please help thank you!!

The solution to the given differential equation is [tex]y = 2x + Ce^{-x/2},[/tex]

where C is an arbitrary constant.

We have,

To solve the given first-order linear differential equation:

dy/dx + (1/2)y = 2 + x

We can use the integrating factor method.

The integrating factor (IF) is given by:

[tex]IF = e^{\int{(1/2)}dx}[/tex]

[tex]= e^{x/2}[/tex]

Multiplying both sides of the equation by the integrating factor:

[tex]e^{x/2} \times dy/dx + (1/2)e^{x/2} \times y = (2 + x)e^{x/2}[/tex]

The left-hand side can be simplified using the product rule of differentiation:

[tex]d/dx (e^{x/2} \times y) = (2 + x)e^{x/2}[/tex]

Integrating both sides with respect to x:

[tex]\int{d/dx} (e^{x/2} \times y) dx = \int{(2 + x)e^{x/2}} dx[/tex]

[tex]e^{x/2} \times y = \int{(2e^{x/2} + xe^{x/2}} dx\\= 2\int{e^{x/2}} dx + \int{xe^{x/2}} dx[/tex]

Using the integration rules, we find:

[tex]e^{x/2} \times y = 4e^{x/2} + 2xe^{x/2} - 4e^{x/2} + C\\= 2xe^{x/2} + C[/tex]

Dividing both sides by [tex]e^{x/2}:[/tex]

[tex]y = 2x + Ce^{-x/2}[/tex]

Therefore,

The solution to the given differential equation is [tex]y = 2x + Ce^{-x/2},[/tex]

where C is an arbitrary constant.

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Answer:

x = 3; y = 4

Step-by-step explanation:

adilenechapa22 wrote the correct answer first so they deserve the Brainiest for the question.  However, I can help you understand how we get to the answer.

Step 1:  Isolate y in 4x + y = 16 by subtracting 4x from both sides to prepare for the substitution:

(4x + y = 16) - 4x

y = -4x + 16

Step 2:  Substitute this equation for y in the first equation to first solve for x:

6x + 5(-4x + 16) = 38

6x - 20x +80 = 38

-14x + 80 = 38

-14x = -42

x = 3

Step 3:  Plug in 3 for x in 4x + y = 16 to solve for y:

4(3) + y = 16

12 + y = 16

y = 4

Optional Step 4:  We can check that we've found the correct solutions by plugging in 3 for x and 4 for y in both equations in the system and seeing that we get 38 and 16:

Checking solutions in first equation:

6(3) + 5(4) = 38

18 + 20 = 38

38 = 38

Checking solutions in second equation:

4(3) + 4 = 16

12 + 4 = 16

16 = 16

Answer:

x = 31°

Step-by-step explanation:

the sum of the 3 angles in a triangle = 180° , that is

x + 54° + 95° = 180°

x + 149° = 180° ( subtract 149° from both sides )

x = 31°

Hello !

(9 + x)/2 = (7 + 3 + x)/3

3(9 + x) =2(7 + 3 + x)

27 + 3x = 14 + 6 + 2x

27 + 3x = 20 + 2x

27 - 20 = 2x - 3x

7 = -x

x = -7

you can check it works for the 2 averages

Apologies for the confusion. The image you provided contains an expression that can be simplified as follows:

√(36 / 12) - 3 × (4 - 3)

Let's break it down step by step:

Simplify the expression inside the square root:

√(36 / 12) = √3

Simplify the multiplication:

-3 × (4 - 3) = -3 × 1 = -3

Subtract the results:

√3 - 3

Therefore, the simplified expression is √3 - 3.


<3

Step-by-step explanation:

y = 15 + x      or   x = y-15

and  x         sum to <= 85

x + y <= 85

(y-15)   + y <= 85

2y <= 100

y <= 50             The maximum value of y is 50   minimum would be 16 ( since it must be 15 greater than a positive number x (which would be 1 ))

Saving for a down payment was worth it for Isabel. It reduces the total amount paid for the car by lowering the loan amount, resulting in less interest paid over time.

When evaluating whether saving for a down payment on a car was worth it for Isabel, we need to consider the difference in the total amount paid for the car in two scenarios: with a down payment and without a down payment. Let's analyze the two scenarios to make a judgment.

Scenario 1: With a down payment

In this scenario, Isabel saved up and made a down payment on the car. As a result, her monthly loan payment fits within her budget. By making a down payment, she reduces the principal amount borrowed and, subsequently, the interest charged on the loan.

