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Q. No. 1: Short Questions a) Find the partial differential equation by eliminating arbitrary constants a and b from the equation (r - a) + (y - b)? + = (02) b) Find the solution of the IVP y' = -4y +2
Answers
a) The partial differential equation (PDE) obtained by eliminating the arbitrary constants a and b is:
F(x^2/2 - ax - y^2/2 + by + C) = 0
b) The solution of the IVP y' = -4y + 2 is given by:
y = (2 - Ce^x) / 4 or y = (2 + Ce^x) / 4, where C is a constant.
a) To eliminate the arbitrary constants a and b from the equation (x - a) ∂u/∂x + (y - b) ∂u/∂y = 0, we can use the method of characteristics.
Let's consider the characteristics defined by the equations:
dx / (x - a) = dy / (y - b) = du / 0
From the first two equations, we have:
(x - a) dx = (y - b) dy
Integrating both sides, we get:
x^2/2 - ax = y^2/2 - by + C
where C is an arbitrary constant.
Now, let's consider the characteristic defined by the equation:
du / 0 = dx / (x - a)
This implies that du = 0, which means u is a constant along this characteristic.
Combining this with the equation x^2/2 - ax = y^2/2 - by + C, we can express u in terms of x, y, a, and b as:
u = F(x^2/2 - ax - y^2/2 + by + C)
where F is an arbitrary function.
Therefore, the partial differential equation (PDE) obtained by eliminating the arbitrary constants a and b is:
F(x^2/2 - ax - y^2/2 + by + C) = 0
b) To find the solution of the initial value problem (IVP) y' = -4y + 2, we can use the method of separation of variables.
First, let's rewrite the differential equation as:
dy/dx = -4y + 2
Now, we separate the variables by moving all terms involving y to one side and all terms involving x to the other side:
dy / (-4y + 2) = dx
Next, we integrate both sides:
∫ (1 / (-4y + 2)) dy = ∫ dx
The integral on the left side can be evaluated using the substitution u = -4y + 2:
∫ (1 / u) du = ∫ dx
ln(|u|) = x + C1
Now, substituting back u = -4y + 2, we have:
ln(|-4y + 2|) = x + C1
Taking the exponential of both sides, we get:
|-4y + 2| = e^(x + C1)
Simplifying further, we have two cases to consider:
1. -4y + 2 = e^(x + C1)
2. -4y + 2 = -e^(x + C1)
For case 1, we can rewrite it as:
-4y + 2 = Ce^x
where C = e^(C1).
Solving for y, we obtain:
y = (2 - Ce^x) / 4
For case 2, we can rewrite it as:
-4y + 2 = -Ce^x
where C = -e^(C1).
Solving for y, we obtain:
y = (2 + Ce^x) / 4
Therefore, the solution of the IVP y' = -4y + 2 is given by:
y = (2 - Ce^x) / 4 or y = (2 + Ce^x) / 4, where C is a constant.
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Related Questions
Convert from Celsius to Fahrenheit.
Answers
Answer:
212° = F
Step-by-step explanation:
We know that,
9/5C + 32 = F
So, if C = 100°, then,
9/5 × 100 + 32 = F
900/5 + 32 = F
180 + 32 = F
212° = F
Hope it helps ⚜
For a monopolist's product, the cost function is c=0.004q
3
+40q+5000 and the demand function is p=450−6q. Find the profit-maximizing output. The profit-maximizing output is (Round to the nearest whole number as needed.)
Answers
The quantity that maximizes the monopolist's profit is approximately 23 units.
To find the profit-maximizing output for the monopolist's product, we need to determine the quantity that maximizes the monopolist's profit.
The profit function is calculated as follows: Profit = Total Revenue - Total Cost.
Total Revenue (TR) is given by the product of the price (p) and the quantity (q): TR = p * q.
Total Cost (TC) is given by the cost function: TC = 0.004q^3 + 40q + 5000.
To find the profit-maximizing output, we need to find the quantity at which the difference between Total Revenue and Total Cost is maximized. This occurs when the marginal revenue (MR) equals the marginal cost (MC).
The marginal revenue is the derivative of the Total Revenue function with respect to quantity, which is MR = d(TR)/dq = p + q * dp/dq.
