It looks like the integral is
[tex]\displaystyle 6 \int yz\cos(x) \, ds[/tex]
With [tex]x=t[/tex], [tex]y=\cos(t)[/tex], and [tex]z=\sin(t)[/tex], the line element is
[tex]ds = \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2 + \left(\dfrac{dz}{dt}\right)^2} \, dt = \sqrt2 \, dt[/tex]
and so
[tex]\displaystyle 6 \int yz\cos(x) \, ds = 6\sqrt2 \int_0^\pi \cos^2(t) \sin(t) \, dt[/tex]
Substitute [tex]u = \cos(t)[/tex] and [tex]du=-\sin(t)\,dt[/tex], then
[tex]\displaystyle 6 \int yz\cos(x) \, ds = -6\sqrt2 \int_1^{-1} u^2 \, du = 12\sqrt2 \int_0^1 u^2 \, du = \boxed{4\sqrt2}[/tex]
where in the second-to-last equality, we have
[tex]\displaystyle -\int_1^{-1} = \int_{-1}^1[/tex]
and
[tex]\displaystyle \int_{-1}^1 u^2 \, du = 2 \int_0^1 u^2\,du[/tex]
by symmetry.
Find the smallest number by which 35280 must be multiplied so that the product is a perfect square.
5 is the smallest number by which 35280 must be multiplied so that the product will be a perfect square.
What number must be an integer multiplied by to find a perfect square?
A number is a perfect square if the following property is satisfied:
a = b², where is a natural number.
Please notice that b can be either a prime number or a product of prime numbers.
Initially, we proceed to factorize 35280 by factorial decomposition, that is, as a product of prime numbers:
35280 = 2⁴ × 3² × 5 × 7²
35280 = (2²)² × 3² × 5 × 7²
35280 = 4² × 3² × 5 × 7²
Then, we must add a 5 to find a product that is a perfect square:
4² × 3² × 5² × 7² = 176400
5 is the smallest number by which 35280 must be multiplied so that the product will be a perfect square.
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9. Safa is a resident of UAE where mobile numbers consist of 7 digits. She tells her friend that her mother's mobile number is the smallest number and father's is the greatest number that can be formed by using the digits 5,7,9,8. Help her friend to find her parents numbers. Each digit must be used at least once.
Answer:
Smallest number: 5555789
Largest number: 9999875
Step-by-step explanation:
Since each digit must be used at least once, the smallest number that can be formed with the digits 5, 7, 8 and 9 is 5555789 (ascending order of digits with smallest digit used four times
The largest number is 9999875 (descending order of digits with largest digit used four times)
If the expression below has a negative value, which inequality represents all possible values of n in the expression?
Answer: B
Step-by-step explanation:
We know N cannot be 0 as the value would not be negative.
We know N cannot be negative as that would make the expression positive.
B is the only answer that works.
Squares and square roots, a perfect square is a number whose square root is an even number
Answer:
false
Step-by-step explanation:
I'm assuming this a true/false statement? Perfect squares can be defined as: [tex]a^2[/tex] where a is an integer, and a^2 is a perfect square. The reason for this is because if you take the square root of a^2, you get a, which should be an integer. Odd numbers are also integers, 3^2 = 9, and taking the square root of 9 results in an integer. So for this reason perfect squares is a number whose square root is an integer, not just even numbers
The exterior surface of a farm silo needs to be painted. if one gallon of paint covers 224 square feet, what is the minimum number of gallons needed to paint the silo? keep in mind that the bottom of the silo is not painted. use π = 3.14
The gallons needed to paint the silo is 10.
How many gallons is needed to paint the silo?A cylinder is a three-dimensional object. It is a prism with a circular base. The total surface area of cylinder can be determined by adding the area of all its faces.
Total surface area of the cylinder excluding its base = 2πrh +πr²
Where:
r = radius h = heightTSA = (2 x 3.14 x 10 x 30) + (3.14 x 10²)
314 + 1884 = 2198 ft²
Number of gallons needed = 2198 / 224 = 9.81 = 10 gallons
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△A ′ B ′ C ′ triangle, A, prime, B, prime, C, prime is the image of △ A B C △ABCtriangle, A, B, C under a rotation about point B BB. A A C C A ′ A ′ C ′ C ′ B B Determine the angles of rotation. Choose all answers that apply:
The rotation of the triangle that's illustrated will be a rotation of 90°.
