Answers
Answer:
Step-by-step explanation:
The general equation for an ellipse with center (h, k) is:
[tex]\boxed{\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1}[/tex]
If a > b, the ellipse is horizontal.
If b > a, the ellipse is vertical.
Given equation:
[tex]\dfrac{(x-5)^2}{4}+\dfrac{(y+5)^2}{9}=1[/tex]
As b > a, the ellipse is vertical. Therefore:
b is the major radius and 2b is the major axis.a is the minor radius and 2a is the minor axis.Vertices = (h, k±b)Co-vertices = (h±a, k)Foci = (h, k±c) where c² = b² - a²Comparing the given equation with the standard form, we get:
[tex]h = 5[/tex][tex]k = -5[/tex][tex]a^2=4 \implies a=2[/tex][tex]b^2=9 \implies b=3[/tex]Therefore:
[tex]\textsf{Center}= (5, -5)[/tex][tex]\textsf{Major axis}=2 \cdot 3 = 6[/tex][tex]\textsf{Minor axis}=2 \cdot 2 = 4[/tex][tex]\textsf{Vertices:} \;\;(h, k \pm b)=(5,-5 \pm 3)=(5,-8)\;\;\textsf{and}\;\;(5,-2)[/tex][tex]\textsf{Co-vertices:}\;\;(h \pm a, k)=(5 \pm 2, -5)=(3, -5)\;\; \textsf{and}\;\;(7, -5)[/tex]To graph the ellipse:
Plot the center at (5, -5).Plot the vertices at (5, -8) and (5, -2). The distance between them is the major axis.Plot the co-vertices at (3, -5) and (7, -5). The distance between them is the minor axis.
Related Questions
A merchant could sell one model of digital cameras at list price and receive $432 for all of them. If he had three more cameras, he could sell each one for $12 less and still receive $432. Find the list price of each camera.
Answers
The list price of each camera is approximately $26.59.
Let's assume the list price of each camera is represented by 'x'.
According to the given information, the merchant sold a certain number of cameras at the list price and received a total of $432. This can be expressed as:
Number of cameras * List price = Total revenue
x * Number of cameras = $432
Now, if the merchant had three more cameras and sold each one for $12 less, the new selling price would be (x - $12). The total revenue would still be $432. This can be expressed as:
(New number of cameras) * (New selling price) = Total revenue
(x + 3) * (x - $12) = $432
Expanding the equation:
x^2 - $12x + 3x - $36 = $432
x^2 - 9x - $468 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring this equation may not yield integer solutions, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -9, and c = -468. Plugging these values into the quadratic formula:
x = (-(-9) ± √((-9)^2 - 4 * 1 * (-468))) / (2 * 1)
x = (9 ± √(81 + 1872)) / 2
x = (9 ± √(1953)) / 2
Calculating the square root of 1953 gives us approximately 44.17. Therefore, the two possible values for x are:
x1 = (9 + 44.17) / 2 ≈ 26.59
x2 = (9 - 44.17) / 2 ≈ -17.59
Since the price of a camera cannot be negative, we discard x2. Therefore, the list price of each camera is approximately $26.59.
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given the line y= -6x +5 what is the line nslope and the y - intercept
Answers
According to slope intersept form : y = mx+b
m is slope and b is y intersept
so y = -6x+5
slope(m) = -6
y intersept = 5
HOPE IT HELPS
PLEASE GIVE BRAINLIEST
PLEASE I NEED HELP I DONT UNDERSTAND THIS
Answers
Answer:
- 15pr
Step-by-step explanation:
- 5p(3r) ← means multiply 3r by - 5p
= - 5p × 3r
= - 5 × 3 × p × r
= - 15 × pr
= - 15pr
a square pyramid has a base with a side length of 3 feet and lateral with a height of 6 feet. what is the area of the pyramid. A.9 square feet B. 27 square feet C. 36 square feet D. 45 square feet
Answers
Answer:
D. 45 square feet
Step-by-step explanation:
To find the surface area of a square pyramid, we need to find the area of the base and the area of the four triangular lateral faces.
Finding the area of the baseFirst, let's find the area of the base (square).
[tex]\boxed{\begin{minipage}{7 cm}Base area = side length $\times$ side length\\ \\Base area = 3 $\times$ 3 \\ \\Base area = 9 square feet\end{minipage}}[/tex]
Finding the area of the triangular lateral facesNow, let's find the area of the triangular lateral faces.
