After solving the following sets in roster form we get B' = {-8,-5,-4,0,1,2,4,5,6,7,8} and
A' U (B ^ C) = {-8, -6, -5 , -4 , 0 , 2, 3, 5, 6 ,7 }
What are sets and roster form?
Either elements or members of a set are terms used to describe the objects that make up a set. A set's components are denoted by curly brackets enclosing them and commas separating them. The sign "" is used to indicate that an element is part of a set. 2 A in the previous illustration. The symbol "" is used to designate an element that is not a part of a set. Here, 3 ∉ A.
The roster notation, in which the components of the sets are contained in curly brackets and separated by commas, is the most popular way to express sets. Set B, for instance, which is made up of the first five even integers, is equal to 2, 4, 6, 8, and 10. The order of the set's components in a roster form is irrelevant; for instance, the first five even integers can equally be described as 2, 6, 8, 10, 4. Additionally, if a set has an infinite number of items, the endless list is defined by adding a string of dots to the end of the final element. For instance, the representation of infinite sets is X = 1, 2, 3, 4, 5,..., where X is the set of natural numbers.
Please see the examples below for a summary of the roster form's notation.
Set A = 1, 2, 3, 4, and 5 in the finite roster notation of sets (The first five natural numbers)
Sets in Infinite Roster Notation: Set B = {5, 10, 15, 20 ....} (The multiples of 5) (The multiples of 5)
a) B' = U - B = {-8,-5,-4,0,1,2,4,5,6,7,8}
b) A' U (B ^ C)
A' = U - A = {-8, -6, -5 , -4 , 0 , 2, 3, 5, 6 ,7 }
B intersection C = ∅
A' U (B ^ C) = A' = {-8, -6, -5 , -4 , 0 , 2, 3, 5, 6 ,7 }
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silicone implant augmentation rhinoplasty is used to correct congenital nose deformities. the success of the procedure depends on various biomechanical properties of the human nasal periosteum and fascia. an article reported that for a sample of 20 (newly deceased) adults, the mean failure strain (%) was 26.0, and the standard deviation was 3.4.a. Assuming a normal distribution for failure strain, estimate true average strain in
a way that converys information about precision and reliability.
b. Predict the strain for a single adult in a way that conveys information about
precision and reliability. How does the prediction compare to the estimate
calculated in part (a)?
A true average strain in a way that converys information about precision and reliability (24.33, 27.67)
From given information we have:
the sample size n = 20
the mean failure strain x(bar) = 26.0
the standard deviation s = 3.4
Assuming a normal distribution for failure strain, we need to estimate true average strain in a way that converys information about precision and reliability.
A 95% confidence interval is necessary; other confidence intervals can be calculated in a similar way.
c = 0.95 or 95%
In the table with the critical values for t distributions in the appendix, locate the row beginning with degrees of freedom.
Here, the degree of freedom :
df = 20 - 1
df = 19
and α = (1-c)/2
= 0.025
To find the t-value:
t_(α/2) = 2.145
Therefore, the error margin is:
E = t_(α/2) × s/√n
E = 2.145× 3.4/√19
E ≈ 1.67
And the confidence interval would be,
(x{bar} - E, x{bar} + E) = (24.33, 27.67)
Therefore, the confidence interval is: (24.33, 27.67)
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Error Analysis.
Use mathematical reasoning and appropriate terminology to describe (with words) and correct the error (with work) in simplifying each rational expression.
Error Analysis 1:(x-3)/(x+7)
Error Analysis 2:(x(x-7)(x+3))/2(x+1)(x+9)(x+5)
What is meant by error analysis?Error analysis is a procedure that is usually used to identify the cause of the issue when pupils continually make mistakes. The method entails going over a student's work and looking for patterns of misinterpretation. Mathematical errors can occur for a variety of reasons and can be factual, procedural, or conceptual. The study of the kind and quantity of mistake, or uncertainty, that might be present in the solution to a problem is known as error analysis in mathematics. This issue is particularly common in practical disciplines like statistics and numerical analysis. The teacher can examine a student's errors on a worksheet, test, or progress monitoring tool to do an error analysis for mathematics. In order to grade each problem, the teacher should mark each wrong digit in the student's response, starting from the RIGHT, going to the LEFT for addition, subtraction, and multiplication problems.
