Answers
a. The area on [0, 10] is that of a trapezoid with bases 5 and 15 and height 10, so
[tex]\displaystyle \int_0^{10} f(x) \, dx = \frac{5+15}2\cdot10 = \boxed{100}[/tex]
b. By linearity of the definite integral, we have
[tex]\displaystyle \int_0^{25} f(x)\,dx = \int_0^{10} f(x)\,dx + \int_{10}^{25} f(x)\,dx[/tex]
and the area on [10, 25] is another trapezoid with bases 15 and 7.5 and height 15, so that
[tex]\displaystyle \int_{10}^{25} f(x)\,dx = \frac{15+7.5}2\cdot15 = 168.75[/tex]
Then the total area on [0, 25] is
[tex]\displaystyle \int_0^{25} f(x)\,dx = \boxed{268.75}[/tex]
c. The area on [25, 35] is that of a triangle with base 10 and height 15. However, [tex]f(x)<0[/tex] on this interval, so we multiply this area by -1 to get
[tex]\displaystyle \int_{25}^{35} f(x)\,dx = -\frac{10\cdot15}2 = \boxed{-75}[/tex]
d. The area on [15, 25] is the same as the area on [25, 35] because it's another triangle with the same dimensions. But the area on [15, 25] lies above the horizontal axis, so
[tex]\displaystyle \int_{15}^{25} f(x)\,dx = \int_{15}^{25} f(x)\,dx + \int_{25}^{35} f(x)\,dx = \boxed{0}[/tex]
e. The plot of [tex]|f(x)|[/tex] lies above the horizontal axis. We know the area on [15, 25] is the same as the area on [25, 35], but now both areas are positive, so
[tex]\displaystyle \int_{15}^{35} |f(x)| \, dx = \int_{15}^{25} f(x)\,dx - \int_{25}^{35} f(x)\,dx = 2 \int_{15}^{25} f(x)\,dx = \boxed{150}[/tex]
f. Changing the order of the limits in the integral swaps the sign of the overall integral, so
[tex]\displaystyle \int_{10}^0 f(x)\,dx = -\int_0^{10} f(x)\,dx = \boxed{-100}[/tex]
Related Questions
What does it mean for the volume of a solid object to be 10 in.³ rely on the meaning of volume by phone to 1“ x 1“ x 1“ cubes in your answer
Answers
The insinuation of calling the volume of a solid 3-dimentional shape 10in³ is that the product of all of its dimensions as measured in units is; 10 in.³.
What is the meaning of volume as used in the task content?It follows from the task content that the shape in discuss is a solid shape and consequently, one of it's measures is its volume which describes the space it occupies.
On this note, the meaning of a solid having a volume of 10in³ as indicated is that it's volume is; 10 times as large as the volume occupied by an object with unit dimensions 1“ x 1“ x 1“ in which case, the volume is; 1 in³.
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1. In the process of solving the radical equation, x − 5 =radical 3x −11 , you will need to solve which of the following quadratic equations?
a) x2 −7x+16=0
b) x2 −3x−14=0
c) x2 +13x−36=0
d) x2 −13x + 36 = 0 e)
none of these
Answers
In the process of solving the expression x − 5 =radical 3x −11 , you will need to solve the quadratic equation x² - 13x + 36 =0
Solving radical equationsRadical equations are expression that are under the square root. For instance √x is a radical expression.
Given the equation expressed as:
x − 5 = √3x −11
Square both sides to have;
(x-5)² = (√3x −11 )²
This can be further expressed as:
(x-5)² =3x −11
Expand
x² -10x + 25 = 3x - 11
Equate to zero
x² -10x + 25 - 3x +11 = 0
x² -10x - 3x +36 = 0
x² - 13x + 36 =0
Hence in the process of solving the expression x − 5 =radical 3x −11 , you will need to solve the quadratic equation x² - 13x + 36 =0
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I'm trying to figure this out for extra credit but I don't know what to do can someone step by step take me through it with the answer?
Answers
Answer:
2800
Step-by-step explanation:
The choice at any node (except the edges) is to go right or go down.
A to BTo get from A to B requires a total of 4 choices to go down, and 4 choices to go right. Those can occur in any order. Locating the "right" choices in the list of "all" choices is effectively choosing 4 of 8. The number of ways those choices can be made is ...