The benefits of saving for a down payment include:

Lower loan amount: By making a down payment, Isabel reduces the total amount borrowed from the lender.

Reduced interest: With a lower loan amount, the interest charged on the loan is also reduced.

Shorter loan term: A down payment can help Isabel secure a shorter loan term, reducing the overall interest paid over time.

Scenario 2: Without a down payment

In this scenario, Isabel does not make a down payment and finances the entire purchase price of the car. As a result, her monthly loan payment may be higher, as she is borrowing a larger sum and potentially paying more interest over the loan term.

The drawbacks of not saving for a down payment include:

Higher loan amount: Without a down payment, the entire purchase price is financed, leading to a higher loan amount.

Increased interest: With a higher loan amount, more interest may accumulate over the loan term, resulting in a higher total cost of the car.

By comparing the two scenarios, we can assess the impact on the total amount paid for the car. Generally, having a down payment reduces the loan amount, decreases interest charges, and may result in a shorter loan term. This, in turn, can lead to substantial savings in the long run.

To determine the exact difference in the total amount paid for the car in each scenario, we would need specific details such as the purchase price of the car, the interest rate, and the loan term. With these figures, we could calculate the total cost of the car, including interest, in both scenarios and compare the results. Based on this analysis, we can then determine if saving for a down payment was worth it for Isabel.

In conclusion, saving for a down payment on a car can offer financial benefits such as lower loan amounts, reduced interest charges, and potentially shorter loan terms. Considering the potential cost savings and improved financial stability, it is generally advisable for Isabel to save for a down payment, making it worth the effort and consideration.

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Question

The down payment makes the monthly loan payment fit into Isabel's budget. Besides this benefit, do you think saving for a down payment on the car was worth it for Isabel? Use each monthly payment with and without a down payment) to find the difference of the total amount paid for the car in each scenario to justify your answer.

a. The mass of the apple is approximately 33.879 grams.

b. The volume of the apple increases by approximately 1.454 cubic inches during the five-week period.

a. To find the mass of the apple, we need to calculate its volume first. The circumference of the apple can be used to determine its radius.

The formula for the circumference of a circle is C = 2πr,

where C is the circumference and r is the radius.

Rearranging the formula, we have r = C / (2π).

Plugging in the given circumference of 6 inches.

we get r = 6 / (2π) ≈ 0.955 inches.

Now, we need to convert the radius from inches to centimeters since the density is given in grams per cubic centimeter.

Since 1 inch is approximately 2.54 centimeters, the radius in centimeters is [tex]0.955 \times 2.54[/tex]  ≈ 2.427 cm.

Next, we can calculate the volume of the apple using the formula for the volume of a sphere: V = (4/3)πr³.

Plugging in the radius in centimeters, we get

V ≈ (4/3)π(2.427)³ ≈ 73.682 cm³.

Finally, we can find the mass of the apple by multiplying the volume by the density:

mass = volume [tex]\times[/tex] density

= 73.682 cm³ [tex]\times[/tex] 0.46 g/cm³

≈ 33.879 grams.

Therefore, the mass of the apple is approximately 33.879 grams.  

b. The volume of a sphere increases with the cube of its radius.

Since the radius increases by 1/8 inch per week for five weeks, the change in radius would be[tex](1/8) \times 5 = 5/8 inch.[/tex]

Now, let's calculate the change in volume during the five-week period. The formula for the volume of a sphere is V = (4/3)πr³.

Initially, the radius was 0.955 inches.

After five weeks, the radius would be 0.955 + 5/8 inches.

To compare the change in volume, we can calculate the new volume and find the difference:

Change in Volume = New Volume - Initial Volume

Initial Volume = (4/3)π(0.955)³

New Volume = (4/3)π(0.955 + 5/8)³

By subtracting the initial volume from the new volume, we can determine how the volume changes during the five-week period.

Please note that the exact calculation will depend on the precise values used for π and the measurements provided.

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After 15 years, the amount in Hank's savings account, with no additional deposits or withdrawals, will be approximately $1,044.50.

To calculate the amount in Hank's savings account after 15 years with monthly compounding interest, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = the future value of the account

P = the principal amount (initial deposit)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the number of years

In this case:

P = $850

r = 1.2% = 0.012 (as a decimal)

n = 12 (monthly compounding)

t = 15 years

Plugging in the values into the formula, we get:

A = 850(1 + 0.012/12)^(12*15)

A = 850(1.001)^180

A ≈ 850 * 1.228824

A ≈ $1,044.50 (rounded to the nearest cent)

Therefore, after 15 years, the amount in Hank's savings account, with no additional deposits or withdrawals, will be approximately $1,044.50.