Error Analysis 1:
(x²-x-6)/(x²+9x+14)
=(x-3)(x+2)/(x+2)(x+7)
=(x-3)/(x+7)
Error Analysis 2:
(x²-6x-27)/(2x²+2x)÷(x²+14x+45)/(x²)
=(x(x-7)(x+3))/2(x+1)(x+9)(x+5)
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44,50,54,66,51,63,43,57,64 and 42 what is the range difference?
A vector field F is shown. Use the interpretation of divergence derived in this section to determine whether div F is positive or negative at P_1 and at P_2.a. div F is negative at both P_1 and P_2. b. div F is positive at P_1 and negative at P_2. c. div F is positive at both P_1 and P_2. d. div F is negative at P_1 and positive at P_2.
As pre the concept of vector, div F is negative at P1 and divF is positive at P2.
In math the term vector refers the quantity that has both magnitude and direction but not position
Here we have to determine whether div F is positive or negative at P1 and at P2.
Let us consider if div F(P)>0, the net flow is outward near P and P is called a source and if div F(P)<0, the net flow is inward near P and P is called a sink.
Here by observing the graph of F, we have identified that the vectors end near the P1 are the larger than the vectors starts near the point P1,
So, the net flow is inward near P1 and div F is negative at P1
Where P1 is called a sink
Whereas, by observing the graph of F we have identified that the vectors end near P2 are shorter than the vectors starts near the point P2,
So, the net flow is outward and div F is positive at P2 and the P2 is called a source
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i need the answers please
(7) Slope of line AB = -6/5 and Slope of the line CD = -6/5 and line AB and line CD are parallel in nature.
(8)Slope of line AB = 3/4 and the Slope of line CD = 1 and line AB and line CD is neither parallel nor perpendicular.
(9) We have drawn the graph passing through the point P and parallel to line AB.
(10)We have drawn the graph passing through the point P and perpendicular to line AB.
What is the property of slopes of parallel lines and perpendicular lines?
If two lines are given and they are parallel to each other then the slopes are equal and for perpendicular lines, the product of their slopes is equal to -1.
(7) The line AB is passing through the points (-1, 5/2) and (3/2, -1/2).
Slope = y2 - y2/x2 - x1
= (-1/2 - 5/2)/(3/2-(-1))
= -6/5
As line AB and line CD are parallel in nature so the slope of line CD is also -6/5.
(8) The line AB is passing through the points (1/2, 1) and (5/2, 5/2).
Slope = y2 - y2/x2 - x1
= (5/2 - 1)/(5/2-1/2)
=3/2/2
= 3/4
Line CD is passing through points (1, 1) and (2, 3).
Slope = y2 - y2/x2 - x1
= (3 - 1)/(2-1)
=2/2
= 1
As the product of their slopes is not equal to -1.
So lines are neither parallel nor perpendicular.
(9) Slope of AB can be computed as (-2, 1) and (-5, 5)
Slope = 5 - 1/-5 + 2
= 4/-3.
The slope of line passing through the point P = -4/3 as we have to draw the parallel line.
Point P = (-1, 3)
Equation of the line :
y - 3 = -4/3(x + 1)
3y +9 = -4x - 4
3y = -4x - 13
Now we can draw the line
(10) Slope of line AB = -4/3
So the slope of the line passing through point P should be 3/4 in order to be perpendicular to line AB.
Equation of the line passing through point P:
y - 3 = 3/4(x - (-1))
y - 3 = 3/4(x+1)
Hence,
(7) Slope of line AB = -6/5 and Slope of the line CD = -6/5 and line AB and line CD are parallel in nature.
(8)Slope of line AB = 3/4 and the Slope of line CD = 1 and line AB and line CD is neither parallel nor perpendicular.