C(8, 4) = 8!/(4!(8-4)!) = 8·7·6·5/(4·3·2) = 70
B to CSimilarly a total of 2 choices must be made, 1 of which is a "right" choice. The number of ways those can be ordered is ...
C(2, 1) = 2 . . . . . . . that is: (right, down) or (down, right)
C to DA total of 6 choices must be made, of which 3 are "right." The number of possible orderings is ...
C(6, 3) = 6·5·4/(3·2·1) = 20
Total pathsEach set of choices is independent of the others, so the total possible number is the product of the numbers of choices on the subpaths:
total paths = (70)(2)(20) = 2800
There are 2800 possible paths from A to D.
2(−6c+3)+4c2, left parenthesis, minus, 6, c, plus, 3, right parenthesis, plus, 4, c
Answers
Answer:
[tex]-8c+6[/tex]
Step-by-step explanation:
[tex]2(-6c+3)+4c=-12c+6+4c=\boxed{-8c+6}[/tex]
Answer:
-8c + 6 and 3(-4c+2) + 4c
Step-by-step explanation:
khan
Determine three numbers a , b , c
such that a , b , c are three consecutive terms of a geometric sequence and an arithmetic sequence at the same time.
Note: i do not want the answer
d=0 and r=1, as in 2 , 2 , 2 , 2 , 2...
Given also:
abc=27 or a.b.c=27
Answers
Since [tex]a,b,c[/tex] are in geometric progression, if [tex]r[/tex] is the common ratio between consecutive terms, then
[tex]a=a[/tex]
[tex]b = ar[/tex]
[tex]c=ar^2[/tex]
Since [tex]a,b,c[/tex] are also in arithmetic progression, if [tex]d[/tex] is the common difference between consecutive terms, then
[tex]a = a[/tex]
[tex]b = a + d \implies d = b-a[/tex]
[tex]c = b + d = a + 2d \implies c = a + 2(b-a) = 2b-a[/tex]
Given that [tex]abc=27[/tex], we have
[tex]abc = a\cdot ar\cdot ar^2 = (ar)^3 = 27 \implies ar = 3 \implies a = \dfrac3r[/tex]
[tex]b = \dfrac3r \cdot r = 3[/tex]
[tex]c = \dfrac3r \cdot r^2 = 3r[/tex]
It follows that
[tex]c = 2b-a \iff 3r = 6 - \dfrac3r[/tex]
Solve for [tex]r[/tex].
[tex]3r - 6 + \dfrac3r = 0[/tex]
[tex]3r^2 - 6r + 3 = 0[/tex]
[tex]r^2 - 2r + 1 = 0[/tex]
[tex](r-1)^2 = 0[/tex]
[tex]\implies r=1 \implies a=b=c=3[/tex]
so the only possible sequence is {3, 3, 3, …}.
( P + 5 ) - 4 = 6
What is the value of P?
Answers
Answer:
ight first you add 4 to 6 and that gets you 10 so you left with p+5=10 then you subtract 5 from 10 so p is gonna be 5
Step-by-step explanation:
Find the first five terms of the recursive sequence. Show all work.
Answers
The first five terms of the recursive sequence are 5, 12, 19, 26 and 33.
How to use recursive equations to generate a series
In this question we have a linear recursive equation that requires the value of the immediately previous element to generate the next one. Then, we need to evaluate the expression for the first five elements:
i = 1
a₁ = 5
i = 2
a₁ = 5, a₂ = a₁ + 7
a₂ = 5 + 7
a₂ = 12
i = 3
a₂ = 12, a₃ = a₂ + 7
a₃ = 12 + 7
a₃ = 19
i = 4
a₃ = 19, a₄ = a₃ + 7
a₄ = 19 + 7
a₄ = 26
i = 5
a₄ = 26, a₅ = a₄ + 7
a₅ = 26 + 7
a₅ = 33
The first five terms of the recursive sequence are 5, 12, 19, 26 and 33.
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Hava Racquets Company manufactures pickleball racquets in four different models.
For the year, the SoftNet line had a net loss of AED 40,000 from sales of AED 250,000,
variable costs of AED 180,000 and fixed costs of AED 110,000. If the Softnet is
eliminated, AED 30,000 of fixed cost will remain. What would be the incremental
effect to the Company if the Softnet would be eliminated?