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Answer:

This is how you simplify it.

Answer:

Step-by-step explanation:

(3x⁵)^3
= {3x}^(15)     (According to rule)

Answer = (3x)^(15)  

Answer:

Option C. 5.5

Step-by-step explanation:

To solve for x in the equation:

[tex]\sf\implies\dfrac{3x - 4}{14} = \dfrac{9}{10}[/tex]

We need to isolate x on one side of the equation.

First, we simplify the left side by solving the division:

[tex]\sf\implies\dfrac{(3x - 4)}{14} = \dfrac{9}{10}[/tex]

Next, we cross-multiply to eliminate the fractions:

[tex]\sf\implies 10(3x - 4) = 14(9)[/tex]

[tex]\sf\implies 30x - 40 = 126[/tex]

Now, we isolate x by adding 40 to both sides of the equation:

[tex]\sf\implies 30x = 166[/tex]

Finally, we solve for x by dividing both sides of the equation by 30:

[tex]\sf\implies x = \dfrac{166}{30}[/tex]

[tex]\sf\implies x = 5.533[/tex]

Therefore, the value of x is approximately 5.533. Thus, the correct option is C. 5.5.

Chi-Chiribi! EmergerOrpheus at your service. Hope it helps!

The probability P(A or B) that either event will occur is 0.67

Probability is the ratio of required outcome to the total possible outcomes . We could solve the question give. thus:

Total possible outcomes , n(Total) = 3+1+2 = 6

P(A) = n(A)/n(Total) = 3/6 = 0.5

P(B) = n(B)/n(Total) = 1/6 = 0.167

P(A or B) defined as P(A) + P(B) will become ;

P(A) + P(B) = 0.5 + 0.167 = 0.667

Hence, the probability of either A or B occuring rounded to the nearest hundredth is , P(A or B) = 0.67

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Answer:

62

Explanation:

Find the area of the region bounded the graph of y=x³+1 and the x-axis over the interval [2,4].

[tex]\text{\underline{Set up:}}\\\\\Longrightarrow \boxed{\int\limits^4_2 {x^3+1} \, dx}[/tex]

[tex]\text{Now solving...}\\\\\Longrightarrow \int\limits^4_2 {x^3+1} \, dx\\\\\Longrightarrow \frac{1}{4}x^4+x \Big]\limits^4_2\\\\\Longrightarrow [\frac{1}{4}(4)^4+(4)]-[\frac{1}{4}(2)^4+(2)]\\\\\Longrightarrow [68]-[6]\\\\\Longrightarrow \boxed{\boxed{=62}}[/tex]

Thus, the problem is solved.

We first need to understand the logistic formula used for modeling population growth. The formula is:
P'(t) = rP(t)(1 - P(t)/K),

where P'(t) represents the rate of change of the population, P(t) is the population at time t, r is the intrinsic growth rate, and K is the carrying capacity of the environment.

(a) To calculate P'(0), we need the values for r, P(0), and K. P(0) refers to the initial population.

(c) The assumption made in the computation is that the population grows continuously and the rate is proportional to both the population size and the remaining capacity for growth.

To verify this assumption, we can plug in the given values and compare the result with the actual data. If the difference is within 1%, the assumption holds true.

Please provide the required values for r, P(0), and K to proceed with the calculations.

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In which direction and by how many units should f(x) be shifted to match g(x): A. Left by 4 units.

In Mathematics and Geometry, the standard form of a quadratic function is represented by the following equation;

ax² + bx + c = 0

Based on the information provided about the parent quadratic function f(x), we can logically deduce that it is given by this equation;

f(x) = x²

Next, we would create a quadratic function for g(x) by using the data points provided in the table above;

g(x) = x² + 8x + 16

g(x) = x² + 4x + 4x + 16

g(x) = x(x + 4) + 4(x + 4)

g(x) = (x + 4)(x + 4)

g(x) = (x + 4)²

Therefore, the parent quadratic function must be shifted left by 4 units to create g(x).

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Check the picture below.

Answer:

3³-8*3 = 3

Step-by-step explanation:

Use order of operations:

P -> Parentheses

E -> Exponents

M -> Multiplication (left to right)

D -> Division (left to right)

A -> Addition (left to right)

S -> Subtraction (left to right)

3³=27 is done first because of the exponent. Next, we would do 8*3=24 because that has multiplication. Finally, we do 27-24=3 because of subtracting being the last step in PEMDAS.