(9) We have drawn the graph passing through the point P and parallel to line AB.
(10)We have drawn the graph passing through the point P and perpendicular to line AB.
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A lamina occupies the region inside circle x2 +y2 = 2y but outside circle x2 +y2 = 1. Find the center of mass if the density atany point is inversely proportional to its distance from theorigin.
the answer is {0, [3√3/2(3√3-π)]}
Ф {0, [3√3/2(3√3-π)]}
The poster above has good diagrams ; so , I will just go through the calculations .
The region of integration is with in the circle x ^2+y^2=2y but outside the circle x^2+y^2=1.
I would convert to polar . With polar , we have x^2+y^2=r^2 ,x=rcosФ and y=rsinФ
We thus have the first circle becomes x^2+y^2=2y ⇒r^2=2rsinФ Divide through by r, and you get r=2sinФ(1).
This is the upper circle in the previous posters diagrams .
The other equation of a circle becomes :
x^2 + y^2 = 1 ⇒ r^2 = 1 ⇒ r = 1 (2)
These two curves intersect at :
r= 2sinФ =1 ⇒ sinФ=1/2 ⇒Ф= [tex]\alpha[/tex]/6 or 5[tex]\pi[/tex]/6
From the previous poster's diagrams , we can see that there is as much mass on either side of the positive. y axis or
Ф=[tex]\pi[/tex]/2 axis in polar ; so , the X coordinate of the center of mass is zero
X=0 We can also see from his diagrams that the mass and moment of mass is symmetrical about the Ф=[tex]\pi[/tex]/2
we only need to integrate from Ф=[tex]\pi[/tex]/6 to Ф=[tex]\pi[/tex]/2 ( the answers from to are the same and double )
The differential of area in polar coordinates is
d A=rdrdФ They say the area mass density is inversely proportional to the distance from the origin .
This can be written as : p=k/r. We can thus say the differential of mass is
d M=PDA=
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help help help help plsssss
Answer:true, false, false, true
Step-by-step explanation:
What is the slope-intercept equation for this line?
Answer:
y = -1/2x - 2
Step-by-step explanation:
y = mx+b, m = -1/2, and b (the y-intercept) is -2.
Eli and Amy brought in 3 packs of stickers to give out to their two classes. They had enough for Mrs. O’s 21 students, Mrs. L's 24 students and even had 15 stickers left over! How many stickers were in each pack?
ill give brainliest semester ends tommorow
Answer: We can set up the equation 21 stickers/pack + 24 stickers/pack + x stickers/pack = 15 stickers.
We can solve this equation to find that x, the number of stickers in each pack, is equal to 6 stickers/pack. Answer: 6
Answer: 20
Step-by-step explanation:
24 + 21 = 45
45 + 15 = 60
60 / 3 = 20
a service center receives an average of 0.7 customer complaints per hour. management's goal is to receive fewer than three complaints each hour. assume the number of complaints follows the poisson distribution. determine the probability that exactly four complaints will be received during the next eight hours.
The probability that exactly four complaints will be received during the next eight hours is 15.41%.
A discrete probability distribution called the Poisson distribution provides the likelihood that an event will occur a specific number of times over the course of a given period of time or location. The formula for this distribution is given by [tex]P(X=x)=\frac{e^{-\mu}\times \mu ^x}{x!}[/tex] where x is the number of successes, μ is the mean, and e is Euler's number.
Given n = 8 hours then, μ = 0.7n = 0.7×8 = 5.6.
Then, the probability that exactly four complaints are received is,
[tex]\begin{aligned}P(X=4)&=\frac{e^{-5.6}\times5.6^4}{4!}\\&=0.1541\end{aligned}[/tex]
The required answer is 15.41%.
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W is the midpoint of SU, TU ⊥ SU and SU ⊥ SV. Complete the proof that Δ SVW ≅ Δ UTW.