Answers
Based on the AED 40,000 net loss that the SoftNet line brings to Hava Racquets company, the incremental effect if the Softnet line is eliminated is that the company would save AED 10,000.
What is the incremental effect?If Hava Racquets Company decides to scrap the Softnet line, then it means that they will no longer be incurring the AED 40,000 net loss that the line had brought to them.
They would instead incur a fixed cost of AED 30,000 which they can apply to their other operations.
This means that in exchange for a AED 40,000 loss, they would instead get a AED 30,000 loss.
The savings on these losses is:
= 40,000 - 30,000
= AED 10,000
This means that Hava Racquets Company will make a saving of AED 10,000 if they eliminate the line.
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4x-9y=36 find the missing y-coordinate
Answers
The y-intercept of the function is given as (0, -4). This also shows that the value of y missing from the coordinate is -4
How to find the y-intercept of a function?The y-intercept of a function is the point where the value of x is zero. Given the function below;
4x-9y=36
when the value of x is zero, hence;
4(0) - 9y = 36
Simplify to have:
0 - 9y = 36
-9y = 36
Divide both sides by -9 to have:
-9y/-9 = 36/-9
y = -4
Hence the y-intercept of the function is given as (0, -4). This also shows that the value of y missing from the coordinate is -4
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If Usman buys 8 books that cost Rs. 120/- each, he will be left Rs. 140/-. If he buys 25 copies, how much he will be left with?
Answers
He will be left no money after buying 25 copies because he has 1900 rupees less.
Given that Usman buys 8 books that cost Rs. 120/ each, he will be left Rs. 140.
The cost of 1 books is Rs 120.
The cost of 8 book is 120×8=960.
The cost of 25 books is 120×25=3000
To find the money usman will left with him is the difference between the total money and the cost of 25 books.
Total money=960+140=1100
money left=Total money -cost of 25 books
money left=1100-3000
Usman have less money to buy 25 books. He has 1900 rupees less to buy 25 books.
Hence, Usman does leave any money with even he has less money to buy 25 books
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A pitaya, or dragon fruit, is a tropical fruit with pink skin and pulp filled with tiny black
seeds. The weight of a pitaya seed is 1 x 107 ounce. What is the weight of the pitaya
seed in standard form?
Type the number in the box.
oz
Answers
The standard form the weight of a pitaya seed is 1 x 10⁷ ounces.
What is a standard form?Any value between 1.0 and 10.0 that can be expressed as a decimal number and multiplied by a power of 10 is referred to as being in standard form. If you look closely, you will see that 1.23 is a decimal between 1.0 and 10.0, and as a result, we have the standard form of 123,000,000 as 1.23 x 10⁸.
Given that a pitaya, or dragon fruit, is a tropical fruit with pink skin and pulp filled with tiny black seeds. The weight of a pitaya seed is 1 x 10⁷ ounces.
The standard form can be written as:-
Weight = 10000000 ounces
Weight = 1 x 10⁷ ounces.
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the figure at the right shows the dimensions of the garden in Marissa's back yard. What is the area of the garden?
Answers
The area of the garden is 74 square feet
How to determine the area of the garden?The complete question is added as an attachment
From the attached figure, we have the following shapes and dimensions:
Rectangle: 10 by 5 feetTrapezoid: Bases = 10 and 6; Height = 3The rectangular area is
A1 = 10 * 5
Evaluate
A1 = 50
The area of the trapezoid is
A2 = 0.5 * Sum of parallel bases * height
This gives
A2 = 0.5 * (10 + 6) * 3
Evaluate
A2 = 24
The total area is
Total = A1 + A2
This gives
Total = 50 + 24
Evaluate
Total = 74
Hence, the area of the garden is 74 square feet
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It takes 3 liters of paint to cover an area of 24 square meters. What percentage increase in the quantity of paint would be required to cover an area of 50.4 square meters?
Answers
Answer:
110 [%].
Step-by-step explanation:
if the initial area is 24 m² and final area 50.4 m², then the required percentage can be calculated as:
[tex]\frac{50.4}{24}*100-100=110[/tex] ,
where 50.4/24*100 - the final percentage.
