This is a separable differential equation, which means we can separate the variables y and x on either side of the equation. To do this, we'll move all the terms involving y to one side and all the terms involving x to the other side.
dy/dx = (1+y²)e^x
We'll divide both sides by (1+y²)e^x:
dy/(1+y²)e^x = dx
Now we'll integrate both sides with respect to their respective variables. The integral of dy/(1+y²) with respect to y is the inverse tangent (tan^-1) function, so we'll use that on the left side:
∫ dy/(1+y²) = ∫dx + C
tan^-1(y) = x + C
On the right side, we'll integrate e^x with respect to x, which gives us e^x. So we have:
tan^-1(y) = e^x + C
We can solve for y by reversing the tan^-1 function:
y = tan(e^x + C)
Where C is an arbitrary constant of integration.
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42. Let f be a function from the set A to the set B. Let S and T be subsets of A. Show that
a) f(S ∪ T)=f(S) ∪ f(T).
b) f(S ∩ T) ⊆ f(S) ∩ f(T).
Start with any random elements of sets and compute LHS and RHS individually.
a) The proof of f(S ∪ T) = f(S) ∪ f(T) is explained below.
b) The proof of f(S ∩ T) ⊆ f(S) ∩ f(T) is expliane dbelow.
What are sets and subsets?A set is a collection of well-defined objects.
A subset contains all the elements or a few elements of the given set.
The improper subset is when it contains all the elements of the given set and the proper subset is when it doesn't contain all the elements of the given set.
Let, The set A = {1, 2, 3, 4, 5} and B = {2, 4, 5}.
Therefore, S = {1, 2, 3} and T = {1, 4, 5}.
Now, f(S ∪ T) = {1, 2, 3, 4, 5} and f(S) ∪ f(T) = {1, 2, 3} ∪ {1, 4, 5}.
Hence, f(S ∪ T)=f(S) ∪ f(T).
Now, f(S ∩ T) = {1} and f(S) ∩ f(T) = {1, 2, 3} ∩ {1, 4, 5} = {1}.
Therefore, f(S ∩ T) ⊆ f(S) ∩ f(T). (⊆ implies improper subset)
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Aaron tracks the time it takes him to mow lawns by writing coordinate points relating x, the time in hours it takes to mow a lawn, and y, the area of land mowed in acres. Two of his points are (3, 1.5) and (5, 2.5). Which statement describes the slope of the line through these two points?
Answer:
The slope of the line through these two points is 0.5, which means that for each hour moving a lawn, 0.5 more lands of area are mowed.
Step-by-step explanation:
Slope of a line:
Suppose we have two points in a line. The slope is the change in y divided by the change in x. It states how much y changes when x changes by 1.
In this question:
x: Time in hours to move a lawn.
y: Area of land.
Two of his points are (3, 1.5) and (5, 2.5).
Change in y: 2.5 - 1.5 = 1
Change in x: 5 - 3 = 2
Slope: 1/2 = 0.5
The slope of the line through these two points is 0.5, which means that for each hour moving a lawn, 0.5 more lands of area are mowed.
Chen tosses a number cube 19 fewer times than Jayden. Chen tosses a number cube 38 times. How many times does Jayden toss a number cube?
Use partial quotients to solve 5,140 ÷ 9 = ___.
The correct answer is 571 quotient and 1 is remainder.We can solve easily by partial quotients.
What are partial quotients division?Partial quotients division is a deviation from the standard method. The divisor is multiplied with a number and the multiple obtained is deducted from the dividend. This multiple of the divisor is as close as possible to the dividend, that is less than or equal to the dividend.
Why do we use partial quotient division?Partial quotient division provides natural differentiation because we can have students solve at a level they are comfortable with and then we can challenge them to find a more efficient way. And this means using fewer steps and solve easily.
Now we solve
first 9 multiply with 500 so we get 4500. 5140-4500=640
then 9 multiply with 70 so we get 630. 640-630=10
then 9 multiply with 1 so we get 9. 10-9=1
Hence 500+70+1=571
And our remainder is 1.
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A parallelogram has two
adjacent angles that are
supplementary. One
angle is 55° less than the
other. What is the
measure of the smaller
angle?