Statement Reason
1. W is the midpoint of SU Given
2. TU ⊥ SU Given
3. SU ⊥ SV Given
4. ∠S ≅ ∠U All right angles are congruent
5. ∠ SVW ≅ ∠ UTW Definition of midpoint
6. Δ SVW ≅ Δ UTW.
The proof that Δ SVW ≅ Δ UTW is given by ASA congruence criterion.
What do you mean congruence of triangle?Two triangles are said to be congruent if all three corresponding sides are equal and all the three corresponding angles are equal in measure. These triangles can be slides, rotated, flipped and turned to be looked identical. If repositioned, they coincide with each other.
The symbol of congruence is’ ≅’.
Given:W is the midpoint of SU
⇒ SW = WUTU ⊥ SU and SU ⊥ SV
To prove:Δ SVW ≅ Δ UTW
Proof:In Δ SVW and Δ UTWSW = UW (given)
∠S ≅ ∠U (given)
∠ SVW ≅ ∠ UTW (vertically opposite angle)
Therefore, Δ SVW ≅ Δ UTW (by ASA congruence criterion)
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The list shows the number of books read by the students in Ms. Alonzo’s class each week for two months.
25, 7, 13, 8, 22, 7, 13, 8
Which stem and leaf plot represents the same data?
stem leaf
0 7, 7, 8, 8
1 3 , 3
2 5, 2
What is stem and leaf plot ?
A stem and leaf plot, also known as a stem and leaf diagram, arranges data in a way that makes it simple to see how often certain sorts of values occur. It is a graph that displays ordered numerical data. A stem and a leaf are divided into each piece of data.
A stem and leaf plot is shown as a customized table with the first or final digit of each data value divided into a stem and the remaining digit in a leaf. The stem and leaf plot key is the symbol " | " that is used to display stem and leaf values.
For instance, the stem and leaf plot key is used to represent 46 as 4 on the stem and 6 on the leaf.
in 2|5 , 2 is the stem side and 5 is the leaf side
in 0|7 , 0 is the stem side and 7 is the leaf side
in 1|3 , 1 is the stem side and 3 is the leaf side
in 0|8 , 0 is the stem side and 8 is the leaf side
in 2|2 , 2 is the stem side and 2 is the leaf side
in 0|7 , 0 is the stem side and 7 is the leaf side
in 1|3 , 1 is the stem side and 3 is the leaf side
in 0|8 , 0 is the stem side and 8 is the leaf side
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Select one or more expressions that together represent all solutions to the equation. Your answer should be in degrees.
Assume n is any integer.
4cos(10x)+2=2
Choose all answers that apply
A. -172° + n*36°
B. -90° + n*360°
C. -9° + n*36°
D. 9° + n*36°
E. 90° + n*360°
The expressions are x=9°+n.36° and x= -9°+n36°.
So, options C and D are correct.
What is meant by an expression?A finite combination of symbols that are well-formed in accordance with context-dependent rules is referred to as an expression or mathematical expression in mathematics. Mathematical symbols can represent variables, operations, functions, brackets, punctuation, and grouping to help specify the logical syntax and order of operations. Lambda expressions are a new class of expressions that Alonzo Church and Stephen Kleene introduced in the 1930s for formalizing functions and their evaluation. In mathematical logic and the theory of programming languages, they serve as the foundation for the formal system known as lambda calculus.
Given,
4cos(10x)+2=2
4cos(10x)=0
cos(10x)=0
10x=90°+n360°
x=(90°/10°)+n(360°/10)
x=9°+n.36°
(or)
10x= -90°+n360°
x= -9°+n36°
Therefore, the expressions are x=9°+n.36° and x= -9°+n36°
Hence, options C and D are correct.
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A train travels at 80 miles per hour. An equation can be written that compares the time (t) with the distance (d). What is the domain and range?
1. The domain is distance (d) and the range is time (t).
2. The domain is time (t) and the range is distance (d).
3. The domain is time (t) and the range is 80.
4. The domain is 80 and the range is time (t).
The required answer is the domain is time (t) and the range is a distance (d) i.e. Option 2.
