) + x - (- 3x) = 6 - (- 2) , that is
x + 3x = 6 + 2
4x = 8 ( divide both sides by 4 )
x = 2
substitute x = 2 into either of the 2 equations and solve for y
substituting into (1)
2y + 2 = 6 ( subtract 2 from both sides )
2y = 4 ( divide both sides by 2 )
y = 2
solution is (2, 2 )
t
f(t) = t - 8
h(t) = −2t
f(h(-5)) =
⇒Let us first find the expression of f(h(t))
⇒It means in the expression of f in the place of t you will plug in the value of what h is equal to.
[tex]f(h(t))=(-2t)-8\\f(h(t))=-2t-8[/tex]
Now f(h(-5)) means in the place of t you will plug in -5 and simplify.
[tex]f(h(-5))=-2(-5)-8\\\\f(h(-5))=10-8\\f(h(-5))=2[/tex]
Ann and Ben translate documents from German to English.
A set of documents that would take Ann 10 days would take Ben 12 days.
Ann starts to translate the document.
After 2 days Ann and Ben both work on translating the documents.
How many more days will it take to complete the work?
They would require more 4 8/22 days to complete the work.
Ann can translate the document in 10days
In 1 day he can translate 1/10 of the document
In 2 days he translates = 2 x 1/10
Ben can translate the same document in 12 days
In 1 day he can translate 1/12
Work left after Ann works for 2 days is 1 - 2/10 = 8/10
When working together they can work in 1 day is= 1/10 + 1/12 = 22/120
They do 22/120 work in 1 day
For 1 work, they require 1/(22/120) = 120/22
for 8/10, they would require (120/22)(8/10) = 96/22 = 4 8/22
Therefore, They would require more 4 8/22 days to complete the work.
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suppose you plot the location of the animals on a number line which animale would represent by the point farthest from 0 on the number line ? explaine
The animal that is farthest from 0 on the number line is the Fanfin anglerfish.
What number is farthest from 0 on the number line?
A number line is a marked straight line that is used to visually represent real numbers.
The values given in the table are negative number. A negative number is a number that has a value less than zero. A negative number is represented with a minus sign. The table has decimals, mixed numbers and fractions.
The larger a number is, the farther the distance of the number from 0 on the number line. In the given table, the largest number is -2 1/4. Thus, -2 1/4 is the farthest from the number line.
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A vehicle purchased for $27500 depreciates at a constant rate of 8% Determine the approximate value of the vehicle 10 years after purchase Round to the nearest whole number.
The approximate value of the vehicle 10 years after purchase is $11,945.68.
What is the approximate value of the vehicle?Depreciation is when the value of an asset falls with the passage of time as a result of wear and tear of the asset. Deprecation would reduce the price of the vehicle each year.
The equation that can be used to determine the value of the vehicle after 10 years is:
FV = P (1 - r)^t
Where:
FV= value of the vehicle in 10 years P = value of the vehicle when it was purchased r = rate of decline t = number of yearsFV = 27,500 (1 - 0.08)^10
FV = 27,500 x (0.92^10)
FV = $11,945.68
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Factor completely x^(3)-2x^(2)-24x
Answer:
x(x - 6)(x + 4)
Step-by-step explanation:
x³ - 2x² - 24x ← factor out x from each term
= x(x² - 2x - 24) ← factor the quadratic
consider the factors of the constant term (- 24) which sum to give the coefficient of the x- term (- 2)
the factors are - 6 and + 4 , then
x² - 2x - 24 = (x - 6)(x + 4)
then
x³ - 2x² - 24x = x(x - 6)(x + 4)
The admission fee to a museum is $12.50 per nonsenior adult, $8 per child, and $6.50 per senior citizen. A tour group consists of m nonsenior adults, (5/4m +6) children, and 8n senior citizens. What is the total admission fee of the group
The total admission fee of the group is 22.5m + 51n + 48.
How to calculate the total cost?Given that the admission fee to a museum is $12.50 per nonsenior adult, $8 per child, and $6.50 per senior citizen.
Also, it was further stated that the tour group consists of m nonsenior adults, (5/4m +6) children, and 8n senior citizens.
The total amount for the fees will be:
= Non senior adults + Children + Senior citizen
= 12.5(m) + 8(1.25m + 6) + 6.50(8n)
= 12.5m + 10m + 48 + 51n
= 22.5m + 48 + 51n
= 22.5m + 51n + 48
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What is the equation of a circle with center (-3, -5) and radius 4?
