Using coefficients ak, the following product may be extended into a power series: the expression is provided in the attached file. For each of the [tex]k 2 N[/tex]phrases, determine the coefficients ak before them. The formula [tex]ak = (-1)k(k+1)/2[/tex] yields the coefficients ak.
To get the coefficients ak, we may first simplify the above formula by factoring out a -x and rearranging terms. This results in the equation: [tex](1-x)/(1+x)2 = -x/(1+x) - x2/(1+x)2.[/tex]
Now, each term in the statement may be expanded into a power series using the formula for the geometric series. This results in: Both[tex]-x/(1+x) and -x2/(1+x)2[/tex] are equal to[tex]-x + x + x + 2 + x + 3 +...[/tex]
By combining like terms and adding these two power series, we can determine that the coefficient in front of [tex]xk is (-1)k(k+1)/2.[/tex] Hence,[tex]ak = (-1)k(k+1)/2[/tex] is the formula for the coefficients ak.
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1. The line segment AB has endpoints A(-5, 3) and B(-1,-5). Find the point that partitions the line segment in
a ratio of 1:3
Answer:
To find the point that partitions the line segment AB in a ratio of 1:3, we can use the following formula:
P = (3B + 1A) / 4
where P is the point that partitions the line segment in a ratio of 1:3, A and B are the endpoints of the line segment, and the coefficients 3 and 1 represent the ratio of the segment we are dividing.
Substituting the values, we get:
P = (3*(-1, -5) + 1*(-5, 3)) / 4
P = (-3, -7)
Therefore, the point that partitions the line segment AB in a ratio of 1:3 is (-3, -7).
Step-by-step explanation:
A square field has a side length of 6x10³ meters. Which of the following is its area in square meter
(1) 6x106
(3) 36x106
(2) 36×10⁹
(4) 6x10⁹
Answer:
36 × 10^6 m²
Step-by-step explanation:
Given the side length of a square = 6 × 10³m,
To solve for the area of a square, use the following formula:
A = S² where:
S = side of the square
Substitute the given value for the side into the formula:
A = S²
A = (6 × 10³)²
A = 36000000 or 36 × 10^6 m²
NOTE:
6 × 10³ is also the same as 6 × 1000 = 6000,
(6 × 10³)² is essentially 6,000² = 36,000,000
Therefore, its area in square meters is 36 × 10^6
please help!!
4.
Two bikers meet at a park. Biker A needs to stop at the store that is 12 miles east of the park. Biker B heads southeast at a 61° angle at the same time for 24 miles. Once biker A leaves the store he heads southwest at an angle of 89° for 21 miles. Do NOT use the law of cosines, use your knowledge from the content of this course.
a. Use your knowledge of triangles to figure out if the two bikers will be able to meet up if each biker travels the distance given.
b. If they do not meet up, how much farther would one of the bikers have to travel to meet the other?
c. What is the measure of the angle between the bikers?
d. What is the relationship between the measure of the angles and the paths the bikers took?
e. Classify the triangle the paths created.
f. How many miles did they travel together?
a) Biker A follows the hypotenuse of the triangle on a straight path.
It is probable that the bikers will meet at the vertex located at the base of the triangle.
How to solve:The bikers have created a triangle with sides measuring 12, 21, and 24 miles and angles measuring 61, 89, and 30 degrees, respectively.
a) Biker A follows the hypotenuse of the triangle on a straight path.
It is probable that the bikers will meet at the vertex located at the base of the triangle.
They cover almost equal distances from their starting points:
24 miles ≈ √12²+21² miles
b) They encounter each other.
c) The angle at one vertex measures 30 degrees.
d) e) As shown in the attached picture, it is an almost right triangle.
f) Together, they cover a total distance of 57 miles (12+21+24).
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calculator may be used to determine the final numeric value, but show all steps in solving without a calculator up to the final calculation. the surface area a and volume v of a spherical balloon are related by the equationA³ - 36πV² where A is in square inches and Vis in cubic inches. If a balloon is being inflated with gas at the rate of 18 cubic inches per second, find the rate at which the surface area of the balloon is increasing at the instant the area is 153.24 square inches and the volume is 178.37 cubic inches.