What are domain and range?
The value range that can be plugged into a function is known as its domain. In a function like f, this set represents the x values f(x). The collection of values that a function can take on is known as its range. The values that the function outputs when we enter an x value are in this set.
From the given question, and the above definition of domain and range,
the time (t) acts as an x-values or input value and the distance (d) acts as a y-value or output value
Hence, the domain is time (t) and the range is a distance (d) i.e. Option 2.
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Reshawn read 25 pages of his book each day until he finished the book. His book was 400 pages long. Which sketch represents this solution
Graph (C) which has points (0, 400) and (16, 0) represent the solution to the given situation of Rashawn.
What is a graph?In mathematics, a graph is a visual representation or diagram that shows facts or values in an ordered way.
The relationships between two or more items are frequently represented by the points on a graph.
Graphs and charts are useful visual aids because they make information accessible and quick to understand.
Therefore, it is not unexpected that print and electronic media frequently use graphs.
When data is displayed as a graph rather than a table, it can often be easier to understand because the graph might show a pattern or comparison.
So, Rashawn consumed 25 pages of his book daily till it was finished. There were 400 pages in his book.
400 / 25 = 16 days is the estimated number of days he needs to finish the book.
He will have 400 pages finished in 0 days and 0 pages in 16 days.
As a result, the situation is represented by a graph with a line segment in the first quadrant.
Days are shown on the x-axis, and Pages left are listed on the y-axis.
The line segment's ends are (0, 400) and (16, 0).
Therefore, graph (C) which has points (0, 400) and (16, 0) represent the solution of the given situation of Rashawn.
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Complete question:
Rashawn read 25 pages of his book each day until he finished the book. His book was 400 pages long.
Which sketch represents this situation?
a) A graph in the first quadrant containing a line segment. The x axis is labeled Days and the y axis is labeled Pages remaining. The endpoints of the line segment are begin ordered pair 0 comma 400 end ordered pair and begin ordered pair 25 comma 0 end ordered pair.
b) A graph in the first quadrant containing a line segment. The x axis is labeled Days and the y axis is labeled Pages remaining. The endpoints of the line segment are begin ordered pair 0 comma 16 end ordered pair and begin ordered pair 400 comma 0 end ordered pair.
c) A graph in the first quadrant containing a line segment. The x axis is labeled Days and the y axis is labeled Pages remaining. The endpoints of the line segment are begin ordered pair 0 commas 400 end ordered pair and begin ordered pair 16 commas 0 ends ordered pair.
d) A graph in the first quadrant containing a line segment. The x axis is labeled Days and the y-axis is labeled Pages remaining. The endpoints of the line segment are begin ordered pair 0 commas 25 ends ordered pair and begin ordered pair 400 commas 0 ends ordered pair.
[(6x2.99)-1] ÷2
Please help me
Answer: Answer is 8.47
Step-by-step explanation:
[(6x2.99)-1] ÷2
16.94 / 2
8.47
Answer:
8.47
Step-by-step explanation:
[(6x2.99)-1] ÷2 16.94 /2 =8.47
Cathy was not watching her dog's eating habits and his weight went from 40 pounds to 48
pounds. By what percentage did the dog's weight increase?
Answer:
Step-by-step explanation:
To find the percentage increase in the dog's weight, you can use the following formula:
percentage increase = (final value - initial value) / initial value x 100%
In this case, the initial value is 40 pounds and the final value is 48 pounds, so the percentage increase is:
percentage increase = (48 - 40) / 40 x 100% = 8 / 40 x 100% = 20%
Therefore, the dog's weight increased by 20%.
19
Select the correct answer.
What is the solution to this equation?
log₃(4 x)-2log₃ x=2
A. 1/36
B. 9/4
C. 4/9
D. 36
Question
1/3 Marks
The prism shown has a volume of 35 cm³.
Work out h, the height of the triangular cross section.
h = Answ cm
13/27 Marks
+
hi
4 cm
7 cm
The height of the triangular cross section is equal to 2.5 cm.