Finding the area of one triangular lateral faceAs stated in the problem, the slant height of the triangle is 6 feet. We can find the area of one triangular lateral face:
[tex]\text{Triangle area = $\frac{1}{2}$ $\times$ base $\times$ height}\\\\\text{Triangle area = $\frac{1}{2}$ $\times$ 3 $\times$ 6} \\\\\text{Triangle area = 9 square feet}[/tex]
Finding the total lateral areaSince there are four triangular lateral faces, we need to multiply this area by 4:
[tex]\text{Total lateral area = 4 $\times$ 9}\\\\\text{Total lateral area = 36 square feet}[/tex]
Total surface areaFinally, we add the base area and the total lateral area to find the total surface area of the pyramid:
[tex]\text{Total surface area = base area + total lateral area}\\\\\text{Total surface area = 9 + 36}\\\\\text{Total surfacel area = 45 square feet}[/tex]
ConclusionSo, the area of the square pyramid is 45 square feet. Which is option D.
________________________________________________________
Answer:
D. 45 square feet
Step-by-step explanation:
You want the surface area of a square base pyramid with a side length of 3 feet and a slant height of 6 feet.
Area formulasThe area of a triangular face is given by ...
A = 1/2bh . . . . . base b and height h
The area of the square base is given by ...
A = s²
The total surface area of the pyramid is the area of the base plus the areas of the 4 triangular faces:
total area = s² +4(1/2bh) . . . . where b = s
This can be simplified to ...
total area = s(s +2h) . . . . . area of a square pyramid with slant height h
ApplicationFor the pyramid with a square base 3 ft on a side, and a slant height of 6 ft, the total area is ...
total area = (3 ft)(3 ft + 2·6 ft) = (3 ft)(15 ft) = 45 ft²
The area of the pyramid is 45 square feet.
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please help me !!!!!!!
Answers
Answer:
F(5)= -1
if 5 is the value on the x axis it corresponds to -1 on the y axis
Pls help I need this answer
Answers
The expression is completed as (x-4)(x -7)
How to determine the valueFrom the information given, we have that the polynomial is given as;
x² - 11x + 28
Using the factorization method, we have;
First, find the product of the coefficient of x squared and the constant value
Then, we have;
1(28) = 28
Now, find the pair factors of the product that adds up to -11, we have;
-7x and -4x
Substitute the values, we have;
x² - 7x - 4x + 28
Group in pairs, we get;
x(x-7) - 4(x - 7)
Then, we have the expressions as;
(x-4)(x - 7)
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Determine the surface area and volume
Answers
Answer:
Volume : 300
Surface Area : 280
Step-by-step explanation:
Volume : 6*5*10
Surface Area : 50+50+30+30+60+60
Kelly started with 2 pennies in her penny jar. She puts 2 more pennies in her penny jar every day. How many pennies will she have on Day 10
Answers
On Day 10, Kelly will have a total of 20 pennies in her penny jar.
To determine how many pennies Kelly will have on Day 10, we need to consider the progression of pennies added to her jar each day.
On Day 1, Kelly starts with 2 pennies. On Day 2, she adds 2 more pennies, resulting in a total of 2 + 2 = 4 pennies. This pattern continues, with 2 more pennies being added each day.
To find the number of pennies on Day 10, we can observe that the number of pennies on any given day can be calculated using the formula:
Number of pennies = Initial number of pennies + (Number of days - 1) * Number of pennies added per day
Using the provided information, we can substitute the values into the formula:
Number of pennies on Day 10 = 2 + (10 - 1) * 2
= 2 + 9 * 2
= 2 + 18
= 20
Therefore, on Day 10, Kelly will have a total of 20 pennies in her penny jar.
To summarize, starting with 2 pennies and adding 2 more pennies each day, Kelly will have a total of 20 pennies in her penny jar on Day 10.
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Consider the following pair of points.
(8,−8)
and (−5,−1)
Step 2 of 2 : Determine the midpoint of the line segment joining the pair of points.
Answers
The midpoint of the line segment joining the pair of points (8, -8) and (-5, -1) is (1.5, -4.5).
To find the midpoint, we need to take the average of the x-coordinates and the average of the y-coordinates of the two given points.
For the x-coordinate: (8 + (-5))/2 = 3/2 = 1.5
For the y-coordinate: (-8 + (-1))/2 = -9/2 = -4.5
Thus, the midpoint of the line segment is (1.5, -4.5).