How to illustrate the information?It should be noted that transformation is a general term for 4 specific ways that can be used to manipulate the shape and the position of a point, line, or a geometric figure. In such a case, the original shape of the object is called the pre-Image while the final shape and the position of the object is simply the image under transformation.
There are typically four common types of transformations and they're translation, rotation, reflection, as well as dilation. It should be noted that rotation about simply illustrates the transformation that takes place regarding the triangle.
For rotation, the shape rotates or turns the pre-image around the axis and there'll be no change in the size or shape of the object.
Here, the counterclockwise rotation was illustrated in the triangle ABC. Therefore, the rotation of the triangle that's illustrated will be a rotation of 90°.
In conclusion, based on the information given, the correct option is A.
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Complete question:
Triangle ABCTriangle ABC is rotated to form △A'B'C' .
The coordinates of point A are (1, 1) , the coordinates of point B are (3, 1) , and the coordinates of point C are (1, 4) .
Which countr clockwise rotation around the origin results in the transformation of △ABC to △A'B'C' ?
Select from the drop down arrow to choose the correct rotation.
(a) rotation of 90
(b) rotation of 180
(c) rotation of 360
If the 4 population assumptions in the hard-weinberg principle are met, what will happen to the allele frequencies?
The Hardy-Weinberg principle states that the frequency of alleles in a large randomly reproducing population will remain constant from generation to generation if certain assumptions are met.
The assumptions are:
No mutations.No migration into or out of the populationNo selection, andNo genetic drift.What is the hardy-Weinberg principle?The Hardy-Weinberg principle, also known as the Hardy-Weinberg equilibrium, model, theorem, or law in population genetics, holds that in the absence of additional evolutionary factors, allele and genotype frequencies in a population would remain constant from generation to generation.A population is not evolving while it is in Hardy-Weinberg equilibrium. Discover how violations of Hardy-Weinberg assumptions result in evolution.
When a population reaches Hardy-Weinberg equilibrium for a gene, it stops evolving and allele frequencies remain constant over generations.Hardy-Weinberg's assumptions include no mutation, random mating, no gene flow, infinite population size, and no selection.If the assumptions for a gene are not met, the population may evolve for that gene (the allele frequencies of the gene may change).Therefore, the Hardy-Weinberg principle states that the frequency of alleles in a large randomly reproducing population will remain constant from generation to generation if certain assumptions are met.
The assumptions are:
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Geometry Question: Determine the lengths of each side of triangles LMN and L'M'N'.
What can we conclude about all of the side lengths?
Using your knowledge about parallel lines,
why must all of the angles be congruent?
The lengths of each side of the triangle LMN are, LM = 4 units, MN = 3.606 units, and NL = 2.236 units.
The lengths of each side of the triangle L'M'N' are, L'M' = 4 units, M'N' = 3.606 units, and N'L' = 2.236 units.
We can conclude that all corresponding sides are equal, and the two triangles are congruent.
All angles must be congruent, as the image L'M'N' has been parallelly translated from the preimage LMN.
The coordinates of the image are L' (-4, 1), M' (-4, 5), and N' (-2, 2). The length of each side of the triangle LMN:
LM = √((2 - 2)² + (-6 - (-2))²),
or, LM = √((0² + (-4)²),
or, LM = √16 = 4 units.
MN = √((2 - 4)² + (-2 - (-5))²),
or, MN = √((-2)² + (3)²),
or, MN = √(4 + 9) = √13 = 3.606 units.
NL = √((4 - 2)² + (-5 - (-6))²),
or, NL = √((2)² + (1)²)
or, NL = √(4 + 1) = √5 units = 2.236 units.
The length of each side of the triangle L'M'N':
L'M' = √((-4 - (-4))² + (5 - 1)²),
or, L'M' = √((0² + (4)²),
or, L'M' = √16 = 4 units.
M'N' = √((-4 - (-2))² + (5 - 2)²),
or, M'N' = √((-2)² + (3)²),
or, M'N' = √(4 + 9) = √13 = 3.606 units.