O A. (x+3)2 + (y + 5)² = 16
O B. (x-3)2 + (y- 5)² = 4
O C. (x+3)2 + (y + 5)² = 4
O D. (x-3)2+(vy- 5)² = 16
The equation of a circle with center (-3, -5) and radius 4 is (x+3)^2+(y+5)^2=16.
In the given equation we have to find the equation of circle with center and radius.
The given center is (-3, -5).
Radius = 4
The standard equation of a circle with center and radius.
(x−a)^2+(y−b)^2=r^2
where a and b represent a point of center and r represent a radius.
So a=-3, b=-5 and r=4.
Putting the value in the standard equation.
(x−(-3))^2+(y−(-5))^2=(4)^2
Simplifying
(x+3)^2+(y+5)^2=16
Hence, the equation of a circle with center (-3, -5) and radius 4 is (x+3)^2+(y+5)^2=16.
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8.26 cats, part i. the following regression output is for predicting the heart weight (in g) of cats from their body weight (in kg). the coefficients are estimated using a dataset of 144 domestic cats
The correlation coefficient is the square root of the coefficient of determination.
A. The linear model is y' = -0.357+4.034x
Where y' is the predicted heart weight and X is the body weight
B. The intercept of -0.357 is actually showing the non existence of the heart weight when the body weight is absent, ie. When X=0
C. The slope indicates that with each unit increase in the body weight x, the heart weight y increases 4.034 times.
D. R^2 = 64.66% indicates that 64.66% variation in the dependent variable that is explained by our model.
E. Given that the coefficient of determination = 0.6466
Thus the correlation coefficient is the square root of the coefficient of determination
Thus, R = √0.6466 = 0.8041
( Since the slope is positive the correlation ought to be positive)
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The question was incomplete. Check below the complete question.
The following regression output is for predicting the heart weight (in g) of cats from their body weight (in kg). The coefficients are estimated using dataset of 144 domestic cats_ Estimate Std. Error t value Pr(> /tl_ 20 (Intercept _ -0.357 0.692 -0.515 0.607 body wt 4.034 0.250 16.119 0.000 ? 15 1.452 R2 64.66% Radj 64.41% 1 Write out the linear model. [ 10 Interpret the intercept_ Interpret the slope. Interpret R?_ 2.0 Calculate the correlation coefficient 2.5 3.0 3.5 Body weight (kg) 4.0.
To show that the point (3/5, 4/5) is on the unit circle, we need to prove that (3/5)^2 + (4/5)^2 =. The distance between (2, -5) and (-7, -5) is The midpoint of the line segment that joins the given points is given by: (, ).
(a) To show that the point (3/5, 4/5) is on the unit circle, we need to prove that (3/5)² + (4/5)² = 1 .
(b) The distance between points (2,-5) and (-7,-5) is 9 units .
(c) The midpoint of the line segment that joins the given points is given by: (-5/2,-5) .
In the question ,
it is given that ,
the points given is (3/5 , 4/5)
(a) To show that the point (3/5, 4/5) is on the unit circle, we need to prove that (3/5)^2 + (4/5)^2 = 1
that is
9/25 + 16/25 = 1
25/25 = 1
1 = 1
hence proved that the point (3/5, 4/5) is on the unit circle .
(b) The distance between (2, -5) and (-7, -5) is calculated using the formula
[tex]=[/tex]√(x₂ - x₁)² [tex]+[/tex] (y₂ - y₁)²
= √(-7 - 2)² + (-5 -(-5))²
= √(-9)² + (-5 + 5)²
= √(-9)² = √81
= 9 units
(c) the mid point joining the points (2, -5) and (-7, -5) is (a,b) that is calculated by
a = (2-7)/2 , b = (-5-5)/2
a = -5/2 , b = -10/2
a = -5/2 and b = -5
the mid point is (-5/2,-5) .
Therefore , (a) To show that the point (3/5, 4/5) is on the unit circle, we need to prove that (3/5)² + (4/5)² = 1 .
(b) The distance between points (2, -5) and (-7, -5) is 9 units .
(c) The midpoint of the line segment that joins the given points is given by (-5/2,-5) .