The marginal cost is the derivative of the Total Cost function with respect to quantity, which is MC = d(TC)/dq.
Setting MR equal to MC, we have:
450 - 6q + q * (-6) = 0.004 * 3q^2 + 40
Simplifying the equation, we get:
450 - 6q - 6q = 0.004 * 3q^2 + 40
450 - 12q = 0.012q^2 + 40
0.012q^2 + 12q - 410 = 0
Using the quadratic formula to solve for q, we find two possible solutions: q ≈ 23.06 and q ≈ -57.06.
Since the quantity cannot be negative in this context, we take the positive solution, q ≈ 23.06.
Rounding this to the nearest whole number, the profit-maximizing output is approximately 23.
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7.
Rewrite the expression (4x² + 5x) ²-5(4x ² + 5x) – 6 as a product of four linear factors.
Answers
Answer:
(x+2)(x+1)(4x+1)(4x-3)
Step-by-step explanation:
(4x² + 5x)² - 5(4x² +5x) - 6
= (4x² + 5x)² - 6(4x² +5x) + 1(4x² +5x) -6
= (4x² + 5x)(4x² +5x -6) +1 (4x² +5x -6)
= (4x²+5x-6)(4x²+5x+1)
= (4x² + 8x - 3x - 6)(4x² + 4x + x +1)
= (4x(x + 2) - 3(x + 2))(4x(x+1) +1(x+1))
= (x+2)(4x-3)(x+1)(4x+1)
Rocky Mountain Tire Center sells 10,000 go-cart tires per year. The ordering cost for each order is $35, and the holding cost is 40% of the purchase price of the tires per year. The purchase price is $25 per tire if fewer than 200 tires are ordered,$17 per tire if 200 or more, but fewer than 8,000, tires are ordered, and $13 per tire if 8,000 or more tires are ordered.
a) How many tires should Rocky Mountain order each time it places an order?
b) What is the total cost of this policy?
Answers
a) Rocky Mountain should order 200 tires each time it places an order.
b) The total cost of this policy is $17,160.
a) To determine how many tires Rocky Mountain should order each time, we need to consider the different price levels and find the point where it is most cost-effective to order. Let's analyze the three price levels:
If fewer than 200 tires are ordered: The purchase price is $25 per tire.
If 200 or more, but fewer than 8,000 tires are ordered: The purchase price is $17 per tire.
If 8,000 or more tires are ordered: The purchase price is $13 per tire.
Since the ordering cost is $35 per order, it is most cost-effective to order the maximum quantity that falls within the second price level, which is 200 tires.
b) To calculate the total cost of this policy, we need to consider the ordering cost and the holding cost. The holding cost is 40% of the purchase price per tire per year. Let's calculate the total cost:
Total holding cost = (Purchase price per tire * Quantity ordered * Holding cost rate) / 2 = (($17 * 10,000 * 0.4) / 2) + (($13 * 2,000 * 0.4) / 2) = $34,000 + $5,200 = $39,200
Total cost = Total ordering cost + Total holding cost = (Ordering cost per order * Number of orders) + Total holding cost = ($35 * (10,000 / 200)) + $39,200 = $1,750 + $39,200 = $40,950
Therefore, the total cost of this policy is $40,950.
Rocky Mountain Tire Center should order 200 tires each time it places an order, resulting in a total cost of $40,950 for this policy. This ordering quantity and cost analysis allows Rocky Mountain to make efficient and cost-effective decisions in managing their inventory.
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how many prime numbers are there between 1 and 100
Answers
There are 25 prime numbers between 1 and 100. There are 25 prime numbers between 1 and 100. Prime numbers are important in mathematics and have numerous applications, including cryptography, number theory, and computer science.
To find the prime numbers between 1 and 100, we can check each number in that range to see if it is divisible by any number other than 1 and itself. A more efficient approach is to use the Sieve of Eratosthenes algorithm. Using this method, we can eliminate multiples of each prime number starting from 2. The remaining numbers after the elimination process are prime.
Using this algorithm, we find that the prime numbers between 1 and 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
There are 25 prime numbers between 1 and 100. Prime numbers are important in mathematics and have numerous applications, including cryptography, number theory, and computer science.