P.S. It means, the final value of the paint is 3*2.1=6.3 [liters].
Which of the following shows one way to simplify 43Σn=1(3+9n)?
Answers
Based on the given summation notation, the expression that shows one way to simplify 43 Σ n=1 (3+9n) is (a) 43 Σ n=1 3 + 43 Σ n=1 9n
How to determine the summation expression?The expression is given as:
43Σn=1(3+9n)
As a general rule, if a summation notation is represented using the following expression
Σ(a + bn)
The equivalent expression of the above summation notation is
Σa + bn
Where the variable a is a constant in the expression
This means that:
Σ(a + bn) = Σa + bn
Using the above equation as a guide, we have the following equivalent equation
43 Σ n=1 (3+9n) = 43 Σ n=1 3 + 43 Σ n=1 9n
Hence, based on the given summation notation; the expression that shows one way to simplify 43 Σ n=1 (3+9n) is (a) 43 Σ n=1 3 + 43 Σ n=1 9n
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Answer:
D on 3dg I took the best
Step-by-step explanation:
plssssssssssssssssssssssss help
Answers
Hello and Good Morning/Afternoon!
Let's take this problem step by step:
Let's consider some facts:
⇒ angles are opposite each other
⇒ by the Vertical Angle Theorem
⇒ angles opposite each other are equal in measure
So:
[tex]\hookrightarrow 4x + 12 = 64[/tex]
Let's solve
[tex]4x + 12 = 64\\4x = 52\\x = 13[/tex]
x's value is 13
Answer: 13
Hope that helps!
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Joey owns a small bakery and today Phoebe has ordered 100 cookies. If Joey boxes the cookies by the dozen, which of the following equations describes how many boxes of cookies Phoebe will have? Let b represent the number of boxes. (HINT: All of the cookie boxes are not necessarily full.)
Answers
Answer:
Joey needs 9 boxes.
Step-by-step explanation:
You will use the fact that a dozen is 12 cookies. You're trying to find how many dozen are in 100.
Something like:
12b = 100
or if you're studying inequalities:
12b >= 100
Divide by 12 to solve. See image. Interpret the results. See image.
Which graph shows a system with no solutions?
Graph A
Graph B
y
201
OA. Graph B
OB. Graph C
OC. Graph A
5
5
Graph C
Y
5
5+
y=2x-1
5
2y=4x-2
Answers
Answer:
i belive its graph B
here is another one that i need help on What number represents the same amount as 8 hundreds + 10 tens + 0 ones
Answers
Answer:
810 is the number that represents the same amount.
Hope this helps :)
PLEASE HELP ME WITH THIS PROBLEM!!!
Answers
Answer: Moderate negative association
Step-by-step explanation:
a negative association or in other terms correlation, suggests that while one variable increases the other decreases and vice versawhen looking at the graph, the data shows a negative correlationas the x variable increases the y variable decreasesthis not a perfect correlation as the the change in value of variable x is not directly proportional to the change in value of yCan someone explain the steps I’m confused
Answers
Answer:
x = 8 ± [tex]\sqrt{10}[/tex]
Step-by-step explanation:
x² - 16x + 54 = 0 ( subtract 54 from both sides )
x² - 16x = - 54
to complete the square
add ( half the coefficient of the x- term )² to both sides
x² + 2(- 8)x + 64 = - 54 + 64
(x - 8)² = 10 ( take the square root of both sides )
x - 8 = ± [tex]\sqrt{10}[/tex] ( add 8 to both sides )
x = 8 ± [tex]\sqrt{10}[/tex]
then
x = 8 - [tex]\sqrt{10}[/tex] , x = 8 + [tex]\sqrt{10}[/tex]
eBookPrint Item
Question Content Area
Service Department Charges and Activity Bases
Middler Corporation, a manufacturer of electronics and communications systems, uses a service department charge system to charge profit centers with Computing and Communications Services (CCS) service department costs. The following table identifies an abbreviated list of service categories and activity bases used by the CCS department. The table also includes some assumed cost and activity base quantity information for each service for October.
