The relationship between the two variables is y=4/3 x+3.
What is the equation of a line?The general equation of a straight line is y=mx+c, where m is the gradient, and y = c is the value where the line cuts the y-axis. This number c is called the intercept on the y-axis.
The coordinate points are (0, 3) and (3, 7)
Here, slope (m) = (7-3)/(3-0)
= 4/3
Substitute m=4/3 and (x, y)=(0, 3) in y=mx+c, we get
3=4/3(0)+c
c=3
So, the equation is y=4/3 x+3
Therefore, the relationship between the two variables is y=4/3 x+3.
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Grade 5 18 is 15% what number
Answer:
120
Step-by-step explanation:
18 = .15x Divide both sides by .15 (15% as a decimal is .15)
[tex]\frac{18}{.15}[/tex] = [tex]\frac{.15x}{.15\\}[/tex]
120 = x
A tent is in the shape of a triangular prism. About how much canvas, including the floor, is used to make the tent? Draw a net if necessary.
The total area of the canvas needed would be 21.4 square yards.
What is the surface area of triangular pyramid?The formula to calculate the total surface area of a triangular pyramid is -
T.S.A. = {1/2}(a × b) + {1/2}(b × s).
Given is a tent is in the shape of a triangular prism.
The total area of the canvas would be -
A{C} = 2{1/2 x 2 x 1.7} + 2{3 x 2} + {3 x 2}
A{C} = (2 x 1.7) + 12 + 6
A{C} = 3.4 + 18
A{C} = 21.4 square yards
Therefore, the total area of the canvas needed would be 21.4 square yards.
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Answer:6
Step-by-step explanation:beacuse it it
the mass of the part of a rod that lies between its left end and a point x meters to the right is 8 x 3 kg. the linear density of the rod at 1 meters is
The rod's linear density at 1 metre from the left end and a point x meters to the right is 8 x 3 kg is = 8(1)2 = 8 kg/m.
We can start by using the definition of linear density, which is the mass per unit length of a rod or wire. Let's assume that the linear density of the rod at a distance of 1 meter from the left end is ρ kg/m.
Then, the mass of the part of the rod that lies between 0 meters and 1 meter to the right is:
m1 = ρ * 1 kg
Similarly, the mass of the part of the rod that lies between 0 meters and x meters to the right is:
m2 = 8x^3 kg
The mass of the part of the rod that lies between 1 meter and x meters to the right is:
m2 - m1 = (8x^3 - ρ) kg
Since we know that this mass is equal to ρ times the length of the rod segment, which is (x-1) meters, we can write:
ρ * (x-1) = 8x^3 - ρ
By condensing this solution, we obtain:
ρ * (x-1+1) = 8x^3
ρ * x = 8x^3
ρ = 8x^2 kg/m
As a result, the rod's linear density at 1 metre from the left end is = 8(1)2 = 8 kg/m.
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PLS PLS PLS PLS PLS PLS PLS PLS PLS PLS PLS ANSWER THESE QUESTIONS, PLEASEEEEEEEEEEEEEEEEEEEEEEEEEEE
I will be your loyal s l a v e for all eternity
1. What is the equation for a line in slope-intercept form that is perpendicular to y= 1/4x-5
2. Write the equation of the line in slope-intercept form that goes through (3,8) and is parallel to y=-3x+1.
Answer:
1) y = - 4x + b;2) y = - 3x + 17.--------------------------------
Question 1Given line:
y = 1/4x - 5We know that perpendicular lines have opposite-reciprocal slopes.
Perpendicular line to the given has a slope of m = - 1 / (1/4) = - 4.
Let the y-intercept be b, then the line is:
y = - 4x + bQuestion 2Given line:
y = - 3x + 1We know parallel lines have equal slopes.
The parallel line to the given has a slope of m = - 3 and passes through point (3, 8).
Set the point-slope equation:
y - 8 = - 3(x - 3)Convert this into slope-intercept form:
y - 8 = - 3x + 9y = - 3x + 9 + 8y = - 3x + 17ken bought a bike that was priced at $450.He was offered a 25% discount what was the discount on the bike.