N'L' = √((-4 - (-2))² + (1 - 2))²),
or, N'L' = √((2)² + (1)²)
or, N'L' = √(4 + 1) = √5 units = 2.236 units.
Now, since LM = L'M', MN = M'N', and NL = N'L', that, is all the corresponding sides are equal, we can conclude that all corresponding sides are equal, and the two triangles are congruent.
Since the triangles are congruent, the image L'M'N' has been parallelly translated from the preimage LMN.
Thus, all angles must be congruent, as the image L'M'N' has been parallelly translated from the preimage LMN.
Every point moves the same amount and in the same direction throughout a translation. As a result of the picture being identical in size and shape to the preimage, a translation is referred to as a rigid transformation or isometry.
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What is the compound interest on 2500 at 6.75 percent compounded daily for 20 days
Answer:
There for the interest on 2500 will be 93
Step-by-step explanation:
A = 2500(1+0.675/365)^0.055
A = 2500(1.0018)^0.055
A = 2500(1.037)
A = 2592.5
To find how much interest on 2500 = 2593-2500 = 93
It cost Taub $5.00 to send 100 text messages. How many text messages did she send if she spent $9.55
[tex]\begin{array}{ccll} \$&messages\\ \cline{1-2} 5 & 100\\ 9.55& x \end{array} \implies \cfrac{5}{9.55}~~=~~\cfrac{100}{x} \\\\\\ 5x=955\implies x=\cfrac{955}{5}\implies x = 191[/tex]
The number of text messages that can be sent with $9.55 is 191.
Given that, it cost Taub $5.00 to send 100 text messages.
We need to find how many text messages did she send if she spent $9.55.
What is the proportion?Proportion, in general, is referred to as a part, share, or number considered in comparative relation to a whole. The proportion definition says that when two ratios are equivalent, they are in proportion. It is an equation or statement used to depict that two ratios or fractions are equal.
Let the number of text messages she sent with $9.55.
Now, 5:100::9.55:x
⇒5x=955
⇒x=955/5
⇒x=191
Therefore, the number of text messages that can be sent with $9.55 is 191.
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Select the reason that best supports statement 2 in the given proof.
Answer:
Transitive property
The reason that best supports statement 2 in the given proof is division property of equality. Therefore, the correct answer is option D.
Given that, 3(m∠A)=90° and m∠B=m∠A.
We need to prove m∠B=30°.
According to the division property of equality, the quotients of an equation remain equal when both sides are divided by a common real integer that is not equal to 0.
Here,
Statement: Reason:
1. 3(m∠A)=90° 1. Given
2. m∠A=30° 2. Division property of equality
3. m∠B=m∠A 3. Reflexive property
4. m∠B=30° 4. Transitive Property
Therefore, the correct answer is option D.
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(d) Given that n(§) = 96, n(A) = 50 and n(B) = 60. Find the maximum and minimum values for n(An B).
how to get the answer? I want the solution
Step-by-step explanation:
the max. value is when the smaller set (A) is completely contained in the larger set (B).
then n(A n B) is n(A) = 50.
the set intersection between A and B cannot get bigger than that. or A gets bigger ...
after all, the intersection means it is a set of all elements that exist in BOTH sets.
but then there must be other elements besides A and B in the universal set too, because n(universal set) = 96, and n(A u B) would be only 60.
the min. value could be the empty set or 0. but because n(universal set) = 96, and n(A) + n(B) = 110 and larger than 96, it means that there have to be some shared elements. at least 110 - 96 = 14 elements.
in this case there cannot be other elements in the universal set than A and B. and n(universal set) = n(AuB) = 96.
2. Select true or false for each statement about the data above.
A. The most common length of ribbon is 11/
B. The sum of the longest and shortest lengths is 20¹
C. The difference between the longest and shortest
lengths is 2.
X
+
11
-
215
Part Two
A group of children picked oranges and made orange juice. The line plot below shows
the amount of juice each child made. Use the line plot below to answer questions 3-7.