Answer:
x + (x + 55) = 180
2x =125
The angle is 62.5 degrees
Double-Check
62.5 + (62.5 + 55) = 180
125 + 55 = 180
62.5 is the smaller angle
Step-by-step explanation:
Answer:
62.5
Step-by-step explanation:
evaluate - [tex]6^{2}- 2(5+1+3[/tex]
The expression 6² - 2(5 + 1 + 3) after evaluation gives 18.
What is BODMAS rule?BODMAS rule is rule for operation of numbers in a specific order when more than one type of operations is involved.
BODMAS is the abbreviated form for Brackets, Order including powers and exponents, Division, Multiplication, Addition and Subtraction.
Given expression is, 6² - 2(5 + 1 + 3).
First we should do the operation in brackets.
6² - 2(5 + 1 + 3) = 6² - 2 (9)
Now we should do the power operations.
6² - 2 (9) = 36 - 2 × 9
Now we do multiplication.
36 - 2 × 9 = 36 - 18
Now we do subtraction.
36 - 18 = 18
Hence the answer of the expression given is 18.
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I think its either B or D but im not sure
Answer:
Calculate the number of whole years it takes for the population to decrease from 2180 to less than 1000Step-by-step explanation:
Answer:
D
Step-by-step explanation:
Every day, Eve runs 12 mile more than the day before. Which equation can be used to discover how many miles Eve runs on x days?
y=x+12y is equal to x plus 1 half
y=12xy is equal to 1 half x
y=5xy is equal to 5 x
y=5+xy is equal to 5 plus x
Answer:
B) y=1/2x
Step-by-step explanation:
y=1/2x
Help pls pls help pls pls help
It's 8, it gave Y so pretty easy from there on
y = 4
y = 1/4 x + 2
______________
4 = 1/4 x + 2
4 - 2 = 1/4 x + 2 - 2
2 = 1/4 x
2 × 4 = 4 × ( 1/4 ) × x
x = 8
Thus the correctly answer is option D.
Find the GCF of the list of terms.
3x^2,6x^6 and -12x^3
Answer:
3x²
Step-by-step explanation:
3x²
They would all reduce to 1, 2x^4, and -4x
A paint store makes lime green paint by mixing 3 parts yellow paint to 1 part blue paint. A clerk mixès 5 parts yellow paint to 2 parts blue paint. did the clerk use the correct ratio of yellow paint to blue paint? Explain.
Answer:
No
Step-by-step explanation:
The ratio used should have been 6 parts yellow paint to 2 parts blue paint
hope this helps
I need help with this, I need to simplify but I don't know how
Answer: solve the same variables
Step-by-step explanation:
1. 3y + 4.1
2. 4x +
3. 5x + 22½
4. 4y²
5. + 25
6. 0
Find perimeter
Simplify answer completely
Answer:
[tex]perimeter = 14 \frac{3}{4} + 4 \frac{1}{2} + 8 \frac{1}{2} + 10 \frac{2}{5} + 10 \frac{2}{5} + 12 \frac{4}{5} \\ = \frac{59}{4} + \frac{9}{2} + \frac{17}{2} + \frac{52}{5} + \frac{52}{5} + \frac{64}{5} \\ = 61 \frac{7}{20} \: units[/tex]
Write an equation and solve for b using the diagram
below. YOU MUST WRITE THE EQUATION IN
ORDER TO GET CREDIT.
(3b + 18)°
93°
Step-by-step explanation:
hope this is right hehe.....
Please help me with this
9514 1404 393
Answer:
3025
Step-by-step explanation:
We observe that increasing x by 1 from 0.5 to 1.5 causes the function value to be multiplied by 55 (from 1 to 55).
So, increasing the x-value by 1 again, from 1.5 to 2.5, will again multiply the function value by 55.
f(2.5) = 55·f(1.5) = 55^2
f(2.5) = 3025
The next model of a sports car will cost 12.8% more than the current model. The current model costs $42,000. How much will the price increase
in dollars? What will be the price of the next model?
Answer:
$5,376
$47,376
Step-by-step explanation:
Given :
Price of current model = $42,000
Percentage increase in price = 12.8%
Price increase in dollars :
12.8% of $42000
0.128 * $42000
= $5,376
Price of the next model :
Price of current model + increase in price
$42000 + $5376
= $47,376
Answer please for brainliest
Answer:
the answer is 0.7 by the way!
Step-by-step explanation:
This is talking about the distance from zero sooo...
which of the following is the solution to the equation 3^X+9=81^X
a) x=-6
b)x=6
c)x=-3
d)x=3
9514 1404 393
Answer:
d) x = 3
Step-by-step explanation:
The given equation resolves to a quartic equation in (3^x). It has a solution near ...
x = 0.541770946714
__
Perhaps you want the solution to ...