Answer:
You just want to find what 25% of 450 is. To do so, you can change 25% into a decimal, and then multiply:
25% as decimal is 0.25.
0.25 * 450 = 112.5
The discount is $112.50.
Step-by-step explanation:
Hope it helps! =D
write an equation of the line passing through the given points. give the final answer in standard form. (1/2,-4) and (3/4,-40
The equation of line in standard form is y = -146x + 69.
A line equation is an algebraic way of representing a set of points that together form a line in a coordinate system. The various points that together form a line of axes can be represented as a set of (x,y) variables and form an algebraic equation, also called the line equation. You can use the straight line equation to determine if a given point lies on a straight line.
Each row of equations is a linear equation of degree 1. Read the entire article to learn about the different forms of linear equations and how to determine them. A line segment can be defined as a connection between two points. Any two points in a 2D geometry can be connected by a line segment or a simple straight line. The equation of a straight line can be obtained in three ways:
slope intercept method
point slope method
Standard method
Given two points lying on a given line, we usually follow the point-gradient method.
The equation for a straight line is y−y1=m(x−x1)
where y1 is the Y-axis coordinate, m is the tilt, x1 Coordinates on the x-axis. m = y2-y1/x2-x1
eqn = y +4 = (-36/1/4)(x-1/2)
y +4 = -146x + 73
y = -146x + 69
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1. It is claimed that less than 25% of adults prefer Italian food as their favorite ethnic food. A
survey of 1122 adults resulted in 314 who say that Italian food is their favorite ethnic food.
Find the value of the test statistic z using
z=^p-p
——-
Square root of pq
—-
n
Oz=2.31
Oz=-2.31
Oz=2.01
Oz=-2.01
The value of the test statistic z is, 2.31.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
It is claimed that less than 25% of adults prefer Italian food as their favorite ethnic food.
And, A survey of 1122 adults resulted in 314 who say that Italian food is their favorite ethnic food.
Now,
⇒ P = 25%
⇒ q = 100% - 25% = 75%
⇒ n = 1122
Hence, We get;
⇒ z = (314/1122 - 25%) / √(25% 75% / 1122)
⇒ z = 2.31
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Let g be the function given by g(x)=limh→0sin(x+h)−sinx/h. What is the instantaneous rate of change of g with respect to x at x=π3?
Answer:
The function g(x) is the derivative of the function sin(x). So, g(x) = cos(x).
Therefore, the instantaneous rate of change of g with respect to x at x = π/3 is cos(π/3), which is approximately equal to 0.5.
Step-by-step explanation:
np and pls mark brainlist:)
Consider the quadratic function f(x) = x2 – 5x + 12. Which statements are true about the function and its graph? Select three options. The value of f(–10) = 82 The graph of the function is a parabola. The graph of the function opens down. The graph contains the point (20, –8). The graph contains the point (0, 0).
The average of any three odd integers is an odd integer. What proof method would you use to show that the statement is true, or false? Prove true by exhaustion. Prove true by contraposition. O Prove false by counterexample. O Prove true by direct proof. Theorem: The difference between any odd integer and any even integer is odd. In an attempted proof of this statement a student wrote the following lines: Suppose n is any odd integer, and m is any even integer. By definition of odd integer, n = 2k + 1 where k is an integer. By definition of even integer, m = 2k where k is an integer. Then n m = (2k + 1) – 2k = 1 and 1 is odd. Therefore, the difference between any odd integer and any even integer is odd. Which one of the following best describes the reason the proof is not correct? O The proof does not prove that 1 is odd. O The proof violates the parity property. The proof violates the zero-product property. The proof assumes m and n are consecutive integers.
The proof method to show that the statement "The average of any three odd integers is an odd integer" is true is "Prove true by direct proof."