X
x x
X
+
N
X
+
+ +
2²/12 2²/ 2/²/
Quarts of Orange Juice Made
3. What is the greatest amount of juice?
True False
416
True False
True False
XX
+
N
516
4. What is the least amount of juice?
5. What is the sum of the least and greatest amount of juice? Show your work.
X
+
M
6. What is the difference between the greatest and least amount of juice? Show your
work.
7. If the children were to share the juice equally, how much would each child get?
Show your work.
A number line can be defined as a type of graph with a graduated straight line which contains both positive and negative numerical values that are placed at equal intervals along its length.
What is a line plot?A line plot can be defined as a type of graph that is used to graphically represent a data set above a number line, while using crosses, dots, or any other mathematical symbol.
Based on the data provided in the complete question, we can logically deduce the following points:
Length of ribbon (11 1/4) = 3.Length of ribbon (10 3/4) = 1.Length of ribbon (10 1/2) = 1.Length of ribbon (10 7/8) = 1.Length of ribbon (10 5/8) = 1.Length of ribbon (9 7/8) = 1.In conclusion, graph C shown in the image attached below is a line plot which correctly shows the data from the table above.
The most common length of ribbon is 11 1/4: True.
The sum of the longest and shortest lengths is 20 1/8: False.
Sum = 11 1/4 + 9 7/8
Sum = 21 9/8 ≠ 20 1/8.
The difference between the longest and shortest lengths is 2 1/8: False.
Difference = 11 1/4 - 9 7/8
Difference = 1 3/8 ≠ 2 1/8 .
Part Two.The greatest amount of juice is equal to 3.
The least amount of juice is equal to 1 4/6.
The sum of the least and greatest amount of juice is:
Sum = 1 4/6 + 3
Sum = 4 4/6 = 4 2/3.
The difference between the greatest and least amount of juice is:
Difference = 3 - 1 4/6
Difference = 8/6 = 4/3.
How much would each child get?If the children were to share the juice equally, the amount of juice each would get should be calculated by determining the mean:
Mean = [F(x)]/n
Mean = (1 4/6 + 2 + 2 + 2 + 2 2/6 + 2 5/6 + 2 5/6 + 3)/8
Mean = 18.7/8
Mean = 2.3375 = 2 27/80.
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Complete Question:
Katie recorded the length, in inches, of the ribbon pieces she was going to use for a project. Use this information for questions 1 and 2.
11 1/4, 11 1/4, 10 3/4, 10 1/2, 10 7/8, 10 5/8, 9 7/8, 11 1/4,
1. Which line plot correctly shows the data?
Bob needs to send out a mailing for his company. Working alone, it will take him 10 hours to get the job done. Ellen can do the job in 8 hours by herself. They work together for 2 hours, and then Ellen finishes the job by herself. Which choice is the best estimate for how long the job will take in all?
Answer:
6 2/5 hrs or 6.4 hours total
Step-by-step explanation:
In two hours, Bob completes 2/10 = 1/5 of the job
Ellen completes 2/ 8 = 1/4
Find amount left
1 - 1/5 -1/4 = 1 - 4/20 - 5/20 = 11/20 left
Ellen will need 8hrs * 11/20 = 88/20 = 4 2/5 hours to finish the job
2 hours to start + 4 2/5 hours to finish = 6 2/5 hours total
What is (-7624928) * 997 + 7624928 * (-3)
Answer:
-7624928000 is the right answer.
Step-by-step explanation:
(-7624928) × 997 + 7624928 × (-3)
= -7602053216 + (-22874784)
= -7624928000 Ans..
Hope its helpful :-)
If so, please mark me as brainlist :-)
Answer:
Step-by-step explanation:
=(-7624928) * 997 + 7624928 * (-3)
=-7602053216+(-22874784)
=-7624928000
A high school principal reports that the average sat math score for last year’s graduating class is 520. a sample is taken of this year’s graduating class is taken and the average sat math score is reported to be 530. how would you compare the last two graduating classes?
Based on the average SAT Math score that the graduating classes got, the comparison is that this year's graduating class did better than last year's graduating class.
How can both classes be compared?The average score is used to show the general trend of scores that students were able to attain.
The average score of 530 that this year's graduating class attained is higher than the average of the previous graduating year which is 520.