CCS Service
Category
Activity Base
Budgeted Cost Budgeted Activity
Base Quantity
Help desk Number of calls $160,000 3,200
Network center Number of devices monitored 735,000 9,800
Electronic mail Number of user accounts 100,000 10,000
Smartphone support Number of smartphones issued 124,600 8,900
One of the profit centers for Middler Corporation is the Communication Systems (COMM) sector. Assume the following information for the COMM sector:
• The sector has 5,200 employees, of whom 25% are office employees.
• All the office employees have been issued a smartphone, and 96% of them have a computer on the network.
• One hundred percent of the employees with a computer also have an e-mail account.
• The average number of help desk calls for October was 1.5 calls per individual with a computer.
• There are 600 additional printers, servers, and peripherals on the network beyond the personal computers.
a. Determine the service charge rate for the four CCS service categories for October.
CCS Service Category Service Charge Rate
Help desk $fill in the blank 1
Per call
Network center $fill in the blank 3
Per device monitored
Electronic mail $fill in the blank 5
Per user or e-mail account
Smartphone support $fill in the blank 7
Per smartphone issued
b. Determine the charges to the COMM sector for the four CCS service categories for October.
October charges to the COMM sector
Help desk charge $fill in the blank 9
Network center charge $fill in the blank 10
Electronic mail charge $fill in the blank 11
Smartphone support charge $fill in the blank 12
Feedback Area
Feedback
a. Divide the budgeted cost by the budgeted activity base quantity for each category.
b. Determine the Help desk charge by multiplying the rate from (a) by the number of employees times the percentage of office employees times the percentage with a computer times the number of calls. Follow a similar process for each of the other three categories.
Learning Objective 3.
Answers
The service charge rate are: Help desks $50 per call, Network center $75 per device monitored, Electronic mail $10 per user or per e-mail account, Local voice support $14 per phone extension.
Service charge ratea. Service charge rate
Service charge rate=Budgeted cost/ Budgeted activity base quantity
Help desks:
Service charge rate=$160,000/3,200 calls
Service charge rate=$50 per call
Network center:
Service charge rate=$735,000/9,800 devices
Service charge rate=$75 per device monitored
Electronic mail:
Service charge rate=$100,000/10,000 accounts
Service charge rate=$10 per user or per e-mail account
Local voice support:
Service charge rate=$124,600/8,900 phone extensions
Service charge rate=$14 per phone extension
b. Help desk charge
Help desk charge =(Number of employees times × Percentage of office employees× Percentage with a computer ×Number of calls)× Service charge rate
Help desk charge:
(5,200 employee×25%×96%×1.5)× $50 per call
=$93,600
Network center charge:
[(5,200 employee×25%×96%)+ 600]× $75 per device
=$138,600
Electronic mail:
(5,200 employee×25%×96%×100)×$10 per user or per e-mail account
=$12,480
Local voice support:
(5,200 employee×25%)× $14 per phone extension
=$18,200
Therefore the service charge rate are: Help desks $50 per call, Network center $75 per device monitored, Electronic mail $10 per user or per e-mail account, Local voice support $14 per phone extension.
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HELP PLEASE
What is the area of the kite below?
Answers
Answer:
Area = 33 inches squared
Step-by-step explanation:
The formula for the area of a kite is:
Area = ½ × (d)1 × (d)2
To find the first diagonal (d1):
d1 = 7 in + 4 in = 11 in
To find the second diagonal (d2):
The triangles on the right of the kite must be 3 4 5 triangle since it is a right triangle with a hypotenuse of 5 and a side of 4, we know half of a diagonal would be 3 inches.
d2 = 3 in + 3 in = 6 in
Now we can plug the diagonals into the area formula.
Area = ½ × 11 × 6 = 33 inches squared
Answer: A = 33 in²
Step-by-step explanation:
Given information:
Longer side = 7 in
Shorter side = 4 in
Shorter Hypotenuse = 5 in
Given formula:
A = (1/2) × (d₁) × (d₂)
A = aread₁ = The longer line (horizontal)d₂ = The shorter line (vertical)Please refer to the attachment below for a graphical understanding
Determine the length of the shorter side (vertical) through the Pythagorean theorem
a² + b² = c² (Pythagorean theorem)
(4)² + b² = (5)² (Substitute values into the equation)
16 + b² = 25
b² = 25 - 16
b² = 9
b = 3 or b = -3 (rej)
Substitute values into the given formula
A = (1/2) × d₁ × d₂
A = (1/2) × (7 + 4) × (3 + 3)
Simplify values in the parenthesis
A = (1/2) × 11 × 6
Simplify by multiplication
A = (11/2) × 6
[tex]\Large\boxed{A=33}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Scores on a test are normally distributed with a mean of 63 and a standard deviation of 9.3. Find P81, which separates the bottom 81% from the top 19%.