To prove that the statement is true by direct proof, we can assume that the three odd integers are a, b, and c, where a = 2k + 1, b = 2m + 1, and c = 2n + 1 for some integers k, m, and n. Then, the average of these three odd integers is:
(a + b + c)/3 = (2k + 1 + 2m + 1 + 2n + 1)/3
= (2k + 2m + 2n + 3)/3
= 2(k + m + n) + 1
Since k, m, and n are integers, k + m + n is also an integer. Therefore, the average of any three odd integers is of the form 2x + 1, where x is an integer, which is an odd integer.
The reason the attempted proof of the statement "The difference between any odd integer and any even integer is odd" is not correct is "The proof violates the parity property." The student incorrectly subtracted an even integer from an odd integer and concluded that the result is odd. However, the difference between an odd integer and an even integer is always odd, which can be proven directly as follows:
Let n be any odd integer and m be any even integer. Then, by definition, n = 2k + 1 and m = 2j for some integers k and j. The difference between n and m is:
n - m = (2k + 1) - 2j
= 2k - 2j + 1
= 2(k - j) + 1
Since k and j are integers, k - j is also an integer. Therefore, the difference between any odd integer and any even integer is of the form 2x + 1, where x is an integer, which is an odd integer.
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Take a picture or describe a place/instance/event that might influence your heart rate. Start with a heart rate of 90bpm, decide how many beats per minute your situation might increase or decrease your heart rate – DO NOT DISCLOSE YOUR NUMBER. Calculate the number of 0.04s boxes there would be between R-R intervals on an ECG trace. Write that number in your answer. - choose something that might increase your heart rate
Using unitary method, we can calculate the number of 0.04s boxes between R-R intervals on an ECG trace for our chosen event.
The unitary method is a mathematical technique that involves finding the value of one unit of a given quantity and then using it to find the value of a different quantity. In this case, we want to find the change in heart rate per second, so we need to determine how many beats per second our chosen event increases our heart rate by.
To calculate the duration of each R-R interval, we can use the following formula:
R-R interval duration = 1 / heart rate (in seconds)
Assuming our starting heart rate is 90 bpm and our event increases our heart rate by y beats per second, we can calculate the new heart rate as follows:
New heart rate = 90 bpm + (y * 60) bpm
We can then calculate the duration of each R-R interval using the formula above.
Finally, to find the number of 0.04s boxes between R-R intervals on an ECG trace, we can convert the R-R interval duration to a number of 0.04s boxes. Each 0.04s box on an ECG trace represents 0.04 seconds, so we can use the following unitary method:
1 second = 25 0.04s boxes
R-R interval duration (in seconds) = x seconds
Number of 0.04s boxes = x * 25
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I need help solving x
Answer:
x = 12
Step-by-step explanation:
The exterior angle is equal to the sum of the opposite interior angles.
5x+20 = 20 + 6x-12
Combine like terms.
5x+20 = 6x +8
Subtract 5x from each side.
5x+20-5x = 6x+8-5x
20 = x+8
Subtract 8 from each side.
20-8 = x+8-8
12 =x
determine the amount of cash collected at the time of making a cash sale of $9,220 worth of merchandise subject to 13 % HST
The total amount of cash collected is $10,418.6.
What is a percentage?A percentage is expressed as a number or ratio that can be expressed as fraction of 100. It is denoted by % symbol.
The amount of cash collected at the time of making a cash sale of $9,220 worth of merchandise subject to 13 % HST is,
Rate of interest on the sales amount is 13%
13% of 9220 = [tex]\frac{13}{100}*9220=1198.6[/tex]
Here, sales price = $9,220
The total amount of cash collected = 1198.6+9220
= 10,418.6
Hence, the total amount of cash collected is $10,418.6.
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The cost of going to the movie theater for Elissa and 11 of her friends is shown in the table. The total cost is divided equally among each person. How much does each person pay?
The cost for each person is $12.
So, each person paid $12 for a movie ticket.
What is division?The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.
Given:
The cost of going to the movie theater for Elissa and 11 of her friends is $144.
The total cost is divided equally among each person.
The number of total people is 12.
So, the cost of each person,
= total cost/ the number of total people
= 144/12
= 12
Therefore, 12 is the required value.