This means that on average, this year's graduating class did better than that of last year because their general trend was higher.
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Can anyone tell me how you could describe this answer to the equation?
[tex]f(x)=\frac{5x}{x-25}[/tex]
The end behavior of the given polynomial is that as x → -∞ or x → ∞, then, f(x) → 5
What is the end behavior of the Polynomial?We are given the polynomial;
f(x) = 5x/(x - 25)
Now, we want to find the limits as x → ±∞. Let us rearrange the given polynomial to get; f(x) = 5/(1 - (25/x))
Thus, applying limits we have;'
lim x → ±∞ [5/(1 - (25/x))]
From algebraic limit laws we know that;
If f(x) = k, then;
lim x → +∞ [f(x)] = k
Also, lim x → -∞ [f(x)] = k
Thus, applying limits at infinity to our polynomial gives;
lim x → ±∞ [5/(1 - (25/x))] = 5/(1 - 0) = 5
This is because lim x → ∞ for 1/x is 0.
Thus, f(x) has horizontal asymptotes at y = 5
Thus, we conclude that the end behavior is that as x → -∞ or x → ∞, then, f(x) → 5
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Currently, tommy is five times as old as bianca. in eight years, tommy will only be three times as old as bianca. How old is tommy right now?
The age of Tommy is currently 40-years old
According to the statement
we have given that the tommy is five times as old as bianca and in eight years, tommy will only be three times as old as bianca.
And we have to find the present age of the tommy.
So, This is a question involving the ratio of ages.
Let the ages of Tommy and Bianca in the present be T and B respectively.
STEP 1: Let:
T = 5B
T + 8 = 3 (B+8)
Then
STEP 2: We can substitute T with 5B since the two variables are equal, to create:
5B + 8 = 3 (B + 8)
Then
STEP 3: Finally, we can expand and solve:
5B + 8 = 3B + 24
2B = 16
B = 8
T = 5B = 5 x 8 = 40
T = 40
Therefore, The age of Tommy is currently 40-years old
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If 2 = x3, then x equals
A 05
B 01
C | 1
D 5
Answer:
D.
Step-by-step explanation:
This table gives a few ( x , y ) (x,y)left parenthesis, x, comma, y, right parenthesis pairs of a line in the coordinate plane. x xx y yy − 72 −72minus, 72 25 2525 − 54 −54minus, 54 13 1313 − 36 −36minus, 36 1 11 What is the y yy-intercept of the line?
The y-intercept of the line is (0,7).
Given the pairs of a line in the coordinate plane are (x, y) (-72,25), (-54,13) and (-36,11).
First we will find the slope of the linear equation
The formula for calculating the slope between two points is equal to
m = (y₂-y₁) / (x₂-x₁)
take points (-54,13) and (-36.,11)
by substituting the values in the above formula, we get
m = (11-13) / (- 36 - (- 54))
m = -2 / 18
m = -1 / 9
Now, we will find the equation for the line in point-slope form
y-y₁ = m (x-x₁)
we have m = -1 / 9 and (x₁, y₁) = (- 36.11)
Further, substitute the values in the above equation, we get
y-11 = (- 1/9) (x + 36) ..... (1)
Convert the equation to the form of a slope intercept
y = mx + b
isolate the variable y in equation (1), we get
y-11 = (- x / 9) -4
y = (- x / 9) - 4 + 11
y = (- x / 9) +7
Here, b=7
Hence, the y-intercept of the line for the pairs of lines in the coordinate plane(x,y) is (-72,25), (-54,13) and (-36,11) is the point (0,7).
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Hurry show your work
Answer: [tex]\huge\boxed{\boxed{15}}[/tex]
Step-by-step explanation:
Given expression
[tex]\large\boxed{\frac{12[30 - (9+4^2)]}{|10|-|-6| } }[/tex]
Simplify the exponents
[tex]\large\boxed{=\frac{12[30 - (9+16)]}{|10|-|-6| } }[/tex]
Simplify values in the parenthesis
[tex]\large\boxed{=\frac{12[30 - 25]}{|10|-|-6| } }[/tex]
[tex]\large\boxed{=\frac{12[5]}{|10|-|-6| } }[/tex]
Simplify absolute values (all positive)
[tex]\large\boxed{=\frac{12[5]}{10-6 } }[/tex]
[tex]\large\boxed{=\frac{12[5]}{4 } }[/tex]
Simplify by division
[tex]\large\boxed{=3~[5]}[/tex]
Simplify by multiplication
[tex]\huge\boxed{\boxed{=15}}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
A circle is shown. Chords A C and B D intersect at point E.