3^(x+9) = 81^(x)
Rewriting in powers of 3, this is ...
3^(x+9) = (3^4)^x = 3^(4x)
Taking logarithms base 3 gives ...
x +9 = 4x
9 = 3x . . . . . . subtract x
3 = x . . . . . . . divide by 3 . . . . matches choice D
_____
Additional comment
The Order of Operations requires that exponential terms be evaluated before addition and subtraction. That means (3^x) must be evaluated before the sum (3^x) + 9. If you want the sum of x and 9 to be evaluated first, it must be in parentheses: 3^(x+9).
In typeset equations, the superscript font serves to group parts of the exponent. In plain text, a grouping symbol (parentheses) must be used.
150, 60, 24,...
Find the 9th term.
y=x-4 and y=4x+2 graph solution of the systems
Answer:
(-2,-6)
Step-by-step explanation:
we can solve this system of equations with the substitution method because y is already by itself.
since they are both equal to y, we can set both equations equal to each other.
4x+2=x-4
subtract 2 from both sides
4x=x-6
subtract x from both sides
3x=-6
divide both sides by 3
x = -2
now that we know x, we can plug this value into one of the original equations and solve for y.
y=-2-4
y=-6
so on a graph, these two equations will intersect at the point (-2,-6)
A theater has 20 rows of seats. If there are 4 seats in the 1st row 12 in the 2nd row, 20 in the 3rd row . How many seats are there in total? Show and explain all work.
Answer:
164 seats
Step-by-step explanation:
you make the equation 4+ 8X to deteremine the seats and x equals the rows and you get your answer.
An arithmetic sequence is a sequence where each consecutive term has a common constant difference. The total number of seats that the theatre has is 156 seats.
What is the sum of terms of an arithmetic sequence?An arithmetic sequence is a sequence where each consecutive term has a common constant difference.
An arithmetic sequence, therefore, is defined by two parameters, viz. the starting term and the common difference.
Let the starting term be 'a' and the common difference be 'd', then we get the arithmetic sequence as:
a, a+d, a+2d, .....
The sum of those 'n' terms is:
[tex]\rm a + (a+d) + (a+2d) + \cdots + (a+(n-1)d) = \dfrac{n}{2}[2a + (n-1)d ][/tex]
The condition is a case of an arithmetic progression. Therefore, the common difference in the progression is of 8 and the first term of the sequence is 4. Therefore, the sum of the series for the first 20 terms will be,
aₙ = 4 + (20-1)8
= 4 + (19)8
= 4 + 152
= 156
Hence, the total number of seats that the theatre has is 156 seats.
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You are given 2 parallel lines on the coordinate plane. You are told to translate x + 3,y-2, rotate 180° clockwise about the origin, then reflect over the y-axis. After the transformations you will have what type of lines?
A Intersecting lines
B. The Same Line
C. Parallel Lines
Since matching angles are formed and the equal corners are congruent when a line joins two parallel lines, one of the choices isc=d.
Parallel lines are what?Geometry refers to parallel lines as beams, straight lines that never overlap. Parallel planes are those that never cross each other while occupying the same three-dimensional space. Curves which are equal with each other and keep a predetermined minimum separation from one another must not overlap or intersect. Parallel lines on a plane are those that always have the same distance between them. A parallel line cannot be crossed. Perpendicular lines are those that intersect directly at a 90 degree angle.
Here,
Given : vertical angles, A = d also represents a solution.
B+d=180 as a result of b+c=180 therefore
B+d=180 as a result of b+c=180 therefore Since a=d, c=d, and b+d=180, the answer is B,C,E.
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Simplify the following expression: -16 -8/2 + 25 ÷ 5 + 1
A. -18
B. 6
C. 10
D. -6
Solve the inequality: −2x > −6
Answer:
x < 3
Step-by-step explanation:
Each statement describes a transformation of the graph of f(x)=3/x. which statement correctly describes the graph of y= f(x- 7) + 3?
A It is the graph of f translated 3 units up and 7 units to the left.
B. It is the graph of f translated 7 units down and 3 units to the right.
C It is the graph of f translated 7 units up and 3 units to the
right!
D. It is the graph of f translated 3 units up and 7 units to the right.
Answer:
Pretty Sure its D!
