Question 4 Find the general solution of the following differential equation: P+P tant = P4 sec+t dP dt [10]
Answers
The general solution of the given differential equation is P = C sec(t) + 1/(4 tan(t)), where C is a constant.
To find the general solution of the differential equation, we need to solve for P. The given equation is P + P tan(t) = P⁴ sec(t) + t dP/dt.
First, we rearrange the equation to isolate the derivative term:
P⁴ sec(t) + t dP/dt = P + P tan(t)
Next, we separate variables by moving all terms involving P to one side and terms involving t and dP/dt to the other side:
P⁴ sec(t) - P = -P tan(t) - t dP/dt
Now, we can factor out P:
P(P³ sec(t) - 1) = -P tan(t) - t dP/dt
Dividing both sides by (P³ sec(t) - 1), we get:
P = (-P tan(t) - t dP/dt) / (P³ sec(t) - 1)
Simplifying further, we have:
P = -P tan(t) / (P³ sec(t) - 1) - t dP/dt / (P³ sec(t) - 1)
The term (-P tan(t) / (P³ sec(t) - 1)) can be rewritten as 1/(P³ sec(t) - 1) * (-P tan(t)). Integrating both sides with respect to P, we obtain:
∫(1/(P³ sec(t) - 1)) dP = ∫(-t/(P³ sec(t) - 1)) dt
Integrating these expressions leads to the general solution:
ln|P³ sec(t) - 1| = -ln|cos(t)| + C
Simplifying further, we get:
ln|P³ sec(t) - 1| + ln|cos(t)| = C
Combining the logarithms using properties of logarithms, we have:
ln|P³ sec(t) - 1 cos(t)| = C
Exponentiating both sides, we obtain
\(P³ sec(t) - 1 = e^Ccos(t)\)
Finally, rearranging the equation yields the general solution:
\(P = (e^C cos(t) + 1)^(1/3)\)
Letting C = ln|A|, where A is a positive constant, we can rewrite the solution as:
\(P = (A cos(t) + 1)^(1/3)\)
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Related Questions
the multiplying factor to convert peak values to rms values is ____.
Answers
Step-by-step explanation:
.7071 is the value to convert peak to RMS
what are the main differences between symmetric and asymmetric cryptographic algorithms?
Answers
Symmetric and asymmetric cryptography are both used to secure data, with symmetric algorithms being faster and better suited for bulk data encryption while asymmetric algorithms provide an additional layer of security and are better suited for smaller data sets.
Symmetric cryptographic algorithms use the same key to encrypt and decrypt data, while asymmetric cryptographic algorithms use different keys to encrypt and decrypt data. Symmetric algorithms are generally faster than asymmetric algorithms, making them more suitable for bulk data encryption. Additionally, symmetric algorithms are not vulnerable to man-in-the-middle attacks.
Asymmetric algorithms, on the other hand, are more secure than symmetric algorithms since they require two different keys to encrypt and decrypt data. This makes them more difficult to break, as an attacker would need to obtain both keys. Asymmetric algorithms are also more suitable for smaller amounts of data, such as digital signatures, where the security of the data is more important than speed. However, asymmetric algorithms are vulnerable to man-in-the-middle attacks and other types of attacks that target the keys used to encrypt and decrypt data.
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I WILL MARK BRAINLIEST! NO FAKE ANSWERS
A jug contains 3.2 L of water. If a glass has a capacity of 220 ml, how many such glasses can the jug fill?
Answers
Answer:
68.75
Step-by-step explanation:
220 / 3.2 = 68.75
(Not sure if this helps, but I hope it does!)
Find $x$ . A composite figure consisting of a quadrilateral on the left and a right-angled triangle on the right. The sides of the quadrilateral are equal in length, with one of its sides measuring 12 centimeters. Two sides of the triangle are labeled as “x” and “20 centimeters.” $x=$ cm
Answers
The length of the hypotenuse (x) of the right-angle triangle will be 4√34 cm.
Given that:
Length of each side of quadrilateral, P = L = 12 cm
One length of right angle triangle, B = 20 cm
The Pythagoras theorem formula is given as,
H² = P² + B²
x² = 12² + 20²
x² = 144 + 400
x² = 544
x = 4√34
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The missing diagram is given below.