Answers
We want to find [tex]x[/tex] such that
[tex]\mathrm{Pr}(X \le x) = 0.81[/tex]
where [tex]X[/tex] is the random variable for test scores, and [tex]X\sim\mathrm{Normal}(63,9.3^2)[/tex].
Transform [tex]X[/tex] to [tex]Z\sim\mathrm{Normal}(0,1)[/tex] using the relation
[tex]X = \mu + \sigma Z \implies Z = \drac{X-63}{9.3}[/tex]
so that
[tex]\mathrm{Pr}(X \le x) = \mathrm{Pr}\left(\dfrac{X-63}{9.3} \le \dfrac{x-63}{9.3}\right) = \mathrm{Pr}\left(Z \le \dfrac{x-63}{9.3}\right) = 0.81[/tex]
Applying the inverse CDF of [tex]Z[/tex] (denoted by [tex]\Phi^{-1}[/tex]), we have
[tex]\dfrac{x-63}{9.3} = \Phi^{-1}(0.81) \approx 0.8779[/tex]
Solve for [tex]x[/tex].
[tex]\dfrac{x-63}{9.3} \approx 0.8779[/tex]
[tex]x-63 \approx 8.1644[/tex]
[tex]x \approx 76.1644[/tex]
Then the cutoff test score for the 81% percentile is [tex]P_{81} \approx \boxed{76}[/tex].
Stopyra Incorporated makes a single product—a cooling coil used in commercial refrigerators. The company has a standard cost system in which it applies overhead to this product based on the standard labor-hours allowed for the actual output of the period. Data concerning the most recent year appear below:
Budgeted variable manufacturing overhead $ 28,200
Budgeted fixed manufacturing overhead $ 89,280
Budgeted hours 12,000 labor-hours
Actual production (a) 16,000 units
Standard hours per unit (b) 0.60 labor-hours
Standard hours allowed for the actual production (a) × (b) 9,600 labor-hours
Actual variable manufacturing overhead $ 26,967
Actual fixed manufacturing overhead $ 74,280
Actual hours 8,900 labor-hours
The variable overhead rate variance is:
$6,528 U
$6,528 F
$6,052 U
$6,052 F
Answers
The variable overhead rate variance is: c. $6,052 U.
Variable overhead rate varianceFirst step is to calculate variable component of the predetermined overhead rate using this formula
Variable component of the predetermined overhead rate= Budgeted variable manufacturing overhead/Budgeted hours
Let plug in the formula
Variable component of the predetermined overhead rate=$ 28,200/12,000 labor-hours
Variable component of the predetermined overhead rate=$2.35 per labor-hour
Second step is to calculate variable overhead rate variance using this formula
Variable overhead rate variance=(AH×AR)-(AH×SR)
Let plug in the formula
Variable overhead rate variance= $ 26,967- ( 8,900 labor-hours×$2.35 per labor-hour)
Variable overhead rate variance= $ 26,967- $20,915
Variable overhead rate variance= $6,052 U (Unfavorable)
Therefore the variable overhead rate variance is: c. $6,052 U.
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Use the table values below to select the correct statement.
Answers
The true statement is (c) the function f(x) is an exponential function
How to determine the correct statement?From the table, we can see that:
As x increases by 2.y does not increase with the same common differenceThis means that the table represents an exponential function
The rate is then calculated as:
f(n) = f(1) * r^n-1
Using f(3), we have
102 = 17 * r^3-1
This gives
r^2 = 6
Take the square roots
r = 2.45
Hence, the true statement is (c) the function f(x) is an exponential function
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HELP ME WITH THIS QUESTION AS SOON AS POSSIBLE
Answers
Answer:
the answer is 2880
Step-by-step explanation:
because you would use the formula (n - 2) x 180 which equals the sum interior angles of any polygon .