Answer:
10.309 in²/s
Step-by-step explanation:
Given A³ = 36πV² and V' = 18 in³/s, you want to know A' when A=153.24 in² and V=178.37 in³.
DifferentiationUsing implicit differentiation, we have ...
3A²·A' = 36π·2V·V'
A' = (36π·2)/3·V/A²·V' = 24πV/A²·V'
A' = 24π·(178.37 in²/(153.24 in²)²·18 in³/s
A' ≈ 10.309 in²/s
The surface area is increasing at about 10.309 square inches per second.
__
Additional comment
There are at least a couple of ways a calculator can be used to find the rate of change. The first attachment shows evaluation of the expression we derived above. The second attachment shows the rate of change when the area is expressed as a function of the volume.
The result rounded to 5 significant figures is the same for both approaches.
can anyone help me please???
thank you xxxx
Using the property of a kite: the angles formed by two unequal sides of a kite are equal. The value of ∠BCD∠BCD is
∠BCD = ∠BAD ⇒ ∠BCD = 106°
Simplify 4 triangles to 16 squares
The expression 4 triangles to 16 squares when simplified is 1 triangle to 4 squares
How to simplify the expressionGiven that
4 triangles to 16 squares
When expressed as ratio, we have
Triangle : Square = 4 : 16
To simplify the ratio Triangle : Square = 4 : 16, we can divide both the numerator and denominator by their greatest common factor, which is 4.
So, we have
Triangle : Square = 4 : 16
Divide both sides by 4:
Triangle/4 : Square/4 = 1 : 4
So the simplified ratio is 1 : 4, which means for every one triangle, there are four squares.
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For triangles ABC and DEF, ∠A ≅ ∠D and B ≅ ∠E. Based on this information, which statement is a reasonable conclusion?
a. ∠C ≅ ∠D because they are corresponding angles of congruent triangles.
b. CA ≅ FD because they are corresponding parts of congruent triangles.
c. ∠C ≅ ∠F because they are corresponding angles of similar triangles.
d. AB ≅ DE because they are corresponding parts of similar triangles.
the triangles are similar, corresponding parts of the triangles are equal in measure. Thus, it is reasonable to conclude that [tex]AB ≅ DE.[/tex]
It is reasonable to conclude that [tex]AB ≅ DE[/tex]because triangles ABC and DEF are similar.
This means that corresponding parts of the two triangles are equal in measure. Specifically, ∠A and ∠D are equal in measure, as are ∠B and ∠E.
Therefore, the corresponding sides AB and DE are equal in measure.
A way to show that the two triangles are similar is by using the AA Similarity Postulate.
This postulate states that if two angles of one triangle are equal in measure to two angles of a second triangle, then the two triangles are similar. In this case, [tex]∠A ≅ ∠D and B ≅ ∠E[/tex], which means the two triangles are similar.
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Jen’s assignment is to read at least 85 pages of a novel. Jen has read 31 pages. How many pages p does Jen have left to read? Write an inequality that represents this situation. Then solve the inequality
Jen has 54 pages left to read to meet her assignment requirement.
The inequality that represents this situation is p ≥ 85 - 31
To find how many pages Jen has left to read, we can subtract the number of pages she has already read from the minimum number of pages she needs to read.
The minimum number of pages Jen needs to read is 85, and she has already read 31 pages. So, the number of pages she has left to read, p, can be found by:
p = 85 - 31
p = 54
Therefore, Jen has 54 pages left to read.
To represent this situation with an inequality, we can use:
p ≥ 85 - 31
This inequality states that the number of pages Jen still needs to read, p, must be greater than or equal to the difference between the minimum number of pages she needs to read (85) and the number of pages she has already read (31).
Solving for p:
p ≥ 85 - 31
p ≥ 54
This means that Jen must read at least 54 more pages to meet her assignment requirement.