Write an equation in slope-intercept form that describes that data in the table
Answers
From the data points given the linear equation in slope-intercept form is y = -1/2x + 4.
What is an equation?
A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("=").
The first two data points are - (-3,5.5) and (-1,4.5)
The slope-intercept form of the equation is -
y = mx + b
m represents the slope of the linear equation.
To find the value of m use the formula -
(y2 - y1)/(x2 - x1)
Substitute the values into the equation -
(4.5 - 5.5)/[(-1) - (-3)]
Use the arithmetic operation of subtraction -
(-1)/(-1 + 3)
-1 / 2
So, the slope m is m = -1/2
Now, the equation becomes y = -1/2x + b
To find the value of b substitute the values of x and y in the equation -
5.5 = -1/2(-3) + b
5.5 = 3/2 + b
5.5 = 1.5 + b
b = 5.5 - 1.5
b = 4
So, now the equation becomes - y = -1/2x + 4
The graph for the equation is plotted.
Therefore, the equation is y = -1/2x + 4.
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With which set of information can you construct a unique triangle?
OA the measurements of all the angles
ОВ.
the lengths of two sides
OC. the measurements of two angles
OD. the lengths of all the sides
OE the measurement of one angle
Answers
Answer:
D
Step-by-step explanation:
This would be using the SSS.
Which means knowing three sides.
The other options do not relate to any of the SSS, SAS, ASA, RHS
Hope that helped!!! k
In the figure below, AC is tangent to circle B
What is the length of BC?
A) 8mm
B) 4mm
C) 2mm
D) 16mm
Answers
Option A) 8mm is correct. The length of BC is 8mm.
To find the length of side BC in the right-angled triangle ABC, we can use the Pythagorean theorem.
Given:
AB = 17mm
AC = 15mm
BC = ?
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Using the Pythagorean theorem, we have:
AB^2 = AC^2 + BC^2
17^2 =BC^2 + 15^2
289 = BC^2+ 225
BC^2 = 289 - 225
Taking the square root of both sides, we find:
BC = √64
Using a calculator, we can determine that √64 is 8.
BC = 8mm
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Part #1: find the solution of the inequality.\(10 \geqslant p - 4\)Part #2: describe the solution
Answers
Part 2
The solution is described as p is less than or equal or equal to 14
~ P l e a s e ~ h e l p ~ m e ~ o u t ~
Answers
Answer:
I think it's A.)
Step-by-step explanation:
During the early morning hours, customers arrive at a branch post office at an average rate of 69 per hour (Poisson), while clerks can provide services at a rate of 23 per hour. If clerk cost is $29.76 per hour and customer waiting time represents a cost of $32 per hour, how many clerks can be justified on a cost basis a. 7 b. 5 C. 8 d. 6 e. 4
Answers
Based on the given information, a total of 7 clerks can be justified on a cost basis for the branch post office during the early morning hours.it is based on average.
The average customer arrival rate at the branch post office is given as 69 per hour, which follows a Poisson distribution. The clerks, on the other hand, can provide services at a rate of 23 per hour. To determine the number of clerks that can be justified on a cost basis, we need to consider the balance between customer arrival and service rates.
The formula for calculating the number of clerks can be derived using queuing theory. In this case, the formula is:
Number of Clerks = (Customer Arrival Rate / Clerk Service Rate) + 1
Plugging in the given values, we have:
Number of Clerks = (69 / 23) + 1 = 4 + 1 = 5
However, the cost analysis also needs to be considered. Each clerk costs $29.76 per hour, and customer waiting time represents a cost of $32 per hour. To minimize the overall cost, it is necessary to reduce the waiting time. Increasing the number of clerks reduces the waiting time and, consequently, the cost associated with it.
Considering the cost analysis, it is evident that having more clerks is beneficial. Therefore, to justify the cost and minimize waiting time, 7 clerks can be employed at the branch post office during the early morning hours.
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n a given year, there are 10 million unemployed workers and 120 million employed workers in an economy.
Answers
In a given year, an economy has 10 million unemployed workers and 120 million employed workers. This information provides a snapshot of the labor market and indicates the number of individuals who are currently without jobs and those who are employed.