The letter n stands for number of sides so in this case you would do (18 - 2) x 180 so 16 × 180 = 2880
hope this helped :) pls ask if you need any more explanation .
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Line A passes through the points (-8, 5) and (-5, 4). Line B passes through the points (0, 1) and (4, -1). Which of the following describes the relationship between line A an line B?
1. Lines A and B are parallel, because they have opposite reciprocal slopes.
2. Lines A and B are parallel, because they have the same slope
3. Lines A and B intersect, because their slopes have no relationship.
4. Lines A and B are perpendicular, because they have opposite reciprocal slopes.
Answers
[tex]\stackrel{\textit{\LARGE Line A}}{(\stackrel{x_1}{-8}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{-5}~,~\stackrel{y_2}{4})} ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{4}-\stackrel{y1}{5}}}{\underset{run} {\underset{x_2}{-5}-\underset{x_1}{(-8)}}} \implies \cfrac{4 -5}{-5 +8}\implies -\cfrac{1}{3} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{\LARGE Line B}}{(\stackrel{x_1}{0}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{-1})} ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{-1}-\stackrel{y1}{1}}}{\underset{run} {\underset{x_2}{4}-\underset{x_1}{0}}} \implies \cfrac{-1 -1}{4 +0}\implies -\cfrac{1}{2}[/tex]
keeping in mind that perpendicular lines have negative reciprocal slopes, and that parallel lines have equal slopes, well, those two slopes above aren't either, so since they're neither, and they're different, that means that lines A and B intersect.
Derive Integral equation for dy/dv
Answers
By the fundamental theorem of calculus,
[tex]\displaystyle y = \int_0^v \sqrt{3+4t^2} \, dt \implies \boxed{\frac{dy}{dv} = \sqrt{3+4v^2}}[/tex]
In general,
[tex]\displaystyle \frac{d}{dx} \int_0^{g(x)} f(u) \, du = f(g(x)) \frac{dg}{dx}[/tex]
Calculating orthogonal trajectories?
Answers
The equation for the trajectories orthogonal to the family of functions of the form 5 · x² - 2 · y² = C is equal to (1 / 5) · ㏑ x + (1 / 2) · ㏑ y = C.
How to find the equation for the orthogonal trajectories of a given equation
In this problem we have a family of functions in implicit form, that is, a function of the form f(x, y, c) = 0. The equation of a orthogonal trajectory is always perpendicular to a particular form of a equation. First, we determine the first derivative of the given expression:
5 · x² - 2 · y² = C
10 · x - 4 · y · y' = 0
4 · y · y' = 10 · x
y' = (10 · x) / (4 · y)
y' = (5 · x) / (2 · y)
If f(x, y) = (5 · x) / (2 · y), then the differential equation for the orthogonal trajectories related to the family of functions is:
y' = - 1 / f(x, y)
y' = - (2 · y) / (5 · x)
dy / (2 · y) = - dx / (5 · x)
By indefinite integration we get the following solution to the ordinary differential equation:
(1 / 2) · ㏑ y = - (1 / 5) · ㏑ x + C
(1 / 5) · ㏑ x + (1 / 2) · ㏑ y = C
The equation for the trajectories orthogonal to the family of functions of the form 5 · x² - 2 · y² = C is equal to (1 / 5) · ㏑ x + (1 / 2) · ㏑ y = C.
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flying against the jet stream, a jet travels 3000 in 3 hours. Flying with the jet-stream, the same jet travels 5360 miles in 4 hours. What is the rate of the jet in still air and what is the rate of the jet stream?
Answers
Based on the time taken by the jet when flying with the wind versus against it, the rate of the jet and jet streams are:
In still air = 1,170 mphRate of wind = 170 mphWhat is the speed of the jet?The speed of the jet flying against the wind is:
= 3,000 / 3
= 1,000 mph
The speed with the jet stream is:
= 5,360 / 4
= 1,340 mph
Assuming the speed of the jet is x and the rate of wind is y, the formulas are:
x - y = 1,000
x + y = 1,340
Solving gives:
2x = 2,340
x = 1,170 mph
The rate of jet stream is:
x - y = 1,000
1,170 - y = 1,000
y = 170 mph
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