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ASAP 25 POINTS gordanden Greenthumbs likes to grow huge plants. He once grew a zucchini that started out at 6 inches, then grew & inches to re 15 inches on the second day. Each day the zucchini grew by the same amount as the day before plus another 3 inches. On which day would the zucchini be longer than 11 fest? Show all worke
Answers
The Zucchini will be longer than 11 feet on the ninth day.
How did the plant grow?Since 12 inches is equal to 1 foot, then 11 feet will be equal to 132 inches. Below, we are going to describe the rate of the plant's growth from day 1.
Day 1 = 6 inches
Day 2 = 6 + 9 inches = 15 inches
Day 3 = 15 + 9 + 3 = 27 inches
Day 4 = 27 + 12 + 3 = 42
Day 5 = 42 +15 + 3 = 60
Day 6 = 60 + 18 + 3 = 81
Day 7 = 81 + 21 + 3 = 105
Day 8 = 105 + 24 + 3 = 132
Day 9 = 132 + 27 + 3 = 162
We are told that on the second day, the tree grew up to 15 inches and subsequently, it grew by the same amount as the day before plus another three inches.
Going by that calculation, on the third day, the tree will grow by 9 inches plus three more inches and this continues. Finally, on the ninth day, the tree would have grown up to 162 inches which is greater than 132 inches or 11 feet.
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5
Write the equation of the line in slope intercept form given the following:
Slope
intercept (0, -8)
1
5
(Level 2)
Answers
Answer:
2388732er
Step-by-step explanation:
A music store sold 103 CDs and 102 CD players. If each CD costs $12
and each CD player costs $35, what was the store's total earnings?
$36,200
$15,500
$24,500
$47,000
Answers
Answer:
Bro this is all multiplication and addition involved
so
If 103Cd was 12$ so we get =
103*12 =1236$
And CD player =102 =102*35=3570$
So the Total is 4806$
Thanks in Advance
Nahom Wondale
Two parallel wires are separated by 0. 06 m, each carrying 3 a of current in the same direction. What is the magnitude of the force per unit length between the wires?
Answers
The magnitude of the force per unit length between the wires is \(3 \times 10^{-5} N/m\).
What is the magnitude of the force?In a wire carrying a current I 2, the force that each wire feels due to the presence of the other depends on the distance between them and the magnitude of the currents.
Force per unit length = magnetic permeability * (current 1) * (current 2) / (2 π distance between the wires).
The below expression for the force per unit length.
Moreover, before using the below formula we have to change the unit centimeter into a meter. So, we just divide the centimeter by 100.
\(\rm \dfrac{Force}{length}=\dfrac{u_0 \times i _1 \times i_2}{2\pi d}\\\\\dfrac{Force}{length}=\dfrac{u_0 \times i _1 \times i_2}{2\pi d}\\\\Where; \ \mu_0 = 4\pi \times 10^{-7} , i_13 . \ i_2 = 3 \ d=0.06\)
Substitute all the values in the formula
\(\rm\dfrac{Force}{length}=\dfrac{u_0 \times i _1 \times i_2}{2\pi d}\\\\Where; \ \mu_0 = 4\pi \times 10^{-7} , i_13 . \ i_2 = 3 \ d=0.06\\\\ \dfrac{Force}{length}=\dfrac{4\pi \times 10^{-7} \times 3 \times3}{2\pi \times 0.06}\\\\ \dfrac{Force}{length}= 3 \times 10^{-5} N/m\)
Hence, the magnitude of the force per unit length between the wires is \(3 \times 10^{-5} N/m\).
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Help a brother out pls
Answers
Answer:
C
Step-by-step explanation:
here for the points and bc no one else wanted to post the answer.
On saturday ajani sold 2 dozen donuts for every 5 dozen cookies. if he sold a total of 84 dozen itmes, how many dozen were donuts?
Answers
Answer:
24 = no. of dozens of donuts
60 = no. of dozens of cookies
Step-by-step explanation:
The ratio of no. of dozens of donuts to no. of dozens of cookies is 2 to 5
Let 2x = no. of dozens of donuts
and 5x = no. of dozens of cookies
Then 2x + 5x = 84
7x = 84
x = 12
So, 2x = 2(12) = 24 = no. of dozens of donuts
and 5x = 5(12) = 60 = no. of dozens of cookies
Check: 24/60 = 2/5 and 24 + 60 = 84
what best describes the relationship between the lines with equations y + 3x = 10 and 2y = -6x + 4?