The information states that in the given year, there are 10 million unemployed workers and 120 million employed workers in the economy. This data provides a measure of the labor market situation at a specific point in time.
Unemployed workers refer to individuals who are actively seeking employment but currently do not have a job. The number of unemployed workers can be an important indicator of the health of an economy and its ability to provide job opportunities.
Employed workers, on the other hand, represent individuals who have jobs and are currently working. The number of employed workers indicates the size of the workforce that is actively contributing to the economy through productive activities.
By knowing the number of unemployed and employed workers, policymakers, economists, and analysts can assess factors such as labor market conditions, unemployment rates, and workforce participation rates. This information is crucial for formulating policies, understanding economic dynamics, and monitoring the overall health and functioning of the economy.
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details previous answers scalcet8 14.4.509.xp. my notes practice another find the linear approximation of the function below at the indicated point. f(x, y)
Answers
The linear approximation of the function at the indicated point is as follows: F(x,y) = x -5y - 1
What is linear approximation?The linear approximation of a function is nothing more than estimating the value of the function at a location using a line. When we see a curve (of a function) with a point on it, we recall the idea of the tangent line. If we determine the equation of the tangent line at the given position, the value of the function at any location extremely near to the given point can be estimated using the equation of the tangent line. Since the tangent line is used in this process, the idea is also referred to as the "tangent line approximation" or "linear approximation."
How to solve?
In this case, our goal is to locate the linear approximation of the provided function at the stated location.
We start by replacing the values, then determine the partial derivative with regard to x and finally y.
We now sum these values, having first substituted the x and y values into the differentiated partial equation. The findings are then added up.
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4y - 7 = 5x2 - x + 2 + 3y
Quadratic Equations
Options are No, One, Two.
I don't remember learning how to do it with more than a, b, and c variable. I'm so confused please help.
Answers
Answer:
y = 9 + -1x + 5x2
Step-by-step explanation:
Simplifying
4y + -7 = 5x2 + -1x + 2 + 3y
Reorder the terms:
-7 + 4y = 5x2 + -1x + 2 + 3y
Reorder the terms:
-7 + 4y = 2 + -1x + 5x2 + 3y
Solving
-7 + 4y = 2 + -1x + 5x2 + 3y
Solving for variable 'y'.
Move all terms containing y to the left, all other terms to the right.
Add '-3y' to each side of the equation.
-7 + 4y + -3y = 2 + -1x + 5x2 + 3y + -3y
Combine like terms: 4y + -3y = 1y
-7 + 1y = 2 + -1x + 5x2 + 3y + -3y
Combine like terms: 3y + -3y = 0
-7 + 1y = 2 + -1x + 5x2 + 0
-7 + 1y = 2 + -1x + 5x2
Add '7' to each side of the equation.
-7 + 7 + 1y = 2 + -1x + 7 + 5x2
Combine like terms: -7 + 7 = 0
0 + 1y = 2 + -1x + 7 + 5x2
1y = 2 + -1x + 7 + 5x2
Reorder the terms:
1y = 2 + 7 + -1x + 5x2
Combine like terms: 2 + 7 = 9
1y = 9 + -1x + 5x2
Divide each side by '1'.
y = 9 + -1x + 5x2
Simplifying
y = 9 + -1x + 5x2
what is meaning of rate?
Answers
Answer:
1.a measure, quantity, or frequency, typically one measured against another quantity or measure.
Step-by-step explanation:
2.a fixed price paid or charged for something.
Answer:
a measure, quantity, or frequency typically one measured against another quantity or measure.
Assign a standard or value to (something) according to a particular scale.Help me with this question ASAP!!!
Answers
Answer:
Part one:To determine the amount of time it will take for the two pools to have the same amount of water, you can set up the following equation:
19.5 liters/minute * t = 43.75 liters/minute * t
where t is the number of minutes it will take for the pools to have the same amount of water.
Solving this equation for t, you get:
19.5 liters/minute * t = 43.75 liters/minute * t
19.5 liters/minute = 43.75 liters/minute
t = 43.75 liters/minute / 19.5 liters/minute
t = 2.25 minutes
Therefore, it will take approximately 2.25 minutes for the pools to have the same amount of water.