Therefore, the final answer is 3

Answer:

The point U' is (-6, -2).

Step-by-step explanation:

The reflection of point U over the x-axis is the point U' with the same x-coordinate but a y-coordinate that is the opposite of U's y-coordinate. Since the y-coordinate of point U is 3, the y-coordinate of the reflected point U' will be -3. Therefore, the coordinates of U' are (-2, -3).

The graph should show U at the point (2, 3) and U' at the point (2, -3) after being reflected over the x-axis.

Answer:  To estimate the value of csc 120° using the graph of y = csc x, we can locate the point on the graph that corresponds to 120° and read the y-coordinate (csc value) from there.

On the graph of y = csc x, we observe that the csc value is undefined at x = 0° and x = 180° due to vertical asymptotes. Additionally, csc x has maximum and minimum points at multiples of 90°.

To estimate csc 120°, we can look for the nearest maximum or minimum point on the graph. The closest maximum point to 120° is at 90°, and the closest minimum point is at 180°.

Since the maximum value of csc x is 1, we can estimate csc 120° to be approximately 1.

Therefore, the estimated value of csc 120° is 1 (rounded to the nearest tenth).

Step-by-step explanation:

Step-by-step explanation:

Calculate the difference:

12 × ( -15/12x ) = 7

Multiplying a positive and a negative equals a negative: (+) × (-) = (-)

-12 × 15/12x = 7

Cancel the greatest common factor 12

-5x = 7

Divide both sides of he equation by -5

Solution: x = -7/5

26 × (7y + 5 +3) = 1300

Divide both sides of the equation by 26

7y + 5 + 3 = 50

add the numbers

7y + 8 = 50

Move the constant to the right hand side and change it's sign.

7y = 50 - 8

Subtract the numbers

7y = 48

Divide both sides of the equation by 7

= y = 6

Convert the fraction into decimal:

37/10 ÷ (1 2/9 - 0.4)

Convert the mixed number into an improper fraction

37/10 ÷ (11/9 - 0.4)

Convert the decimal into fraction

37/10 ÷ (11/9 - 2/5)

Subtract the fractions

37/10 ÷ 37/45

To divide by a fraction, multiply by the reciprocal of that fraction.

37/ 10 × 45/37

Cancel out the greatest common factor 37

1/10 × 45

Cancel out the greatest common factor 5

1/2 × 9

Calculate the product

Solution: 9/2

Convert the decimal into fraction and mixed fraction into improper fraction.

1/2 + 1/18

________

(7/6 - 7/18) ÷ 14/5

Add and subtract the fractions

5/9

____

7/9 ÷ 14/5

To divide by a fraction, Multiply by the reciprocal of that fraction.

5/9

____

7/9 × 5/14

Cancel out the greatest common factor 7.

5/9

____

1/9 × 5/2

Multiply the fractions

5/9

____

5/18

Simply the complex fraction

Solution: 2

Multiply the brackets

0.5x - 6 + 0.2x + 1.4 = 0.3x + 3.4

Collect like terms and simplify

0.7x - 4.6 = 0.3x + 3.4

Move the variable to the left hand side and move the constant to the right hand side and change the signs.

0.7x - 0.3x = 4.6 + 3.4

Simplify

0.4x = 8

Divide both sides of the equation by 0.4

Solution: x = 20

Step-by-step explanation:

Using prime factorization:

3 = 3 x 1

5 = 5 x1

6 = 2 x 3

9 = 3 x 3       take the bold numbers and multiply them 3 x 5 x 2 x 3 = 90

Answer:

This question asks which numbers (-25, 31, -100) are integers. The answer is that all of the given numbers are integers.

Step-by-step solution:

An integer is a whole number that can be positive, negative, or zero.

The given numbers are -25, 31, and -100.

To determine whether these numbers are integers, we must check whether they are whole numbers.

Whole numbers are numbers that do not have any fractions or decimals.

-25 and -100 do not have any decimal parts, so they are whole numbers and thus integers.

31 has no decimal parts, so it is also a whole number and an integer.

Therefore, all of the given numbers (-25, 31, -100) are integers.

A correct answer is option (d), "All of the answers".

The true equations for the given value of x are C, D and E

3x = 5

x = 5/3

x = 1.667

Substituting x=1.667 in the equations given in the option

The equations C, D and E are true for the value of x in the equation given.