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A system of equations is graphed on the coordinate plane.
A student concludes that the solution of the system is (-0.5, 1.5).
Is this correct? Justify your response.
The correctness of the student's solution, we need to have the equations of the system.
A system of equations is graphed on the coordinate plane. A student concludes that the solution of the system is (-0.5, 1.5). Is this correct? Justify your response.To conclude that a system of equations has a solution in the coordinate plane, a set of ordered pairs (x, y) should satisfy both equations in the system of equations. That is, the system of equations should have a point (x, y) that is a solution of both equations.Only by testing the solution in the given system of equations can we know if the student's conclusion is correct. If the solution is satisfied by the system of equations, then the answer is true. Otherwise, it is false. However, since no system of equations is provided in the question, we cannot test the student's solution.To justify the correctness of the student's solution, we need to have the equations of the system.
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The length of a rectangular room is 9 feet longer than twice the width. If the room's perimeter is 150 feet, what are the room's dimensions?
Answer:
Length = 53 feet
Width = 22 feet
Step-by-step explanation:
Perimeter = 2(length + width)
Then:
a = 2w + 9 Ec. 1
150 = 2(a + w) Ec. 2
a = length
w = width
From Eq. 1:
a - 9 = 2w Eq. 3
From Eq. 2:
150 = 2*a + 2*w
150 = 2a + 2w
150 - 2a = 2w Eq. 4
Equalizing Eq. 3 and Eq. 4
a - 9 = 150 - 2a
a + 2a = 150 + 9
3a = 159
a = 159/3
a = 53
From Eq. 1:
a = 2w + 9
53 = 2w + 9
53 - 9 = 2w
44 = 2w
44/2 = w
w = 22
Check:
From Eq. 2
150 = 2(a+w)
150 = 2(53+22)
150 = 2*75
3. Find the distance from A to B. A(-2,2) B (4,-2)
The distance from points A(-2, 2) to B (4, -2) is equal to √52 units.
How to calculate the distance between the two points?Mathematically, the distance between two (2) points that are on a coordinate plane can be calculated by using the following mathematical expression:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Where:
x and y represents the data points (coordinates) on a cartesian coordinate.
Substituting the given points into the distance formula, we have the following;
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Distance = √[(4 - (-2))² + (-2 - 2)²]
Distance = √[(6)² + (-4)²]
Distance = √(36 + 16)
Distance = √52 units.
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A price was first decreased by 12%, then it was decreased again by an additional 5%. What is the percent of the total decrease?
The answer is not 17%.
Please help!!!!
10 PTS
Answer:
83.6%
Step-by-step explanation:
Let's say the original price is x
x decreased by 12% means 88% of x is left.
0.88x decreased by 5% means 95% of 0.88x is left.
This means the answer is: 0.88x * 0.95 = 0.836
The percent of the total decrease is 83.6%
Hope this helps :)
Have a great day!
Estimating the within-group variance. Refer to the previous exercise. Here are the cell standard deviations and sample sizes for cooking enjoyment: Find the pooled estimate of the standard deviation for these data. Use the rule for examining standard deviations in ANOVA from Chapter 12 (page 560) to determine if it is reasonable to use a pooled standard deviation for the analysis of these data.
In the following question, among the given options, the statement is said to be, The pooled estimate of the standard deviation for the data given is √(54.14^2/10 + 24.26^2/10) = 22.74.
According to the rule for examining standard deviations in ANOVA from Chapter 12 (page 560), the within-group standard deviation should be no more than twice the size of the between-group standard deviation. In this case, the between-group standard deviation is 44.85 and the within-group standard deviation (22.74) is less than twice the size of the between-group standard deviation, so it is reasonable to use a pooled standard deviation for the analysis of these data.
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give three examples of contracts you are currently a part of or have been a part of in the past. identify whether they are unilateral or bilateral; express or implied; executed or executory.