The given question is incomplete , the complete question is
(a) To show that the point (3/5, 4/5) is on the unit circle, we need to prove that (3/5)² + (4/5)² = .
(b) The distance between (2, -5) and (-7, -5) is .
(c) The midpoint of the line segment that joins the given points is given by ( , ) .
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What is an equation of the line that passes through the point (-4,-7) and is parallel to the line 3x-y=4.
Solve it find the slope!
Write an equation the line perpendicular to y-3=-10(x-2) with an x-intercept of (-20, 0) in slope-intercept form.
please show work as detailed as possible
Answer:
y=1/10x+2
Step-by-step explanation:
y-3=-10(x-2) This equation is in point-slope form so the slope is -10
The perpendicular slope is the negative reciprocal so 1/10.
Slope intercept form is y=mx+b
Plug in the points of the x-intercept and slope into y=mx+b
0=-20(1/10)+b
0=-2+b
b=2
y=1/10x+2
A ramp is elevated to a height of 7 inches from the base. The horizontal distance between the lower end of the ramp and the base is 19 inches, as shown in the figure. What is the length of the ramp?
The length of the ramp is [tex]\sqrt{410}[/tex] inches.
According to the question,
We have the following information:
A ramp is elevated to a height of 7 inches from the base. The horizontal distance between the lower end of the ramp and the base is 19 inches.
So, it will be a right-angled triangle.
Now, we can use Pythagoras theorem to find the length of the ramp which is the hypotenuse of the triangle.
Let's denote the length of the ramp to be l, elevated height to be p and horizontal distance to be b.
[tex]l^{2} =p^{2} +b^{2}[/tex]
[tex]l^{2} =7^{2} +19^{2}[/tex]
[tex]l^{2}[/tex] = 49+361
[tex]l^{2}[/tex] = 410
l = [tex]\sqrt{410}[/tex] inches
Hence, the length of the ramp is [tex]\sqrt{410}[/tex] inches.
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Is 8/20 equal to 4/5
Answer: no
Step-by-step explanation:
8 and 20 can both be divided by 4
8 divided by 4 = 2
and 20 divided by 4 = 5
based on this, we can tell that 8/20 is not equivalent to 4/ but rather 2/5
I hope that helps
Answer:
No
Step-by-step explanation:
Because if you divide the numerator by something you have to do the same for the denominator. This means the if you do 8÷2 to get 4 then you do 20÷2 you get. 8/20=4/10/2/5
What is the argument of (1+i)^4
The fourth power of the complex number 1 + i has an argument of π radians.
How to determine the argument of the power of a complex number
In this problem we find a complex number of the form z = (a + i b)ⁿ, where n is the grade of the power of the complex number, whose expanded form is obtained by the De Moivre's theorem:
(a + i b)ⁿ = rⁿ · (cos nθ + i sin nθ)
Where:
r - Norm of the complex number.θ - Argument of the complex number, in radians.And the norm and the argument of the complex number are, respectively:
Norm
r = √(a² + b²)
Argument
θ = tan⁻¹ (b / a)
First, determine the components of the complex number and its power:
a = b = 1; n = 4
Second, calculate the argument of the complex number:
θ = tan⁻¹ 1
θ = π / 4 rad.
Third, find the argument of the power of the complex number according to De Moivre's theorem:
θ' = 4 · (π / 4)
θ' = π
The argument of the power of the complex number is π radians.
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Rewrite the value of each expression without using absolute value notation
|2b| where b<0
The rewritten form of the given expression such that it is without absolute value notation is; 2b.
What is the rewritten form of the given expression?It follows from the task content that the rewritten form of the given expression; | 2b | is to be determined.
On this note, since it is given that the variable b is less than zero; i.e b < 0.
It follows that the expression evaluation of 2b yields a negative number.
However, the rewritten form of the expression without the absolute value notation since the value of b is unknown still remains; 2b.
Therefore, the required rewritten form is; 2b.
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If p + q = 11 and 2p + q = 1 it says that 3p + 2q = 12 but I can't understand how they got to the answer. Please can someone explain? Thank you!
The value of p and q in the equation are -10 and 21 respectively.
How to solve system of equation?The value of p and q will be found in the equation before we can form another equation with p and q.
Therefore,
p + q = 11
2p+ q = 1
using elimination method
p = 1 - 11
p = -10
q = 11 + 10
q = 21
Therefore,
3p + 2q = 12 (let's check if its correct)
3(-10) + 2(21) = -30 + 42 = 12
Therefore, 3p + 2q = 12 is correct.