In more detail, the midpoint formula is derived by averaging the x-coordinates and y-coordinates separately. For a line segment with endpoints (x1, y1) and (x2, y2), the midpoint (xm, ym) is given by:
xm = (x1 + x2)/2
ym = (y1 + y2)/2
In this case, the x-coordinates are 8 and -5, and their average is (8 + (-5))/2 = 1.5. The y-coordinates are -8 and -1, and their average is (-8 + (-1))/2 = -9/2 = -4.5.
Therefore, the midpoint of the line segment joining the two given points is (1.5, -4.5).
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What function has the same range as f(x) = -2 x - 3 + 8
Answers
Answer:
Any equation that has the power of x at 1
Step-by-step explanation:
Since the function f(x) = -2x -3 +8 has an infinite range, so any other equation that only contains x^1 would work
What is the solution to the system of equations graphed below?
A. (0, -1)
OB. (0,1)
OC. (-1,0)
D. (1,0)
S
PRE
y=-x-1
y=x+1
y=x+1
y=-x-1
Answers
The point of intersection is (-1, 0). Therefore, the correct answer is C. (-1, 0).
To find the solution to the system of equations graphed below, we need to identify the point where the two lines intersect.
The equations of the lines are:
y = -x - 1
y = x + 1
To find the point of intersection, we can set the two equations equal to each other and solve for x:
-x - 1 = x + 1
Adding x to both sides:
-1 = 2x + 1
Subtracting 1 from both sides:
-2 = 2x
Dividing both sides by 2:
x = -1
Now that we have the x-coordinate, we can substitute this value back into either equation to find the y-coordinate. Let's use the second equation:
y = x + 1
y = -1 + 1
y = 0
Therefore, the point of intersection is (-1, 0).
Among the given options, the closest point to (-1, 0) is option C: (-1, 0).
Therefore, the correct answer is C. (-1, 0).
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Find the regression line associated with the set of points. (Round all coefficients to four decimal places.) HINT [See Example 2.] (4, 6), (6, 10), (10, 14), (12, 2) y(x) =
Answers
The y-intercept, b, can be calculated as:
b = (Σy - mΣx) / n
To find the regression line associated with the set of points (4, 6), (6, 10), (10, 14), and (12, 2), we can use the least squares method. The regression line represents the best-fit line that minimizes the sum of the squared differences between the observed y-values and the predicted y-values on the line.
The equation for the regression line, y(x), can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
Using the given points, we can calculate the slope, m, and the y-intercept, b, to obtain the equation of the regression line.
The slope, m, is calculated as:
m = (nΣxy - ΣxΣy) / (nΣ[tex]x^2[/tex] - (Σ[tex]x)^2[/tex])
where n is the number of points, Σxy is the sum of the product of x and y values, Σx is the sum of the x-values, and Σy is the sum of the y-values.
Similarly, the y-intercept, b, can be calculated as:
b = (Σy - mΣx) / n
By substituting the given values into the formulas and performing the calculations, the equation for the regression line can be obtained.
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17 students are present in a class. in how many ways ,they can be made to stand in 2 circles of 8 and 9 students
Answers
Answer:
24310
Step-by-step explanation:
17 choose 8= 17 choose 9
17 choose 8= 17*16*15*14*13*12*11*10/8!= 24310
Annette Creighton opened Creighton consulting she rented a small office and paaid a part time worker to ensure to answer the phone and make deliveries
Answers
Annette Creighton opened Creighton Consulting and rented a small office space. She also hired a part-time worker to handle phone calls and make deliveries.
1. Annette Creighton decided to start her own consulting firm and named it Creighton Consulting.
2. As part of setting up the business, she began by renting a small office space. This would serve as the headquarters for her consulting operations.
3. To ensure smooth communication and timely deliveries, Annette realized the need for assistance. She hired a part-time worker to handle incoming phone calls and make necessary deliveries on behalf of the company.
4. The part-time worker would be responsible for answering the phone and addressing client inquiries or forwarding them to Annette if necessary. This step was crucial to maintaining good customer service and professionalism.
5. Additionally, the worker would also handle deliveries, ensuring that any documents, packages, or materials were transported promptly to clients or other relevant destinations.
6. By hiring a part-time worker, Annette aimed to streamline her operations and focus on core consulting tasks, such as client meetings, project management, and providing expert advice to clients.