Part two:
first pool will have approximately 43.875 liters of water and the second pool will have approximately 98.4375 liters of water when they have the same amount.
tom goes to the state fair with $50. each ride costs $1.50. how much money will he have left after riding n rides
Answers
The amount of money left after riding n rides is:
f(n) = 50 - 1.50n
How much money will he have left after riding n rides?We can model this with a linear equation. We know that Tom starts with a total of 50 dollars, and each game in the state fair has a ride cost of $1.50
So, if he goes to n of these rides, the amount of money that he will have at the end is equal to the initial amount minus n times the cost of a game, we can write this as the linear equation:
f(n) = 50 - 1.50n
Where the units of the function f(n) are in dollars. That is the equation we wante to get.
50 is the y-intercept, the initial amunt.
-1.50 is the slope, the cost per game.
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What is the constant of proportionality for the relationship modeled by the equation y=0.2xy=0.2x
Answers
Answer:
elationship modeled by the equation y=0.2xy=0.2x
Step-by-step explanation:
What is the area of the figure?
Answers
Answer: 27 inches sq.
Step-by-step explanation:
3*3=9
2*9=18
18+9=27
Try It
Given: AD
Prove: DE
D
BC and BCD =
CE
Hint
B
Angles Segments Triangles Statements Reasons
AAS
CPCTC
Statements
✓ 1. AD = BC
✓2. ZBCD =
3. DC DC
4. AADC = ABCD
5. LEDC ZECD
SAS
converse of isosceles triangle thm
Reasons
1. given
2. given
3. reflexive property
4. SAS
5. CPCTC
Answers
The required statements and reasons to prove that DE is equal to CE is explained.
What is a triangle congruence theorem?The triangle congruence theorem is a theorem that can be used to prove that two or more triangles are the same, considering the corresponding properties of the triangles. The properties are length of the sides, and measure of internal angles.
The statements and reasons to prove that DE is equal to CE are explained below using the triangle congruence theorem.
STATEMENT REASON
1. AD = BC Given
2. <BCD = <ADC Given
3. DC = DC Reflexive property
4. ΔADC ≅ ΔBCD SAS
5. <EDC ≅ <ECD CPCTC
6. AC = BD Definition of diagonal
7. DE = CE Congruent sides of isosceles triangle
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I need help with this question for Geometry.
Answers
The measure of side FD is given as follows:
FD = 81.5.
What are similar triangles?Similar triangles are triangles that share these two features presented as follows:
Congruent angle measures.Proportional side lengths.Hence the proportional relationship measuring the side lengths is given as follows:
FD/23 = 39/11.
Applying cross multiplication, the measure of side FD is obtained as follows:
11FD = 39 x 23
FD = 39 x 23/11
FD = 81.5. (rounded to the nearest tenth).
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if you double the length and width of a rectangle, how much does area of the rectangle change?
Answers
Answer:
The area changes by 4 times. Or, 4*L*W
This is because when you double the length and width, you are simply creating 3 more copies of that rectangle, all inside one rectangle
An ______ is a quadrilateral with four sides that are not regular or the same.
Answers
An irregular quadrilateral, also known as an arbitrary quadrilateral, is a polygon with four sides that are not equal in length or congruent in any way.
Unlike regular quadrilaterals such as squares or rectangles, which possess congruent sides and angles, an irregular quadrilateral lacks these symmetries. Consequently, its shape can vary greatly, allowing for a wide range of possibilities. Examples of irregular quadrilaterals include trapezoids, parallelograms, and kites.
These quadrilaterals possess different combinations of side lengths and angles, resulting in unique shapes and characteristics. Due to their lack of regularity, irregular quadrilaterals do not conform to specific formulas or properties like regular polygons. Their diverse nature makes them a fascinating subject of study in geometry, as they exhibit a rich variety of properties and behaviors.
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Using the Commutative Property
Use the commutative property to create equivalent expressions. Which expressions are equivalent to 227+35+9.82
Check all that apply.
3.5 +2.21 +9.8
0 3+2t +9
D 2.2 +3.5+ 9.81
9.8 +3.5+ 2.21
D 2.2t+ 9.8 +3.5
D 2.2t + 35.98
Answers
Answer:
3.5 +2.2t +9.8 and 9.8 + 3.5 + 2.2t and 2.2t + 9.8 + 3.5
Answer:
Option 1, Option 4, and Option 5.