The three examples of contracts are:
Employment ContractRental AgreementPurchase AgreementContracts are legal agreements between two or more parties that involve the exchange of goods, services, or money. They can be classified as unilateral or bilateral, express or implied, executed or executory.
Here are three examples of contracts that a person can be a part of:
Employment Contract: An employment contract is a bilateral, express contract between an employer and an employee. It defines the terms and conditions of employment, including salary, benefits, and job responsibilities. An employment contract is executed when both parties have agreed to the terms of the agreement and have signed the contract.Rental Agreement: A rental agreement is a unilateral or bilateral, express or implied, executory contract between a landlord and a tenant. It outlines the terms of the lease, such as the duration of the tenancy, rent, security deposit, and maintenance responsibilities. A rental agreement can be either oral or written. It is considered executed when the tenant moves in and starts paying rent.Purchase Agreement: A purchase agreement is a bilateral, express contract between a buyer and a seller. It outlines the terms of the sale, including the price, payment terms, delivery method, and warranty. A purchase agreement is executed when the buyer pays the agreed-upon amount and the seller delivers the product or service.To know more about the "contracts":https://brainly.com/question/5746834
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How to do matrix multiplication in MIPS?
To perform matrix multiplication in MIPS, we can use nested loops to iterate over the rows and columns of the matrices.
The outer loop iterates over the rows of the first matrix, while the inner loop iterates over the columns of the second matrix. We then perform the dot product of the corresponding row and column, which involves multiplying the elements and summing the products.
To perform multiplication efficiently, we can use MIPS registers to store intermediate values and avoid accessing memory unnecessarily. We can also use assembly instructions like "lw" and "sw" to load and store values from memory, and "add" and "mul" to perform arithmetic operations.
In summary, matrix multiplication in MIPS involves nested loops, efficient use of registers and assembly instructions, and arithmetic operations.
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Solve each inequality (show work)
Answer:
5 less than or equal to x
Step-by-step explanation:
make x subject
60 identical machines in a factory pack 150 crates of limes per day
between them.
a) Write the ratio of the number of machines to the number of crates
packed per day in the form 1: n.
b) How many crates of limes would 70 of these machines pack per day?
Give any decimals in your answers to 1 d.p.
a) To write the ratio of the number of machines to the number of crates packed per day in the form 1: n, we need to find the number of crates packed per day per machine. We can do this by dividing the total number of crates packed per day by the number of machines:
Number of crates packed per day per machine = 150 crates/day ÷ 60 machines = 2.5 crates/machine/day
Therefore, the ratio of the number of machines to the number of crates packed per day in the form 1: n is 1:2.5 or 2:5.
b) To find out how many crates of limes 70 of these machines would pack per day, we can use the ratio from part (a) to set up a proportion:
1 machine : 2.5 crates/day = 70 machines : x crates/day
Solving for x, we get:
x = (70 machines × 2.5 crates/day) / 1 machine = 175 crates/day
Therefore, 70 of these machines would pack 175 crates of limes per day.
Step-by-step explanation:
a) The ratio of the number of machines to the number of crates packed per day can be written as:
60 : 150
To simplify this ratio, we can divide both sides by 10:
6 : 15
Finally, we can divide both sides by 3 to get the ratio in the form 1 : n:
1 : 2.5
Therefore, the ratio of the number of machines to the number of crates packed per day is 1 : 2.5.
b) If 60 machines can pack 150 crates per day, then one machine can pack:
150/60 = 2.5 crates per day
So, 70 machines can pack:
70 × 2.5 = 175 crates per day
Therefore, 70 machines can pack 175 crates of limes per day.
Can someone please help me with these, I wasn’t able to attend school cause I have Covid and I don’t know what’s going on. Please please help me
Answer: The answer for the first question goes as follows: Which one(s) have both pairs of opposite sides that are parallel? Rectangle, Rhombus, and square. Which one(s) have both pairs of opposite sides that are congruent? Rectangle, square, and rhombus. Which one(s) have all four sides congruent? Square and rhombus.