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Alexander buys ice cream and oranges at the store.
• He pays a total of $34.07.
• He pays $5.65 for the ice cream.
• He buys 7 bags of oranges that each cost the same amount.
Which tape diagram could be used to represent the context if x
represents how much each bag of oranges costs?
If x represents the cost of each bag of oranges then the equation will be 7x+5.65 = 34.07 and the value of x is $4.06.
According to the question,
We have the following information:
Alexander buys ice cream and oranges at the store.
• He pays a total of $34.07.
• He pays $5.65 for the ice cream.
• He buys 7 bags of oranges that each cost the same amount.
Let's take the cost of each bag of oranges to be $x.
We have the following equation:
7x+5.65 = 34.07
Subtracting 5.65 from both sides:
7x = 34.07-5.65
7x = 28.42
x = 28.42/7
x = 4.06
Hence, if x represents the cost of each bag of oranges then the equation will be 7x+5.65 = 34.07 and the value of x is $4.06.
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What is the equation of the function
The equation of line will be given as y = 3x + 24
Given,
The graph of a linear function passes through the points (4, 24) and (6, 30).
The equation of the line will be given as:-
y - y₁ = m ( x - x₁)
Slope m will be calculated as:-
m = (y₂ - y₁) / (x₂ - x₁) = (30 - 24) / (6 - 4) = 6/2 = 3
The equation will be:-
y - 24 = 3 ( x - 4 )
y = 3x - 12 + 24
y = 3x + 24
Therefore the equation of the line will be given as y = 3x + 24.
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Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of the given equation. the ordered pair (2, - 2) y = -x - 2.
The equation of the parallel line is y = -x
How to determine the line equation?The equation is given as
y = -x - 2
The point is also given as
point = (2, -2)
The equation of a line can be represented as
y = mx + c
Where
slope = m
By comparing the equations, we have:
m = -1
This means that the slope of y = -x - 2 is -1
The slopes of parallel lines are equal
This means that the slope of the other line is -1
The equation of the parallel line is then calculated as
y = m(x - x₁) +y₁
Where
m = -1
(x₁, y₁) = (2, -2)
So, we have
y = -1(x - 2) - 2
Open the brackets and evaluate
y = -x
Hence, the parallel line has an equation of y = -x
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2 lines 3y-2x=21 and 4y+5x=5, intersect at the point Q. Find the coordinates of Q
Answer: (-3,5)
Step-by-Step Solution:
1. Set 1 equation equal to y
3y-2x=21
3y=2x+21
y=2/3x+7
2. Set the other equation equal to y
4y+5x=5
4y=-5x+5
y=-5/4x+5/4
3. Set the equations equal to each other, and solve for x
2/3x+7=-5/4x+5/4
2/3x+23/4=-5/4x
(Multiply by 12 on both sides to get rid of fractions)
8x+69=-15x
8x=-15x-69
23x=-69
x=-3
4. Solve for y
(Choose any of the equations)
y=2/3x+7
y=2/3(-3)+7
y=-2+7
y=5
ANSWER: (-3,5)
1) Sarah received three fifths of her sister's book collection. If Sarah received 18
books, how many books did her sister have before giving any to Sarah?
The number of books that her sister has before giving any to Sarah is 30 books
Sarah received three fifths of her sister's book collection
Number of books that Sarah received = 18 books
The fraction of the books that Sarah received = 3/5
Consider the total number of books as x
Then we have to find the total number of books
So the equation will be
x × (3/5) = 18
Divide the terms in the equation
x × 0.6 = 18
Move 0.6 to the right hand side of the equation
x = 18 / 0.6
Divide the terms
x = 30 books
Hence, number of books that her sister has before giving any to Sarah is 30 books
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If you can help me that would be great
Answer:
x - 6 = -45y = -5Step-by-step explanation:
For the first one, find the value of x:
[tex]3x-8=-2\\3x=8-2=6\\x=2[/tex]
Then, use the value of x to find the answer:
[tex]x-6=2-6=-4[/tex]For the second one, find the value of y:
[tex]-2(3y+5)=-4\\-6y-10=-4\\-6y=10-4=6\\y=-1[/tex]
Then, use the value of y to find the answer:
[tex]5y=5(-1)=-5[/tex]
Please help me, I don't understand a thing
Answer:
Step-by-step explanation:
Let M(x,y) be the median of ΔABC through A to BC.M will be the midpoint of BC.x1=5,y1=3x2=3,y2=−1By midpoint formula, x=(x1+x2)/2∴x=(5+3)/2=8/2=4By midpoint formula, y=(y1+y2)/2∴y=(3+(−2))/2=2/2=1Hence the co-ordinates of M are (4,1).By diatnce formula, d(AM)=[(x2−x1)2+(y2−y1)2]x1=7,y1=−3x2=4,y2=1
John uses 4 cups of flour for every 4/7 cups of chocolate chips. How much flour will he use for one cup of chocolate chips?