Step-by-step explanation:
Answer this! I WILL GIVE BRAINLIEST TO PERSON WHO GIVES CORRECT ANSWER!!!!!!!!!
Answer:
An 18-month loan with a 4.75% annual simple interest rate
Step-by-step explanation:
have a good day/night
may i please have a branlliest
To determine the inverse of function f,
change f(x) to y, switch
and y,
and solve for
The resulting function can be written as
f(x) = (x - 3.
Answer:
To determine the inverse of function f, change f(x) to y, switch x and y, and solve for y.
[tex]\frac{1}{8} (x-4)^3[/tex]
Step-by-step explanation:
f(x) = [tex]\sqrt[3]{8x} +4[/tex]
Change f(x) to y: y = [tex]\sqrt[3]{8x} +4[/tex]
Switch x and y: x = [tex]\sqrt[3]{8y} +4[/tex]
Solving for y: x - 4 = [tex]\sqrt[3]{8y}[/tex]
(x-4)^3 = 8y
y = [tex]\frac{1}{8}[/tex][tex](x-4)^3[/tex]
Therefore: inverse of function f = [tex]\frac{1}{8} (x-4)^3[/tex]
The inverse function of f(x) will be f⁻¹(x) = 1/8(x - 4)³.
What is a function?A statement, principle, or policy that creates the link between two variables is known as a function. Functions are found all across mathematics and are required for the creation of complex relationships.
The function f(x) is given below.
f(x) = ∛(8x) + 4
Then the inverse function of f(x) will be
Put x = f⁻¹(x) and f(x) = x. Then we have
x = ∛{8f⁻¹(x)} + 4
∛{8f⁻¹(x)} = x - 4
Cube on both sides, then we have
8f⁻¹(x) = (x - 4)³
f⁻¹(x) = 1/8(x - 4)³
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Get it right I need this bro
Based on past experience, the daily revenue, X, for a coffee shop in a city is a normally distributed random variable with a mean daily revenue of $900. The first co-owner of the shop feels that the shop is known throughout the city and as such they could reduce the amount of money spent on advertising without causing a decrease in the population mean daily revenue. The second co-owner is concerned that a reduction in advertising would in fact result in a decrease in the population mean daily revenue. To settle the issue, the first co-owner decides to decrease the advertising dollars for a period of time, during which she will select 64 days at random and compute the sample mean daily revenue. She will then perform a hypothesis test to determine if the sample mean daily revenue provides significant evidence to conclude that the population mean daily revenue is less than $900 when the advertising dollars are reduced. The first co-owner observes a sample mean daily revenue of $892.50 with a sample standard deviation of $24 when the advertising dollars have been reduced. What is the p-value for her test
Answer:
The p-value for her test is 0.0075.
Step-by-step explanation:
She will then perform a hypothesis test to determine if the sample mean daily revenue provides significant evidence to conclude that the population mean daily revenue is less than $900 when the advertising dollars are reduced.
At the null hypothesis, we test that the mean is less than 900, that is:
[tex]H_0: \mu = 900[/tex]
At the alternate hypothesis, we test that the mean is less than 900, that is:
[tex]H_a: \mu < 900[/tex]
The test statistic is:
We have the standard deviation for the sample, which means that the t-distribution is used to solve this question.
[tex]t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}[/tex]
In which X is the sample mean, [tex]\mu[/tex] is the value tested at the null hypothesis, s is the standard deviation of the sample and n is the size of the sample.
900 is tested at the null hypothesis:
This means that [tex]\mu = 900[/tex]
Sample of 64 days. The first co-owner observes a sample mean daily revenue of $892.50 with a sample standard deviation of $24 when the advertising dollars have been reduced.
This means that [tex]n = 64, X= 892.50, s = 24[/tex]
Value of the test:
[tex]t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}[/tex]
[tex]t = \frac{892.50 - 900}{\frac{24}{\sqrt{64}}}[/tex]
[tex]t = -2.5[/tex]
What is the p-value for her test?
The pvalue of the test is the probability of a sample mean below 892.5, which is a left-tailed test for t = -2.5 with 64 - 1 = 63 degrees of freedom.
Using a calculator, the p-value for her test is 0.0075.
Write 1 6/7 as an improper fraction.
Answer:
13/7
Step-by-step explanation:
multiply the 7 x 1 then add 6. 13 would be the numerator.
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