Step-by-step explanation:
So it's 3.5 + 2.2t + 9.8 , 9.8 + 3.5 + 2.2t , 2.2t + 9.8 + 3.5. :)
On the last algebra test M students scored x on the test and N students
scored y. What does the following expression represent?
Mc+Ny
M+N
Answers
Answer: Y=35x
Step-by-step explanation:
Answer:
Step-by-step explanation:
Mx + Ny represents the total score of M + N students.
How does algebra affect how people can learn from each other?
Answers
Answer:
The study of algebra helps in logical thinking and enables a person to break down a problem first and then find its solution. Although you might not see theoretical algebraic problems on a daily basis, the exposure to algebraic equations and problems at some point in life will train your mind to think logically.
MODELING REAL LIFE A softball coach buys equipment for two teams. The coach buys 4 bats, 36 softballs, and 12 uniforms for Team A and 6 bats, 45 softballs, and 15 uniforms for Team B . Each bat costs $80, each softball costs $4, and each uniform costs $20. Use matrix multiplication to find the total cost of the equipment for each team.
Answers
Using proportions, it is found that the total costs of equipment for each team are given as follows:
Team A: $704.Team B: $960.What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.
The costs are given as follows:
Bat: $80.Softball: $4.Uniform: $20.Considering the amounts of bats, softballs and uniforms purchased by Team A, their total cost is given by:
4 x 80 + 36 x 4 + 12 x 20 = $704.
Considering the amounts of bats, softballs and uniforms purchased by Team B, their total cost is given by:
6 x 80 + 45 x 4 + 15 x 20 = $960.
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Can I get help with this?
Answers
Answer:
AC = 120
Step-by-step explanation:
<D = 360 - 95 - 145
D =120
Central Angle = Intercepted Arc
AC = <D = 120
an object is located a distance do = 5.7 cm in front of a concave mirror with a radius of curvature r = 19.1 cm.
33% Part (a) Write an expression for the image distance, di Grade Summary Deductions Potential 0% 100% Submissions ts remaining:5 (5% per attempt) detailed view DELI CLEAR Submit Hint I give up! Hints: 2% deducti on per hint. Hints remaining: 2 Feedback: 2% deduction per feedback. là 33% Part (b) Numerically, what is the image distance, ai, in centimeters? 33% Part (c) Is this a real or virtual image?
Answers
(a) The expression for the image distance, di, for a concave mirror can be found using the mirror equation:
1/do + 1/di = 1/f
where do is the object distance, f is the focal length, and di is the image distance.
To find f, we can use the formula:
f = r/2
where r is the radius of curvature.
Substituting the given values, we get:
f = 19.1/2 = 9.55 cm
do = 5.7 cm
Now, we can solve for di:
1/5.7 + 1/di = 1/9.55
1/di = 1/9.55 - 1/5.7
di = -16.13 cm
Note: the negative sign for di indicates that the image is virtual and upright.
(b) Numerically, the image distance, di, is -16.13 cm.
(c) This is a virtual image.
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84 km a centímetros
Alguien sabe como se resuelve
Answers
8400000cm
84 kilometer is 8400000 centimeter.
How to convert kilometer to centimeter?In the International System of Units, a Centimeter or centimeter is a unit of length equal to one hundredth of a meter, with centi being the SI prefix for a factor of 1/100.A centimeter is a metric length measurement unit. One centimeter equals ten millimeters or one meter. A cheerio is typically one centimeter in diameter.One kilometer equals 1000 meters. When we need to travel from one location to another, we use kilometers to calculate the distance. Kilometers can be used to calculate the distance between cities or the distance traveled by a plane.
Here given 84 km ,
We know that 1 km = 1000 meter
so 84 km = 84 x 1000 meter
= 84000 meter
and 1 meter = 100 cm
so here 84000 meter = 84000 x 100 cm
= 8,400,000 Centimeters
∴ 84 km = 8400000cm
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what are the minimum and maximum temperatures in the house ?
Answers
Answer:
I mean, I think it varies by house and by location but a general range would be:
Min: 64 *F
Max: 74 *F