Answer:
7
Step-by-step explanation:
4 divided by 4/7 = 7
if 4 points are indicated on a line and 5 points are indicated on another line that is parallel to the first line, how many triangles can be formed whose vertices are among the 9 points?
There are 19 triangels can be formed whose vertices are among the 9 points.
A triangle is a polygon with three sides having three vertices. The angle formed inside the triangle is equal to 180 degrees. It means that the sum of the interior angles of a triangle is equal to 180°. It is a polygon having the least number of sides. In other words, a triangle is defined as a three-sided two-dimensional figure whose interior angles are equal to 180°.
In geometry, parallel lines can be defined as two lines in the same plane that are at equal distance from each other and never meet. They can be both horizontal and vertical.
From the question, we know that we have 9 point from 2 line parallel. There are 19 triangels (the same shape is not counted twice) can be formed whose vertices are among the 9 points that shown in the following picture.
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Show that a quadrilateral with vertices (0,0), (5,0), (8,4) and (3,4) is a rhombus using distance
formula.
The quadrilateral with vertices (0,0), (5,0), (8,4), and (3,4) is a rhombus.
What is a rhombus?
A quadrilateral with all equal sides is a rhombus. Rhombuses are a particular kind of parallelogram in which all of the sides are equal since the opposite sides of a parallelogram are equal.
What is a distance formula?
D = √(x₂ - x₁)² + (y₂ - y₁)²
Here, we have
Given vertices A(0,0), B(5,0), C(8,4), and D(3,4)
By applying the distance formula, we get
AB = √(5 - 0)² + (0 - 0)² = 5
BC = √(8 - 5)² + (4 - 0)² = 5
CD = √(3 - 8)² + (4 - 4)² = 5
Hence, it's all sides are equal so it is a rhombus.
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9. [-10, ∞)
Interval notation
The inequality of the interval notation [-10, ∞) is x ≥ -10
The given interval notation is
[-10, ∞)
We have to convert the interval notation to inequality
The inequality is the mathematical statement that shows the relationship between two expression with inequality sign. The inequality signs are less than, less than or equal, greater than, greater than or equal. The inequality consist of different variables, numbers and mathematical operators.
Interval notation is defined as the method of writing subsets of the real number line
The given interval notation is
[-10, ∞)
Convert the interval notation to inequality
x ≥ -10
Hence, the inequality of the interval notation [-10, ∞) is x ≥ -10
The complete question is
Convert the interval notation [-10, ∞) to inequality
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An object is dropped from a height of 1, 600 feet. The amount of time, in seconds, the object takes to hit the ground can be
Found by solving the equation -16r2+1, 600 = 0. How many seconds will it take to hit the ground?
10?,50?,10 or -10?,100?
The time taken by the object to hit the ground is 10 seconds , the correct option is (a) .
In the question ,
it is given that ,
the object is dropped from the height of 1600 feet ,
So , the height h(t) = 1600
the equation representing the height of the object at time t is ,
h(t) = 16t² + 155
Substituting the height h(t) = 1600
16t² + 155 = 1600
16t² = 1600 - 155
16t² = 1445
t² = 1445/16
t² = 90.31
t = √90.31
t = 9.503
t ≈ 10 seconds
Therefore , The time taken by the object to hit the ground is 10 seconds .
The given question is incomplete , the compete question is
An object is dropped from a height of 1,600 feet. The amount of time, in seconds, the object takes to hit the ground can be found by solving the equation 16t² + 155 = h(t) . How many seconds will it take to hit the ground ?
(a) 10
(b) 50
(c) 100
(d) 500
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