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The maximum amount you can finance is $896.

We are given that;

The monthly gross income = $3,200

Now,

28% of your gross monthly income on your mortgage, including taxes and insurance

The percentage = 28%

3200* 28/100

=32*28

=896

Therefore, by percentage the answer will be $896.

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Answer:

[tex]\huge\boxed{\sf y=\frac{9}{5} x + 2}[/tex]

Step-by-step explanation:

Point = (x,y) = (-5, -7)

So,

x = -5, y = -7

Where m is slope and b is y-intercept.

Along with the already given point, let us take any other point of the line.

Point 1 = (x₁,y₁) = (-5,-7)

Point 2 = (x₂,y₂) = (0,2)

So,

x₁ = -5

y₁ = -7

x₂ = 0

y₂ = 2

[tex]\displaystyle Slope = \frac{y_2-y_1}{x_2-x_1} \\\\Slope = \frac{2-(-7)}{0-(-5)} \\\\Slope = \frac{2+7}{0+5} \\\\\boxed{Slope = \frac{9}{5} }[/tex]

Here, x = 0 when y = 2.

So,

Put slope and y-intercept in general form of slope-intercept equation.

So,

[tex]\displaystyle y=\frac{9}{5} x + 2\\\\\rule[225]{225}{2}[/tex]

The first order predicate calculus is the mathematical logic that is used to express and analyze propositions involving objects, characteristics, and relations

A foundational framework for mathematical logic that is used to express and analyze propositions involving objects, characteristics, and relations is known as the first-order predicate calculus.

It offers a formal language for expressing statements and a set of inference guidelines for producing reliable results. By incorporating quantifiers like "for all" and "there exists," which allow statements about variables and predicates to be quantified over a domain of objects, first-order logic expands propositional logic.

It enables exact reasoning about intricate systems and structures and forms the basis for mathematical proofs, computer science, artificial intelligence, and different branches of philosophy.

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[Top Question] Answer: A. |-6| + |3|

Step-by-step explanation:

       First, we realize they are both on the same y-axis as they both have y-values of 4. This means that we can "ignore" the y-value and focus on the x-value. To find the distance between these two points, we will find the absolute value of the x-values and add them together. This gives us answer option A;

                   A. |-6| + |3|

[Bottom Question] Answer: 36 [tex]\frac{3}{4}[/tex] units²

Step-by-step explanation:

       To find the area of a parallelogram, we multiply the base by the height. I will change 8 [tex]\frac{3}{4}[/tex] into 8.75 and 4 [tex]\frac{1}{5}[/tex] into 4.2 to make multiplying with a calculator easier.

A = bh

A = (8.75 units)(4.2 units)

A = 36.75 units² = 36 [tex]\frac{3}{4}[/tex] units²

what is needed to be calculated please tell

The expression of the statement that is given above can be written as follows: 262144y³ + 27.

To determine how to express the given statement the following should be carried out.

When a number is said to be cubed, it means that the number should be times by itself three consecutive times.

That is (64y + 3)³. This means that various components should be multiplied by itself three times. That is,

= 64×64×64(y)³ + 3×3×3

= 262144y³ + 27.

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The value of x that satisfies the equation -8 + 5(2x + 1) = -(7x + 9) + x is x = 3/4.

To solve the equation -8 + 5(2x + 1) = -(7x + 9) + x, we will simplify the expression step by step and solve for the value of x.

Distribute 5 to the terms within the parentheses: -8 + 10x + 5 = -(7x + 9) + x

Combine like terms on both sides of the equation: -8 + 5 + 10x = -7x - 9 + x

Simplifying further, we have: -3 + 10x = -6x - 9

To eliminate the negative sign on the right side of the equation, we can multiply every term by -1: -3 + 10x = -6x - 9 * -1

This simplifies to: -3 + 10x = -6x + 9

Next, we want to isolate the x terms on one side of the equation and the constant terms on the other side. We can do this by adding 6x to both sides: -3 + 10x + 6x = -6x + 9 + 6x

Simplifying further: 16x - 3 = 9

To isolate the x term, we need to eliminate the constant term on the left side. We can do this by adding 3 to both sides: 16x - 3 + 3 = 9 + 3

Simplifying further: 16x = 12

Finally, we solve for x by dividing both sides of the equation by 16: (16x)/16 = 12/16

The equation simplifies to: x = 3/4

Therefore, the value of x that satisfies the equation -8 + 5(2x + 1) = -(7x + 9) + x is x = 3/4.

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