The answers to the second question are:
Circumference = pi * diameter. So, the distance around it is 100pi.Area of a circle = pi * r^2. So the area of the theater is 7854.0 ft^2.Step-by-step explanation:
Essentially, all of these parallelograms have different properties and properties that they share. For example, a property they share by the definition of a parallelogram is that all of them have two pairs of opposing sides that are parallel. Another property they share from the definition of a parallelogram is that opposite sides are congruent.
As such, rectangles, squares, and rhombuses have the properties that all parallelograms have because they are all parallelograms. However, they also have their own properties.
Rectangles: A rectangle has the properties of a parallelogram, as well as the property of having all of its angles be right angles. This also makes all of its angles congruent.Rhombuses: Rhombuses have the properties of a parallelogram, and they also have all four of their sides congruent. Squares: Squares are like rhombuses and rectangles put together. They have all of their angles as right angles and they have all four of their sides congruent.Second Question Explanation:
1. The question is describing a theater that has a circular base. Therefore, you can use the formula for the circumference of a circle, pi * diameter, to find the circumference (or distance around) the theater. The diameter is given to you: 100 ft. So just multiply that by pi.
2. The question is asking for the area of the theater. This means that you can use the formula for the area of a circle, A = pi * r^2. Note that it is asking you for the area, not the volume, so you would find the area of the circle, not the volume of the hemisphere. r represents the radius of the circle/theater. The diameter is given as 100 ft, and the radius is half of the diameter, so it is 50 ft. Now just plug the radius into the equation and you get the answer I showed.
give the position of C on this number line
The position of C on the number line is 1/8th position.
Define the term number line?A number line is a visual representation of the real numbers as points or marks on a straight line. The number line is usually represented horizontally, with zero in the middle and positive numbers to the right of zero and negative numbers to the left of zero.
From the given number line, there are total 8 points between 0 to 1
That means, it's fractions of 8.
Location of point C is on the 1st point,
So, position of C on this number line = (1/8) ÷ (1-0) = 1/8
Therefore, In the number line, C is located in the 1/8th place.
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(x+2)^2=-16 This equation had no solution, why not?
Answer:
It has no solution within Real numbers. Its solutions are Complex/imaginary, since its discriminant is negative (-64).
Step-by-step explanation:
The normalized form of the equation
(x+2)^2= - 16
x^2 + 4x + 4 = - 16
x^2 + 4x + 20 = 0
We have in the form ax^2 + abx + c = 0
a = 1
b = 4
c = 20
Discriminant is b^2 - 4ac = 4^2 - 4*1*20 = 16 - 80 = - 64
Discriminant is negative, therefore, no Real solutions.
three cards are drawn with replacement from a standard deck of 52 cards. find the the probability that the first card will be a diamond, the second card will be a black card, and the third card will be a face card. express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.
The probability of drawing a diamond, then a black card, and then a face card from a standard deck of 52 cards with replacement is 3/104 or 0.028846 .
What is the probability?The probability that the first card will be a diamond, the second card will be a black card, and the third card will be a face card is given by the expression, `(13/52) × (26/52) × (12/52)`.
In a standard deck of 52 cards, there are 13 diamonds, 26 black cards (13 clubs and 13 spades), and 12 face cards (4 Jacks, 4 Queens, and 4 Kings).
To calculate the probability that the first card will be a diamond, the second card will be a black card, and the third card will be a face card, we use the formula of probability:
`P(E) = n(E) / n(S)`
where, P(E) = Probability of an event
n(E) = Number of favorable outcomes
n(S) = Total number of outcomes
Total number of outcomes = 52
First card will be a diamond
Number of diamonds in a deck of 52 cards = 13
Total number of outcomes after drawing the first card = 52
Probability of drawing a diamond in the first attempt = P(diamond)`= 13/52
Probability of drawing a black card in the second attempt, given that the first card is a diamond= `P(black/diamond)`= (26/52) = `(1/2)`
Probability of drawing a face card in the third attempt, given that the first card is a diamond and second card is a black card= `P(face/diamond and black)`= `(12/52)` = `(3/13)`
Therefore, probability that the first card will be a diamond, the second card will be a black card, and the third card will be a face card`= P(diamond) × P(black/diamond) × P(face/diamond and black) = (13/52) × (1/2) × (3/13)= 3/104`
Therefore, the required probability is 3/104 or 0.028846 rounded to the nearest millionth.