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A rectangle has a length of 30 feet less than five times its width if the area of the rectangle is 360 ft.² find the length of the rectangle
Answer:
[tex]l = 30[/tex] OR [tex]l=-60[/tex]
Step-by-step explanation:
We can solve this problem with a system of equations by creating 2 equations from the information given.
1. "a length of 30 feet less than 5 times its width"
[tex]l = 5w - 30[/tex]
2. "the area of the rectangle is 360 ft²"
[tex]l\cdot w = 360[/tex]
To solve for the rectangle's length, we can create a substitution for w by dividing both sides of the second equation by length ([tex]l[/tex]).
[tex]l\cdot w = 360\\\overline{\ \ l\ \ } \ \ \ \: \ \overline{\ l \ \ }[/tex]
[tex]w = \dfrac{360}{l}[/tex]
Then, we can substitute this value back into the first equation and solve for [tex]l[/tex].
[tex]l = 5(\dfrac{360}{l}) - 30[/tex]
↓ distribute the 5
[tex]l(l) = \left(\dfrac{1800}{l} - 30\right)l[/tex]
↓ multiply both sides by [tex]l[/tex]
[tex]l^2 = 1800 - 30l[/tex]
↓ move all [tex]l[/tex]'s to one side
[tex]l^2 + 30l = 1800[/tex]
To solve this quadratic, we can complete the square.
↓ add (30/2)² to both sides
[tex]l^2 + 30l + \left(\dfrac{30}{2}\right)^2 = 1800 + 15^2[/tex]
↓ simplify 15²
[tex]l^2 + 30l + 225 = 1800 + 225[/tex]
↓ factor the left side as a perfect square
[tex](l + 15)^2 = 2025[/tex]
↓ take the square root of both sides
[tex]l + 15 = \pm \sqrt{2025}[/tex]
[tex]l + 15 = \pm 45[/tex]
↓ split into 2 equations
[tex]l + 15 = 45[/tex] OR [tex]l + 15 = -45[/tex]
[tex]\boxed{l = 30}[/tex] OR [tex]\boxed{l=-60}[/tex]
How do you find the volume of the parallelepiped with adjacent edges pq, pr, and ps where p(3,0,1), q(-1,2,5), r(5,1,-1) and s(0,4,2)?
The volume of the parallelepiped is 190.
Parallelepiped VolumeThe volume V of a parallelepiped with adjacent edges pq, pr, and ps can be found using the scalar triple product as:
V = |pq · (pr x ps)|
where · denotes the dot product and x denotes the cross product.
First, we need to find the vectors pq, pr, and ps:pq = q - p = (-1-3, 2-0, 5-1) = (-4, 2, 4)
pr = r - p = (5-3, 1-0, -1-1) = (2, 1, -2)
ps = s - p = (0-3, 4-0, 2-1) = (-3, 4, 1)
Next, we need to find the cross product of pr and ps:pr x ps = det([i, j, k; 2, 1, -2; -3, 4, 1]) = i(-4) - j(7) + k(10) = (-4, -7, 10)
Finally, we can find the volume of the parallelepiped:V = |(-4, 2, 4) · (-4, -7, 10)| = 190
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The triangle on the grid will be translated two units left.
54321
47
C
2 3 4 5 X
Which shows the triangle when it is translated two units left?
The coordinates of the triangle when translated 2 units left are A'(-3, -1), B'(-3, -5) and C'(-1.5, -5).
What is Translation?Translation is a type of transformation of geometrical figures. After translation, the original figure is shifted from a place to another place without affecting it's size.
The given triangle has the coordinates A(-1, -1), B(-1, -5) and C(0.5, -5).
Translation is done 2 units left.
A(-1, -1) becomes A(-1 - 2, -1) = (-3, -1)
B(-1, -5) becomes B(-1 - 2, -5) = (-3, -5)
C(0.5, -5) becomes C(0.5 - 2, -5) = (-1.5, -5)
Hence the coordinates are A'(-3, -1), B'(-3, -5) and C'(-1.5, -5).
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make w the subject of the formula y-aw=2w-1
dont answer y+1/a+2 cuz it doesnt work
How many solutions does the system of
equations below have?