How to calculate the volume of a rectangular prism?Mathematically, the volume of a rectangular prism can be calculated by using this formula:
Volume = L × W × H or Volume = base area × length
Where:
L represents the length of a rectangular prism.W represents the width of a rectangular prism.H represents the height of a rectangular prism.Next, we would determine the base area of this rectangular prism by using the formula for the area of a triangle:
Area of triangle = 1/2 × base area × height
Area of triangle = 1/2 × 4 × h
Area of triangle = 2h
Now, we can determine the height of this rectangular prism as follows;
Volume = base area × length
35 = 2h × 7
Height, h = 35/14
Height, h = 2.5 cm.
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-3/2 passes through (-4, 3)
what is the y intercept and equation
(in slope intercept form)
Refer to the above diagram. Economies of scale: A) are evident over the entire range of output. B) occur over the OQ range of output. C) begin at output 03. D) occur only over the
Economies of scale occur when average cost declines with an increase in output. This occurs up to output level Q1.
Diseconomies of scale occur when average cost increases with an increase in output. This begins at output level Q3.
The phenomenon known as economies of scale occurs when the size or magnitude of the production generated by a business increases while the average cost per unit of output decreases.
The Long Run Average Cost Curve demonstrates how a firm's average expenses change over time. The shift from point A to point B illustrates how the cost per unit decreases as the curve slopes down. Because of economies of scale, the average cost grows less proportionally to output over the downward sloping region of the curve.
Minimum efficient scale is the level of output where economies of scale have been fully exploited.
A competitive firm produces at the minimum point of ATC in the long run and thus achieves productive efficiency.
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Can someone help me with this?
omgggg thats what she said
If sin =−0.093 and 00 ≤ <3600, then what is to the nearest tenth of a degree?
A. 5.3⁰ or 174.7⁰
B. 185.3⁰ or 354.7⁰
C. 174.7⁰ or 185.3⁰
D. 2.2⁰ or 0.9⁰
There are two angles for the sine function such that sin θ = - 0.093: (i) 185.3°, (ii) 354.7°.
What are the approximate values of a trigonometric function?
In this problem we must determine two values of the angle of the sine function such that its result is - 0.093. According to the features of the trigonometric function, sine is negative for angles between 180° and 360°.
Since trigonometric functions are trascendent functions, we can only obtain approximate values of the angle by means of inverse trigonometric functions and numerical methods. There are two possible solutions:
sin⁻¹ - 0.093
θ₁ ≈ 185.336° or θ₂ ≈ 354.664°
θ₁ ≈ 185.3° or θ₂ ≈ 354.7°
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What is a selective industrialization is for an Expo
A selective industrialization is a phenomenon in which those sectors or elements where it is desired to centralize or focus the industrialization itself are selected.
Analysis on selective industrializationThis has been a type of industrialization where industrialization is projected in sectors that are previously selected and defined.
Some of the characteristics of selective industrialization are:
It focuses on the industrial development of a sector and selected elements.The main objective has been to promote an economic sector that has a high potential.¡Hope this helped!
Selective industrialization is a strategy that involves focusing on the development of specific industries or sectors in a country's economy.
This approach is often used by developing countries to promote economic growth and development by targeting key industries that have the potential to drive growth and create jobs.
In the context of an Expo, selective industrialization could refer to showcasing and promoting specific industries or sectors that a country or region specializes in or wants to develop further.
This could involve exhibiting products, technologies, and innovations related to those industries, as well as networking with potential investors, partners, and customers to explore opportunities for collaboration and growth.
Four friends paid a total of $32 for bowling. What is the ratio $32 for 4 people written as a unit rate?