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You purchased 40 shares for $3.95/sh. If you sold the shares for a total of $200. Did you net a profit or a loss?
Answer: profit
Step-by-step explanation:
3.95 (price of one share) multiplied by 40 (the amount of shares bought) would have cost $158. so selling all for $200 would be a $42 profit
Sarah took the advertising department from her company on a round trip to meet with a potential client. Including Sarah a total of 15 people took the trip. She was able to purchase coach tickets for $220 and first class tickets for $910. She used her total budget for airfare for the trip, which was $8130. How many first class tickets did she buy? How many coach tickets did she buy?
A cat gave birth to 333 kittens who each had a different mass between 147147147 and 159\,\text{g}159g159, start text, g, end text. Then, the cat gave birth to a 4^{\text{th}}4 th 4, start superscript, start text, t, h, end text, end superscript kitten with a mass of 57\,\text{g}57g57, start text, g, end text.
The answer to the question is 334 kittens.
Given that a cat gave birth to 333 kittens who each had a different mass between 147 g and 159 g. Then the cat gave birth to a 4th kitten with a mass of 57 g.
First of all, we will find out the range of the mass of kittens. The range is given as follows;Range = Maximum Value - Minimum Value Range = 159 g - 147 g Range = 12 g
Now, the cat gave birth to a 4th kitten with a mass of 57 g, we can say that the minimum value of kitten's mass is 57 g.So, the maximum value of kitten's mass can be calculated as follows;Maximum Value = 57 g + Range Maximum Value = 57 g + 12 g Maximum Value = 69 g Now, we can say that all kittens with a mass of 69 g or less would be born because the minimum value of kitten's mass is 57 g and the range of mass is 12 g.
Therefore, the answer to the question is 334 kittens.
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Gary's backpack weighs 1.2 pounds. His math textbook weighs 3.75 pounds, and his science textbook weighs 2.85 pounds. How much will his backpack weigh with the math and science textbooks in it?
Answer:
To find out how much Gary's backpack will weigh with the math and science textbooks in it, we need to add the weight of the textbooks to the weight of the backpack:
Total weight = backpack weight + math textbook weight + science textbook weight
Total weight = 1.2 + 3.75 + 2.85
Total weight = 7.8 pounds
Therefore, Gary's backpack will weigh 7.8 pounds with the math and science textbooks in it.
Answer: His backpack will weigh 7.8 pounds
Step-by-step explanation:
Gary's backpack already weights 1.2 pounds without the science and math textbook, now we add the weights of both the math and science textbook.
1.2 + 3.75 = 4.95
4.95 is the weight with only his math textbook in his bag
now we add the science textbooks weight to 4.95
4.95 + 2.85 = 7.8
7.8 is the weight of his backpack with both his science and math textbook in his bag
A production facility contains two machines that are used to rework items that are initially defective. Let X be the number of hours that the first machine is in use, and let Ybe the number of hours that the second machine is in use, on a randomly chosen day. Assume that X and Y have joint probability density function given by 3. f(x) = { 3/2(x^2 +y^2) 0
Answer:
Step-by-step explanation:
Square ABCD is similar to square EFGH. The ratio of AB:EF is 1:4. The area of square EFGH is 14,400ft ft squared by 2. What is AB?
The Length of AB in square ABCD is 30 feet.