Chords AC and BD intersect at E, with BD = 7 units, DE = 2 units, and AE = 4 units.
What is the length of segment CE?
units
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Given : Chords A C and B D intersect at point E. Length of Different segments is given in terms of x
To find : Length of chord BD
Solution:
Chords A C and B D intersect at point E.
=> AE * CE = BE * DE
AE = x
CE = x + 12
BE = x + 2
D = x + 5
=> x(x + 12) = (x + 2)(x + 5)
=> x² + 12x = x² + 7x + 10
=> 5x = 10
=> x = 2
BD = BE + DE
= x + 2 + x + 5
= 2x + 7
putting x = 2
=> BD = 2 (2) + 7
Applying the intersecting chord theorem, the length of segment CE in is: 2.5 units.
What is the Intersecting Chords Theorem?In geometry, the intersecting chords theorem is the theorem that defines the relation of the four line segments formed when two chords of a circle intersect each other at a point in the circle.
According to the intersecting chords theorem, the products of the lengths of the line segments that are formed on each chord are congruent or equal to each other.
In the diagram given, we have chords AC and BD intersecting at point E to form the following line segments:
BE = BD - DE = 7 - 2 = 5 units
DE = 2 units
AE = 4 units
CE = ?
Based on the intersecting chord theorem, we have:
(BE)(DE) = (AE)(CE)
Plug in the values
(5)(2) = (4)(CE)
10 = 4(CE)
Divide both sides by 4
10/4 = 4(CE)/4
2.5 = CE
CE = 2.5 units
Thus, applying the intersecting chord theorem, the length of segment CE in the circle given is: 2.5 units.
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Guess my number when i tripple my number snd add five,i get 26 what is my number
A graph of a line goes through the points (zero, four) and (negative six, zero).
Find the equation of the line using exact numbers.
The equation of the line is y = 2/3x + 4
How to find the equation of the line using exact numbers?The points on the line are given as:
(zero, four) and (negative six, zero).
Rewrite the points properly as
(0, 4) and (-6, 0)
The slope of the line is calculated using:
m =(y2 - y1)/(x2 - x1)
Substitute the known values in the above equation
m = (0 - 4)/(-6 - 0)
Evaluate
m = 2/3
The equation of the line is then calculated as:
y = mx + b
This gives
y = 2/3x + b
So, we have:
4 = 2/3 * 0 + b
Evaluate
b = 4
Substitute b = 4 in y = 2/3x + b
y = 2/3x + 4
Hence, the equation of the line is y = 2/3x + 4
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The average (mean) mass of an apple is 14g. The total mass of the
apple basket is 182g. So, how many apples are there in the basket?
which expression is equivalent to (125^2/ 124^4/3)
The answer is 25.
What is numerator and denominator?The value above the horizontal line is the numerator in a fraction. It represents the number of pieces removed from the whole. The denominator of a fraction is the numerical value that comes before the vinculum. It indicates the overall total of all of the equal portions.For instance, the fraction bar is the line that separates the numbers 4 and 5, and 45 is a fraction. Here, the numerator is the number above the fraction bar, and the denominator is the number below the fraction bar. The denominator, or the number of pieces out of the whole, is represented by a numerator.Multiplying the numerator times the numerator and denominator. Denominator. The answer should be lowered to a mixed number in the simplest form.The expression is equivalent to [tex]\frac{125^{2} }{125^{\frac{4}{3}} }[/tex] :
[tex]\frac{125^{2} }{125^{\frac{4}{3}} } =125^{2-(\frac{4}{3}) }[/tex]
[tex]=125^{\frac{2}{3} }[/tex]
[tex]125=5x^{3}[/tex]
[tex](5^{3} )^{\frac{2}{3}} =5\frac{6}{3} =5^{2} =25[/tex]
Therefore, The answer is 25.