-2x - 3y + z = 7
-x - y + 3z = -10
-2x - y - z = 13
no solution
one solution
infinitely many
solutions
The system of equations has infinitely many solutions.
What is a system of equations?
A finite set of equations for which common solutions are sought is referred to in mathematics as a set of simultaneous equations, often known as a system of equations or an equation system.
Here, we have
Given:
-2x - 3y + z = 7
-x - y + 3z = -10
-2x - y - z = 13
We have to how many solutions the system of equations has.
Arranging the above equations in form of a matrix and finding the determinant, we get,
Δ = [tex]\left|\begin{array}{ccc}-2&-3&1\\-1&-1&3\\-2&-1&-1\end{array}\right|[/tex]
= -2(1+3) +3(1+6) +1(1-2)
= - 4 + 21 -1 = 16
Δ₁ = [tex]\left|\begin{array}{ccc}7&-3&1\\-10&-1&3\\13&-1&-1\end{array}\right|[/tex]
= 7(1+4) +3(10-39) +1(10+13)
= 35 - 87 + 23
= -29
Hence, the system of equations has infinitely many solutions.
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Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u,v, and w.u=i+jv=j+kw=i+k
The volume of the parallelepiped formed by the adjacent edges u=i+j, v=j+k, and w=i+k is: 1.
The triple scalar product can be used to find the volume of the parallelepiped formed by three non-coplanar vectors u, v, and w. The triple scalar product is defined as:
u · (v × w)where × represents the cross product and · represents the dot product.
Given that u=i+j, v=j+k, and w=i+k, we can first find the cross product of v and w as:
v × w = (j+k) x (i+k)= i x (j+k) + k x (j+k)
= i x j + i x k + k x j + k x k
= -k + i -j
Next, we can find the dot product of u with the cross product of v and w as:
u · (v × w) = (i+j) · (-k+i-j)= i · (-k) + i · i + i · (-j) + j · (-k) + j · i + j · (-j)
= -ik + i^2 - ij - jk + ji - j^2
= i^2 - j^2 - k(i+j)
= 1 - 0 - (1+0)
= -1
The absolute value of this result gives the volume of the parallelepiped, so the volume is:
|u · (v × w)| = |-1| = 1Therefore, the volume of the parallelepiped formed by the adjacent edges u=i+j, v=j+k, and w=i+k is 1.
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What is the difference of 8/3x+5 -2x/3x+5
The difference of 8/3x+5 -2x/3x+5 is (8 - 2x) / (3x+ 5).
The correct option is (A).
What is Expression?Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows: Expression: (Math Operator, Number/Variable, Math Operator)
We have the Expression as:
8/3x+5 -2x/3x+5
Now, Simplifying the expression as:
= 8/3x+5 -2x/3x+5
= (8 - 2x) / (3x+ 5)
= 8/(3x+ 5) - 2x/ (3x+ 5)
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An individual who has automobile insurance from a certain company is randomly selected. Let Y be the number of moving violations for which the individual was cited during the last 3 years. The pmf of Y is the following.
y 0 1 2 3
p(y) 0.50 0.25 0.20 0.05
(a) Compute E(Y).
E(Y) =
(b) Suppose an individual with Y violations incurs a surcharge of $110Y2. Calculate the expected amount of the surcharge.
$
Expert A
(a) The value of E(Y) will be = 0.8
(b) The expected amount of the surcharge is $165.
(a) The expected value of Y is given by:
E(Y) = 0(0.5) + 1(0.25) + 2(0.2) + 3(0.05)
= 0.25 + 0.4 + 0.15
= 0.8
Therefore, the expected number of moving violations for which the individual was cited during the last 3 years is 0.8.
(b) The amount of the surcharge for an individual with Y violations is $110[tex]y^{2}[/tex]. The expected surcharge is given by:
E($110[tex]y^{2}[/tex]) = $110 * E([tex]y^{2}[/tex])
To calculate E([tex]y^{2}[/tex]), we use the formula:
E([tex]y^{2}[/tex]) = Σ [tex]y^{2}[/tex] p(y)
where the sum is taken over all possible values of Y.