Answer:
$8 per person
Step-by-step explanation:
32 divided by 4 is 8
32 : 4 = 8 : 1
Jeremy has 3/4 pound of fish food if fish food is used to feed 24 fish how much food is there for each fish
Answer:
There is 1/32 pound food for each fish
Step-by-step explanation:
24 fish = 3/4 pound food
1 fish = (3/4)/24 pound food = 1/32 pound food
(Problem solved by using unitary method)
A die is rolled. Find the probability of the given event. Write your answers as whole numbers or reduced fractions. (a) The number showing is a 6 P(6)= (b) The number showing is an even number P(even) = (c) The number showing is greater than 5 P(greater than 5) =
Answer:lol
Step-by-step explanation:lol
(a) Find a vector function, r(t), that represents the curve of intersection of the two surfaces.
The cylinder
x2 + y2 = 25
and the surface
z = xy
r(t)=??
(b)Find a vector function, r(t), that represents the curve of intersection of the two surfaces.
The paraboloid
z = 7x2 + y2
and the parabolic cylinder
y = 2x2
r(t)=???
(a) A vector function, r(t), that represents the cylinder x^2+y^2 = 25 and the surface z = xy.
(b) A vector function, r(t), that represents the paraboloid z = 7x^2 + y^2 and the parabolic cylinder y = 2x^2 is r(t) = <t, 2t^2, 7t^2+4t^4.
(a) We have to find a vector function, r(t), that represents the curve of intersection of the two surfaces.
The cylinder x^2+y^2 = 25 and the surface z = xy.
The given cylinder is x^2+y^2 = 25
We can write it as
x^2+y^2 = 5^2………………(1)
The given surface is z = xy………………..(2)
Let x = 5cost and y = 5sint, then x and y satisfy the equation 1
Now putting x=5cost and y=5sint in equation (2), we get
z = (5cost)(5sint)
z = 25sintcost
Now x=5cost, y=5sint and z=25sintcost
Hence, r(t) = <x(t),y(t),z(t)>
r(t) = <5cost, 5sint, 25sintcost>
(b) We have to find a vector function, r(t), that represents the curve of intersection of the two surfaces.
The paraboloid z = 7x^2 + y^2 and the parabolic cylinder y = 2x^2.
The given paraboloid z = 7x^2+y^2………………………(1)
The parabolic cylinder y = 2x^2……………………….(2)
Let x = t, then from equation 2
y = 2t^2
Now putting the value of x and y in equation (1)
z = 7t^2+(2t^2)^2
z = 7t^2+4t^4
Now x=t, y=2t^2 and z=7t^2+4t^4
Hence, r(t) = <x(t),y(t),z(t)>
r(t) = <t, 2t^2, 7t^2+4t^4>
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Write a function that matches the graph.
A function that will match the given graph is; f(x) = 2(x - 5)²(x + 2)
How to interpret Polynomial Graphs?In polynomial graphs, we know that the roots are the points where the graph crosses or touches the x-axis.
In this given graph, we see that the graph touches the x-axis at x = 5 and then crosses the x-axis at x = -2. Thus;
The roots of the equation are -2 and 5.
However, when the graph changes sign at a point on the x-axis, it means that point represents double roots. Thus, we will now have three roots which are -2, 5 and 5. The factors of the polynomial are;
(x + 2), (x - 5)²
Now, we see that when x = 0, y = 100 from the graph. Thus;
a(0 + 2)(0 - 5)² = 100
a(50) = 100
a = 100/50
a = 2
Thus, the equation would now be;
f(x) = 2(x - 5)²(x + 2)
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Find the value of k given that both points lie on a line passing through the origin. (−12,−2) and (k, 8)
Using the slope, the required value of k is 48.
What is the slope?In mathematics, the steeper direction of a line is overall processed by the slope or gradient of the line.
A line's slope can be used to determine how steep it is.
Theoretically, the gradient is calculated as "rise over run" (change in y divided by change in x).
Given is a line that crosses through the origin:
y = mx
Let,s take x = -12 and y = -2.
y = mx
-2 = -12m
m = -2/12
m = -1/6
m is the slope.
The equation now:
y = mx
y = -1/6x
Now, insert:
x = k and y = 8
y = mx
8 = -1/6x
x = 6*8
x = 48
Therefore, using the slope, the required value of k is 48.
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