7. The worker's role would involve effective communication skills, organization, and attention to detail to represent Creighton Consulting in a professional manner.
8. Annette understood that investing in administrative support would free up her time and allow the business to operate more efficiently, ultimately benefiting both her and her clients.
9. With the office space secured and the part-time worker in place, Annette was ready to embark on her consulting journey, confident in her ability to handle client inquiries and maintain smooth business operations.
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15
25
15
23
15
23
17
21
21
19
15
a.) The standard deviation is(round to two decimal places)
b.) The variance is(round to one decimal place)
c.) The range is
Answers
The base of a triangle is 21 inches and the height is 12 inches. Which of these expressions correctly shows how to calculate the area of a triangle?
A. (21 × 12) × 2
B. (21 + 12) ÷ 2
C. (21 + 12) × 2
D. (21 × 12) ÷ 2
Answers
The correct expression to calculate the area of a triangle with a base of 21 inches and height of 12 inches is (21 × 12) ÷ 2.
Explanation:The subject of your question is Mathematics, specifically dealing with the topic of how to calculate the area of a triangle. The formula to calculate the area of a triangle is 1/2 multiplied by the base multiplied by the height. So, in your question where the base of the triangle is 21 inches and the height is 12 inches, the correct choice would be D. (21 × 12) ÷ 2 which applies the formula correctly.
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are statistical questions?
Which subjects do the students What is the number of students
in my class like?
in my class?
How many servings of fruit did
I eat each day this month?
What is my height?
What is my favorite color?
What is the highest temperature
of each month this year?
Reset
Next
What is the height of each
student in my class?
What is my mother's favorite
fruit?
How many students from each
school in this city love football?
Answers
The statistical questions for this problem are given as follows:
Which subjects do the students like?How many servings of fruit did I eat each day this month?What is the highest temperature of each month this year?What is the height of each student in my class?How many students from each school in this city love football?What is an statistical question?A question is classified as statistical if it can receive answers of data that vary, that is, questions that do not have an exact answer.
When the answer is exact, it must be composed by a set of data, such as the number of students that like football in each school, the number will vary for each school.
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Two vacationing families leave New York at the same time. They take 20 and 6 days, respectively, to reach their destination and return to New York. The vacationing families each take continuous trips to and from New York. How many days will pass before the two vacationing families leave New York on the same day again?
Answers
Answer:
Step-by-step explanation: evagline has 3/5 of box of nuts.she uses it to fill 6 bowls
a sector has radius 10 mm and area 40/3 pie mm^2. what is the measure, in degrees, of the central angle of the sector?
Answers
The central angle of the sector is 540°.
The area of the sector formula is A = (1/2) r² θ Where, A = Area of sector r = Radius of sectorθ = Central angle of sector Given that the radius of the sector is 10mm and area is 40/3 pie mm²
To find the measure of central angle θ, plug the given values in the formula as shown; A = (1/2) r² θ40/3 pie = (1/2)(10)² θ40/3 pie = (1/2)100 θ40/3 pie = 50θ (multiply by 3/40)θ = (3/40) × 40πθ = 3π.
So, the measure, in degrees, of the central angle of the sector is;θ = (180/π) × 3πθ = 540°Therefore, the central angle of the sector is 540°.
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Find the volume of this cylinder give your answer to one decimal place 11height 14 Length
Answers
To find the volume of the given cylinder, we need to use the formula of volume of a cylinder which is given by V = πr²h, where V is the volume of the cylinder, r is the radius of the cylinder and h is the height of the cylinder. So, the volume is 1078.1 cubic units.
Given, Length = 14 units, Height = 11 units. Now, we know that the formula for the volume of a cylinder is given as; V = πr²h.
Here, the radius of the cylinder is half the length of the cylinder, which is; Radius = Length/2
Radius = 14/2
Radius = 7 units
Substituting the value of the radius and height, we have; V = π(7)²(11)
V = 1078.1 cubic units. Therefore, the volume of the cylinder is 1078.1 cubic units. Rounded off to one decimal place, the volume is 1078.1 cubic units.
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1 Express 12 + 5i in polar form (i.e in form of \[z=r\cos\theta + i\sin\theta\]
A. [13(\cos 22.6 - i\sin 22.6)\]
B [13(\cos 22.6+i\sin 22.6)\]
C. [13(\cos 23.5 - i\sin 23.5)\]
D. [13(\cos 23.6 - i\sin 23.6
Answers
The correct option is A. [13(cos 22.6 - isin 22.6)] in which the modulus is 13 and the argument is 22.6 degrees.