Since the squares ABCD and EFGH are similar, their corresponding sides are proportional, so we can set up the following relation:
AB/EF = 1/4
We can also use the fact that the ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding sides. Therefore,
AB²/EF² = (Area of square ABCD)/(Area of square EFGH)
Substituting the given values:
AB²/EF² = (Area of square ABCD)/(14400)
Since the areas of squares are proportional to the square of their sides, we can write,
Area of square ABCD/Area of square EFGH = (AB/EF)²
Substituting this into the above equation and solving for AB, we get,
AB²/EF² = (AB/EF)²
AB² = (AB/EF)² * EF²
AB² = (1/4)² * 14400
AB² = 900
AB = 30 feet
Therefore, the length of the side AB of square ABCD is 30 feet.
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suppose the minimum volume of a clown is 60,000 cm3 and the volume of my car is 3 million cm3. 55 is______ the maximum number of clowns that can fit in my car.
a. An upper bound on
b. Not a bound
c. A lower bound on
d. exactly
When the minimum volume of a clown is 60,000 cm³ and the volume of a car is 3 million cm³, 55 is an upper bound on the maximum number of clowns that can fit in a car. The correct answer is Option A.
What are bounds?Bounds are the maximum and minimum limits that are permitted, within which something can or must be performed. Bounds are used to refer to an acceptable range of values that provide safe operation or performance. In mathematics, a set of bounds can define the limits of the amount of things or objects.
The minimum volume of a clown is 60,000 cm³, so to fit 55 clowns in a car we need:
55 clowns × 60,000 cm³/clown = 3,300,000 cm³
This is lower than the volume of the car. Therefore, 55 is an upper bound on the maximum number of clowns that can fit in the car.
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g a second unit vector which is also orthogonal to both 8,-8,8 and 0,5,5 is the unit vector which points in the direction opposite to u1 -2/6.-1/6,1/6 this is the vector u2
The vector u2 is (0, √3/3, 2/3). It is a unit vector that is orthogonal to both 8,-8,8 and 0,5,5.
Given two vectors 8,-8,8 and 0,5,5, we need to find another unit vector that is orthogonal to both the given vectors. Let's call this vector u1.The vector u1 can be obtained by taking the cross product of the two given vectors:u1 = (8,-8,8) × (0,5,5)u1 = (-40,-40,40)
To get a unit vector, we need to normalize u1 by dividing it by its magnitude:|u1| = √((-40)² + (-40)² + 40²) = 60u1 = (-40/60, -40/60, 40/60) = (-2/3, -2/3, 1/3)Now we need to find another unit vector that is orthogonal to u1.
One way to do this is to take the cross product of u1 with another vector, and then normalize the result. We can choose any vector that is not parallel to u1. For example, we can choose the vector (1,0,0).u2 = u1 × (1,0,0)u2 = (-2/3, -2/3, 1/3) × (1,0,0)u2 = (0,1/3,2/3)
To get a unit vector, we need to normalize u2 by dividing it by its magnitude:|u2| = √(0² + (1/3)² + (2/3)²) = 1/√3u2 = (0, √3/3, 2/3)
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You put $200 at the end of each month in an investment plan that pays an APR of 4. 5%. How much will you have after 18 years? Compare this amount to the total deposits made over the time period.
a.
$66,370. 35; $43,200
c.
$66,380. 12; $43,000
b.
$66,295. 23; $43,000
d.
$66,373. 60; $43,200
As per the given APR, the sum of amount after 18 years is $66,373. 60, and the total deposits made over the time period is $43,200. (option d).
To calculate this, we can use the formula for future value of an annuity:
FV = PMT x (((1 + r)⁻¹) / r)
where FV is the future value, PMT is the monthly payment, r is the monthly interest rate (which is calculated by dividing the APR by 12), and n is the number of payments (which is 18 x 12 = 216 in this case).
Plugging in the numbers, we get:
FV = $200 x (((1 + 0.045/12)²¹⁶ - 1) / (0.045/12)) = $66,373.60
Therefore, you would have approximately $66,373.60 in your investment plan after 18 years.
Now let's compare this amount to the total deposits made over the time period. In this case, the total deposits would be:
$200 x 12 x 18 = $43,200
Hence the correct option is (d).
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