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In a mathematics class, half of the students scored 79 on an achievement test. With the exception of a few students who scored 49, the remaining students scored 78. Which of the following statements is true about the distribution of scores?
Option C. The mean is less than the median in the distribution of the scores.
How to solve for the medianThe median is given as 79 + 78 / 2
= 78.5
The mode is 79 this is the number that has the highest frequency in the data.
The median is 78.5 this is the middle number in the set of scores of these students.
Hence the mean is less than the mode and the median given that it was it was brought down by the few scores of 49.
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please help me with 3.2
The measure of the angle ∠PQR is 90 degrees
How to prove that ∠PQR is 90 degrees?The equation of the line PQ is given as:
3x - y - 2 = 0
The coordinates of the QR are given as:
(0, -2) and (6, -4)
Make y the subject in 3x - y - 2 = 0
y = 3x - 2
The slope of the above line is
m1 = 3
Next, we calculate the slope (m2) of points Q and R.
So, we have:
m2 = (y2- y1)/(x2 - x1)
This gives
m2 = (-4 + 2)/(6 - 0)
Evaluate
m2 = -1/3
The slopes of perpendicular lines are opposite reciprocals.
m1 = 3 and m2 = -1/3 are opposite reciprocals.
This means the lines PQ and QR are perpendicular lines.
The angle at the point of perpendicularity is 90 degrees
Hence, the measure of the angle ∠PQR is 90 degrees
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Sarmad has a backyard pool that is filled with 10800 gallons of water the pool has a leak and is loseing about 2 gallons of water every second
a) Write an expression to represent that amount of water in the pool assume the pool starts with a full quantity of 10800 gallons.
b) How much water is the pool after 187 seconds
11) describe the transformations that have been applied to the function y=x Use proper mathematical terminolgy.
a) y=-2x b) y=x-2
The function that represents the amount of water is y = -2x + 10800 while the water in the pool after 187 seconds is 10426 gallons
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
A linear equation is in the form:
y = mx + b
Where m is the slope and b is the y intercept
Let y represent the amount of water in the pool after x seconds, hence:
y = -2x + 10800
After 187 s:
y = -2(187) + 10800 = 10426 gallons
The function that represents the amount of water is y = -2x + 10800 while the water in the pool after 187 seconds is 10426 gallons
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Can you help me in my math work
The LCM of 18 and 24 using the prime-factorization method is 72.
The LCM of 27, 81, and 54, using the division method is 162.
In the first question, we are asked to find the LCM of the given numbers using the prime factorization method.
(i) 18, 24.
Prime factorization gives:
18 = 2*3², and 24 = 2³*3.
The LCM is the product of the highest power of each prime factor.
Thus, LCM = 2³*3² = 8*9 = 72.
Thus, the LCM of 18 and 24 using the prime-factorization method is 72.
Doing similarly for other sub-parts, The LCM of:
(ii) 16, 40 is 80.
(iii) 30, 36 is 180.
(iv) 28, 44 is 308.
(v) 20, 32 is 160.
(vi) 20, 135 is 540.
(vii) 45, 75 is 225.
(viii) 36, 84 is 252.
(ix) 12, 18, 24 is 72.
(x) 25, 35, 45 is 1575.
(xi) 9, 15, 21 is 315.
(xii) 25, 50, 75 is 150.
In the second question, we are asked to find the LCM of the given numbers, using the division method.
(i) 27, 81, 54.
The division method gives:
[tex]\underline{2|27,81,54}\\\underline{3|27,81,27}\\\underline{3|9,27,9}\\\underline {3|3,9,3}\\\underline{3|1,3,1}\\{1|1,1,1}[/tex]
The LCM is the product of all the numbers used for division.
LCM = 2*3*3*3*3*1 = 162.
Thus, the LCM of 27, 81, and 54, using the division method is 162.
Doing similarly for other sub-parts, The LCM of:
(ii) 18, 45, 63 is 630.
(iii) 35, 55, 100 is 7700.
(iv) 210, 140, 315 is 1260.
(v) 112, 120, 150 is 8400.
(vi) 144, 180, 300 is 3600.
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