Given the complex number z = 12 + 5i. We have to express this complex number in the polar form which is\[z=r\cos\theta + i\sin\theta\]where r is the modulus and θ is the argument of the complex number.
The modulus of the complex number is given by,|z|=√(12²+5²)=√(144+25)=√169=13
Therefore, the modulus of the complex number is 13.
Now, we need to find the argument of the complex number, which is given byθ=tan⁻¹(b/a)Where a and b are the real and imaginary parts of the complex number z.θ=tan⁻¹(5/12)So, θ=22.6 degrees. (approximate value)
Thus, the complex number z = 12 + 5i can be expressed as\[z=13\cos(22.6^{\circ}) + i\sin(22.6^{\circ})
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Solve for the measure of the indicated arc.
O 127°
165°
164°
157°
52 °
K
53°
L
M
C
Answers
Answer:
? = 157°
Step-by-step explanation:
the measure of the secant- secant angle KLM is half the difference of the intercepted arcs , that is
[tex]\frac{1}{2}[/tex] (CJ - KM) = ∠ KLM , that is
[tex]\frac{1}{2}[/tex] (? - 53) = 52° ( multiply both sides by 2 to clear the fraction )
? - 53° = 104° ( add 53° to both sides )
? = 157°
Please awnser asap I will brainlist
Answers
The system has exactly one solution. The solution is (13, 8)
How to calculate the solution to the system of equationsFrom the question, we have the following parameters that can be used in our computation:
Country A: -x + 20y = 147
Country B: -x + 10y = 67
Country C: y = 8
So, we have
-x + 20y = 147
-x + 10y = 67
y = 8
Substitute 8 for y in the first and second equations
So, we have
-x + 20 * 8 = 147
-x + 10 * 8 = 67
Evaluate the products
-x + 160 = 147
-x + 80 = 67
So, we have
x = 160 - 147
x = 80 - 67
Evaluate
x = 13
x = 13
Hence, the solution to the system of equations is (13, 8)
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(−x² − 1) ÷ (x + 1)
Help
Answers
Please please help me!!
What can you conclude about the population
density from the table provided?
Region A
Region B
Region C
Region D
Population
20,178
1,200
13,475
6,980
Area (km²)
521
451
395
426
Answers
Answer:
Region A: 20,178 people/521 km²
= 38.7 people/km²
Region B: 1,200 people/451 km²
= 2.7 people/km²
Region C: 13,475 people/395 km²
= 34.1 people/km²
Region D: 6,980 people/426 km²
= 16.4 people/km²
The regions, in order from the most densely populated to the least densely populated: A, C, D, B
(24)³-(13)³-(11)³ Without actually calculating the cubes, evaluate.
Answers
By applying the algebraic identity, the expression (24)³ - (13)³ - (11)³ can be simplified as (11)(1057) - (11)³ = 10396, without actually calculating the cubes.
How to Evaluate the expression without Calculating the Cubes?The expression (24)³ - (13)³ - (11)³ can be evaluated without calculating the cubes by applying the concept of algebraic identities.
By using the identity (a³ - b³) = (a - b)(a² + ab + b²), we can rewrite the expression as (24 - 13)(24² + 24*13 + 13²) - (11)³.
Simplifying further, we have (11)(576 + 312 + 169) - (11)³.
Combining like terms, the expression evaluates to (11)(1057) - (11)³.
Finally, we can simplify to obtain the result of 11 multiplied by 1057 minus 11 cubed, without actually calculating the cubes. The simplified expression (24)³ - (13)³ - (11)³ evaluates to 10396.
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Tell whether the information in the diagram allows you to conclude that c is on the perpendicular bisector of an
Answers
11. Yes, the information provided can be used to conclude that C is on the perpendicular bisector of AB because CE bisects AB.
12. Yes, the information provided can be used to conclude that C is on the perpendicular bisector of AB because C is equidistant from AB.
What is a perpendicular bisector?In Mathematics and Geometry, a perpendicular bisector can be used for bisecting or dividing a line segment exactly into two (2) equal halves, in order to form a right angle with a magnitude of 90° at the point of intersection.
Question 11.
By critically observing the geometric shape, we can logically deduce that the information provided can be used to conclude that C is on the perpendicular bisector of AB because CE bisects AB:
AC ≅ BC
CD ≅ CD
AE ⊥ EC
Question 12.