Answers
Which statement represents the equation 4 x 2 = 8?
O 4 is 8 times as many as 2.
4 times as many as 8 is 2.
O 8 is 4 times as many as 2.
O 8 times as many as 2 is 4.
Answers
Answer:
2×4=8
Step-by-step explanation:
you take two and multiply by 4
By selling an exhaust fan for Rs 1470,Mr Ekaraj gains 1/6 of its cost price. find its cost price.
Answers
sorry ur on your own
A cheetah traveling at 31 m/s chased its prey for 11 seconds. What distance did the cheetah travel during the chase?
Answers
Answer:
341 meters
Step-by-step explanation:
you just multiply 31 by 11.
Answer:
341 meters
Step-by-step explanation:
Jusr do 31 times 11
J PLS HELP 8 MINUTES LEFT Ted Thumper, who bats at number eight, had an average (mean) of 14 last year. So far this year he has had these scores.
20,0,14,19,26,12,14,23,7
What is Jed's average this season so far?
What are the total runs Jed would need to get in his next three innings to get his average up to 16?
Answers
Answer:
answer is 15 and second one i dont know yet sorry am doing scoo
Step-by-step explanation:
What is the solution for the following equation 2(y+3) squared
Answers
Answer:
2y² + 12y + 18
Step-by-step explanation:
2(y + 3)²
First use formula (a + b)² = a² + 2ab + b² to expand (y + 3)²:
(a + b)²
(y + 3)²
y² + 2*y*3 + 3²
y² + 6y + 9
Dont forget the 2:
2(y² + 6y + 9) <== distribute and multiply
2y² + 12y + 18
Hope this helps!
A group of 300 students were divided into small groups based on their year of birth.
• One-fourth of the students were born in the year 2000
• The number of students who were born in 1999 is twice the number of students who were born in
2000.
• The number of students who were born in 2001 is one-third of the number of students who were
born in 1999.
• The number of students who were born in 1998 is half of the number of students who were born in
2001.
Answers
Answer:
Students born in 2000= 300*(1/4)=75 students
Students born in 1999= 75*2=150 students
Students born in 2001=150*(1/3) =50 students
Students born in 1998=50*(1/2) =25 students
Which of the following is written in a scientific notation?
Answers
Answer:
Its either A or D
Step-by-step explanation:
Answer:
option a
explanation:
its the correct answer because if you know what a scientific notation it would help you a lot with problems like this one.
Does a school has provided a teacher with a budget to purchase math notebooks hacker rank solution?
Answers
A teacher may receive funding from the school to buy arithmetic amount notebooks for the pupils in their class. Materials like notebooks, pens, calculators, and other essential items can be purchased with this money.
A teacher may receive funding from the school to buy arithmetic notebooks for the pupils in their class. Materials like notebooks, pens, calculators, and other essential items can be purchased with this money. It is crucial to provide each student with enough math notebooks so that they have enough supplies to work with. These notebooks can be used by the students to keep track of key information, practise equations and other mathematical ideas, and stay organised and study-focused. To make sure that kids have the resources they need to excel in the classroom, provide the instructor a budget to buy math notebooks.
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The school park has a circular pond with a diameter of 8 yards. What is the pond's droumference? Use 3.14 for x. If necessary, round your answer to the nearest hundredth
Answers
Answer:
The circumference is 8π yards, or about 25.13 yards. Since we need to use 3.14 for π, the circumference of the pond is about 8 × 3.14 = 25.12 yards.
Factoring without Combining Like Terms
Try it
Complete the steps to factor 2x2 + 6x + 5x + 15 by grouping.
2x2 + 6x + 5x + 15
Step 1: Group the first two terms and the second two terms.
(2x2 + 6x) + (5x + 15)
Group 1 Group 2
Step 2: What is the greatest common factor of Group 1?
2
Ox
O 2x
Answers
Answer:
(x + 3)(2x + 5)
Step-by-step explanation:
Given
2x² + 6x + 5x + 15 ← grouping the terms
= (2x² + 6x) + (5x + 15) ← factor each group
= 2x(x + 3) + 5(x + 3) ← factor out (x + 3) from each term
= (x + 3)(2x + 5) ← in factored form
Answer:
EDG 2021
Step-by-step explanation:
Step 2: C. 2x
Step 3: B. 5
Step 4: C. x + 3
Step 5: A. (x + 3) (2x + 5)
12.5 x n = 32
(solve for n plz, thanks!!)