E([tex]y^{2}[/tex]) = [tex]0^{2}[/tex](0.5) + [tex]1^{2}[/tex](0.25) + [tex]2^{2}[/tex](0.2) + [tex]3^{2}[/tex](0.05)
= 0 + 0.25 + 0.8 + 0.45
= 1.5
Therefore, the expected surcharge is:
E($110[tex]y^{2}[/tex]) = $110 * E([tex]y^{2}[/tex])
= $110 * 1.5
= $165
So, the expected amount of the surcharge for an individual with moving violations is $165.
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Discuss what can be done to raise a credit score, as well as what you might do to lower it.
Use the following key points to start your discussion:
- factors affecting a person's credit score
- ways people can improve their credit history
a) The factors that impact a person's credit score and history include:
Credit mixNew creditType of creditPayment historyLength of credit historyAmounts of indebtedness.b) To raise or improve a credit score and history, you must undertake the following actions:
Ensuring timely paymentsRegular review of credit reportsLimiting opening new credit accountsMaintaining a low credit utilization rateKeeping old credit accounts operational.What is a credit score?A credit score is a numerical rating or expression of an individual's credit history to determine their creditworthiness.
Credit scores are presented on credit reports, which are regularly prepared by credit bureaus.
On the other hand, a credit history is an official record of an individual's credit management.
Thus, one must understand the credit history and meet payment obligations to improve a person's credit score.
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A bond has a 1,9% interest rate compounded quarterly. A stock has a 7.2% interest rate compounded
annually.
Which of the following shows how to transform the function representing the value of the bond from
quarterly compounding interest to annually compounding interest? Assume t is the time in quarters, and
that there is an initial investment of $1 in each account.
The way to transform the function such that the bond goes from quarterly compounding interest to annually compounding interest is = 1 + ( 1 + rate / 4 )ⁿ ˣ ⁴.
How to find the value ?The bond is said to be compounded quarterly. There would therefore be four compounding periods in a single year because there are four quarters in a year.
The formula would have been:
= Initial investment + ( 1 + rate ) ^ number of years
The adjusted formula to account for the quarters becomes :
= 1 + ( 1 + rate / 4 )ⁿ ˣ ⁴
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The original price of a lawn mower was $199.99. During a midsummer sale, the lawn mower is on sale for $139.99. What is the discount rate?
Answer:
60 dollars
Step-by-step explanation:
I say this because it's simple math 199.90 - 139.99 its obviously 60
mike tyson is high no ee aiqidj
Answer:
Jakaur7889 on inta
Step-by-step explanation:
Babahaja
i need this ASAP!!!!
In accordance with Side-Angle-Side theorem of congruence, the triangles seen in choices A and C are congruent.
How to find two congruent triangles
In this problem we need to determine what triangles are congruent, that is, two triangles whose sides and angles are the same in measure and distribution. We can prove congruence by means of Side-Angle-Side theorem, where given angle is between two given sides.
Then, by direct inspection, the triangles from choices A and C are congruent, because sides and in-between angle are the same in measure and distribution.
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The triangles that are certainly congruent in the diagram are triangle A and triangle C.
What are congruent triangles?Congruent triangles are triangles that have the same size and shape, such that the lengths of the three sides in one triangle are equivalent or have same length as the lengths of the three sides of the other triangle.
The congruent triangles can be obtained by using the congruence rules, or conditions for congruence, which includes; ASA, SAA, SSS, SAS
Where;
ASA is an acronym for Angle-Side-Angle
SAA is an acronym for Side-Angle-Angle
SSS is an acronym for Side-Side-Side
SAS is an acronym for Side-Angle-Side
The lengths of two sides and the measure of the included angle in triangle A are congruent to the length of two sides and the measure of the included side between the two sides in triangle C.
Therefore, the triangle in option A is congruent to the triangle in option C by the Side-Angle-Side, SAS, congruency rule.