By critically observing the geometric shape, we can logically deduce that the information provided can be used to conclude that C is on the perpendicular bisector because C is equidistant from line segment AB.
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The Vilas County News earns a profit of $20 per year for each of its 3,000 subscribers. Management projects that the profit per subscriber would increase by 1¢ for each additional subscriber over the current 3,000. How many subscribers are needed to bring a total profit of $123,525?
Answers
In order to achieve a profit of $123,525, the Vilas County News would require a subscriber base of 6,355,500 individuals.
Let's break down the problem step by step to find the number of subscribers needed to bring a total profit of $123,525.
First, we know that the Vilas County News earns a profit of $20 per year for each of its 3,000 subscribers. This means that the current profit from the 3,000 subscribers is $20 x 3,000 = $60,000.
Management projects that the profit per subscriber would increase by 1¢ for each additional subscriber over the current 3,000. This means that for every additional subscriber, the profit increases by $0.01. Therefore, we need to find how many additional subscribers are required to reach a total profit of $123,525 - $60,000 = $63,525.
To find the number of additional subscribers needed, we divide the additional profit required by the increase in profit per subscriber: $63,525 / $0.01 = 6,352,500.
However, we need to remember that this number represents the total number of additional subscribers needed, not the final total number of subscribers. To find the final total number of subscribers, we add the additional subscribers to the current number of subscribers: 6,352,500 + 3,000 = 6,355,500.
Therefore, to bring a total profit of $123,525, the Vilas County News would need a total of 6,355,500 subscribers.
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Let V=W+W* be a vector space, being the direct product of the (finite dimensional) vector space W and its dual space W*. Now, let us define a bilinearform B: VxV -> R by
<(a,p), (b,q)> := q(a) + p(b).
Now let us suppose we have both e_1, …, e_n Basis of W and e*_1,….,e*_n Basis of W*.
What is the matrix of this bilinear form?
(I know how these matrices usually look like, but the inner product makes me very confused about the layout of this matrix).
Answers
M[i,j] := B((e_i, e*_i), (e_j, e*_j))
For simplicity, let's assume that dim(W) = n and let's write the basis vectors as row vectors. Then we have
e_i = [0 ... 1 ... 0]
|__ i-th position __|
and
e*_i(w) = [x_1 ... x_n] [0 ... 1 ... 0]^T
|__ i-th position __|
where x_i is the i-th coordinate of the dual basis vector e*_i and the superscript T denotes matrix transposition.
Using this notation, we can compute the matrix entries as follows:
M[i,j] = B((e_i, e*_i), (e_j, e*_j))
= (e*_i(e_j) + e*_j(e_i))(e*_i(e_j) + e*_j(e_i))^T
= (e*_i(e_j) + e*_j(e_i))[e*_i(e_j) e*_j(e_i)]
= e*_i(e_j)e*_i^T(e_j) + e*_j(e_i)e*_j^T(e_i)
= [e*_1(e_j) e*_2(e_j) ... e*_n(e_j)][e*_1(e_i) e*_2(e_i) ... e*_n(e_i)]^T
+ [e*_1(e_i) e*_2(e_i) ... e*_n(e_i)][e*_1(e_j) e*_2(e_j) ... e*_n(e_j)]^T
where we have used the definition of B and the fact that e*_i^T(e_j) = delta_ij (Kronecker delta).
Therefore, the matrix M has entries given by
M[i,j] = e*_i(e_j)e*_i^T(e_j) + e*_j(e_i)e*_j^T(e_i)
This gives us the general form of the matrix, where the (i,j)-entry is determined by the values of the dual basis vectors on the corresponding basis vectors. However, without explicit knowledge of the basis vectors and the dual basis vectors, it is not possible to write down the matrix in a more explicit form.
If the vertical height of the ramp is 4 feet, how long must the ramp (x) be?
Show all of your work. Round your answer to the nearest foot. (Picture is
not drawn to scale.
Answers
To find h, we can use the tangent function:
tan(5 degrees) = 4 / h
Rearranging the equation to solve for h:
h = 4 / tan(5 degrees)
Using a calculator, we find that tan(5 degrees) ≈ 0.0874887. Plugging in this value:
h = 4 / 0.0874887 ≈ 45.74 feet
Therefore, the length of the ramp (x) is approximately 45.74 feet when the height is 4 feet and the angle is 5 degrees.