Answers
Answer:
n=2.56
Step-by-step explanation:
A cone has a volume of 21 cubic meters and a radius of 3 meters. What is its height?
Answers
Step-by-step:
The volume of cone= (1/3)* π*radius^2*height
Height of cone= volume/(1/3)* π*radius^2
=21/(1/3)* π*3^2
= 2.23 cm (2 d.p)
Question 1
25 pts
What is the total cost to run the cookie shop when rent is $50.00, insurance is $50.00 and the cost
to make the cookies is $100.00?
O $50.00
O $100.00
O $200.00
O $4.00
Answers
Answer:
200
Step-by-step explanation:
50 for rent and 50 for insurance is 100 plus 100 to make them
y = 130(0.45)x
What is the growth factor, decay factor, or slope?
What is the initial amount?
Blank 1:
Blank 2:
Answers
Answer:
initial amount is 130
decaying at 55%
Step-by-step explanation:
the length of a rectangle is 2cm longer than its width. If the perimeter of the rectangle is 60 cm, find its length and width
Answers
Answer:
Let the Width(W) = X cm
then the length(L)= (2+X)cm
Perimeter(P)= 60cm
P =L+W
L+W =P
(2+X)cm +X cm = 60 cm
2X+2=60
2X=60-2
2X=58
dividing both sides by 2
X=29cm
Width =X
Width =29cm
the width is 29cm
Length= (2+X)cm
= (2+29)cm
= 31cm
the length is 31cm
We will let w represent the width of the rectangle (in centimeters).
We know that the length is 2 cm longer than the width.
So the length is w + 2 .
We can label the rectangle as follows.
The perimeter is given as 60 cm.
So we solve for w as follows.
w + (w + 2) + w + (w + 2) = 60
w + w + 2 + w + w + 2 = 60
4w + 4 = 60
4w = 56
w = 14
Using this value of w, we can find w + 2.
w + 2 = 14 + 2 = 16
ANSWER
length: 16 cm
width: 14 cm
Jessica starts skiing 1.2 miles East of the ski lodge, at an elevation of 1.8 miles above sea level. She finishes 0.3 miles East of the ski lodge, at an elevation of 1.2 miles above sea level.
What is the slope of the ski slope
Answers
Therefore , the slope of the ski slope is 39.197°.
What is slope?In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.
Here ,
Given:
She begins her skiing at an elevation of 1.8 miles above sea level, 1.2 miles east of the ski lodge. At an altitude of 1.2 miles above sea level, she comes to an end 0.3 miles to the east of the ski lodge.
a = 1.2 miles
b = 1.8 miles
x = 1.2 miles
y= a-b = 1.8-1.2 = 0.6 miles
=>y tan A = A1
=> arctan(-) = arctan( 0.6 )
=> 0.68413 radian
A = A1 → degree= 0.68413 * 180 /π = 39.197°
Therefore, the slope of the ski slope is 39.197°.
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How can you find f(2) if f(x) = -3x2 – 7?
Answers
Answer:
-3(2) x 2 -7 = -6x2-7 =-19
Step-by-step explanation:
The value of f(2) for the given function f(x) = -3x² – 7 will be -19.
What is a function?A certain kind of relationship called a function binds inputs to essentially one output.
The machine will only accept specified inputs, described as the function's domain, and will potentially produce one output for each input.
As per the given function,
f(x) = -3x² – 7
Put x = 2
f(2) = -3(2)² - 7
f(2) = -2(4) - 7
f(2) = -8 - 7 = -15
Hence "The value of f(2) for the given function f(x) = -3x² – 7 will be -19".
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How can you identify and write equivalent expressions?
Answers
You can identify equivalent expressions if one of the expressions can be written like the other and it is usually written by combining any like terms on each side of the equation such as x-terms with x-terms and constants with constants.
What is an Equivalent expression?These are the type of expressions that have similar value or worth but do not look the same.
This type of expression can be identified if one of the expressions can be written like the other and example can be seen below:
3(x + 3) and 3x + 9
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