The angle 62° is the non included angle to the 14 m and 10 m long side in option B and D, while the 14 m long side is the adjacent side to the 62° angle in option B. In option D the 10 m long side is adjacent to the 62° angle, which the orientation of triangles B and D to be different
The only two congruent triangles are triangles in option A and option C
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use the fundamental theorem of calculus to find the exact value of the definite integral `\int {1}^{\pi}\left(3x^{-1} \sqrt{x}\right)\ dx`. a correct solution must: - clearly state an antiderivative of the integrand, - show where the fundamental theorem of calculus is used and - give an exact value (do not approximate).
By using fundamental theorem of calculus The exact value of the definite integral [tex]\int {1}^{\pi}\left(3x^{-1} \sqrt{x}\right)\ dx[/tex] is 6(√(π) - 1).
First, we can find an antiderivative of the integrand by using the power rule and the constant multiple rule of integration:
∫(3x^(-1/2)) dx = 3 ∫(x^(-1/2)) dx = 3 (2x^(1/2)) + C = 6√(x) + C
where C is the constant of integration.
Now, we can use the fundamental theorem of calculus to evaluate the definite integral \int {1}^{\pi}\left(3x^{-1} \sqrt{x}\right)\ dx. The theorem states that if f(x) is continuous on the interval [a, b] and F(x) is an antiderivative of f(x), then
∫[a, b] f(x) dx = F(b) - F(a)
In this case, f(x) = 3x^(-1/2) and F(x) = 6√(x) + C. Therefore,
∫{1}^{π} (3x^(-1/2)) dx = F(π) - F(1) = 6√(π) + C - 6√(1) - C = 6√(π) - 6√(1) = 6(√(π) - 1)
So the exact value of the definite integral [tex]\int {1}^{\pi}\left(3x^{-1} \sqrt{x}\right)\ dx[/tex] is 6(√(π) - 1).
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Determine the ratio of the volume of a paraboloid (a parabola rotated about the y axis) to the volume of a cylinder with the same height and base radius:
The ratio of the volume of a paraboloid (a parabola rotated about the y axis) to the volume of a cylinder with the same height and base radius is 1/2
We know that the formula for the volume of a paraboloid is:
V = 1/2 × π × r² × h
where r is the Radius of Paraboloid
and h is Height of Paraboloid
Let us assume that V₁ represents the volume of a paraboloid (a parabola rotated about the y axis)
So, V₁ = 1/2 × π × r² × h
Also, we know that the formula for the volume of a cylinder is:
V = π × r² × h
Let us assume that V₂ be the volume of a cylinder with the same height h and base radius r.
So, V₂ = π × r² × h
consider the ratio of the volume of a paraboloid to the volume of a cylinder with the same height and base radius,
V₁ / V₂
= (1/2 × π × r² × h) / (π × r² × h)
= 1/2
Therefore, the required ratio is 1/2
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Prove that a nonempty set W is a subspace of a vector space V if and only if ax + by is an element of W for all scalar a and b and all vectors x and y in W.In one direction, assume W is a subspace and show by using closure axioms that ax + by is an element of W. In the other direction, assume ax + by is an element of W for all scalars a and b and all vectors x and y in W, and verify that W is closed under addition and scalar multiplication.(i) If W is a subspace of V, then use scalar multiplication closure to show that ax and by are in W. Now, use additive closure to get the desired result.(ii) Conversely, assume ax + by is in W. By cleverly assigning specific values to a and b, show that W is closed under addition and scalar multiplication.
If W is a subspace of V, then ax + by is an element of W since it satisfies the closure axioms of a vector space. Specifically, scalar multiplication closure confirms that ax and by are in W, and additive closure confirms that their sum, ax + by, is also in W.
Conversely, if ax + by is an element of W for all scalars a and b and all vectors x and y in W, then W must be closed under both addition and scalar multiplication.
To demonstrate this, we can assign a particular value to a or b, set the other to 1, and substitute the corresponding vectors x and y from W to get the equation ax + by = a(x) + b(y). This equation confirms that W is closed under addition and scalar multiplication, making it a subspace of V.
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