.
An exponential function is written as F(x) = a b, where the coefficient a is
the base bis positive but not equal to 1, and the exponent x is any
number.
A. an exponent
B. a constant
C. a variable
D. an integer

Answer:

a) To solve the differential equation y''-2y'-3y= e^4x, we first find the characteristic equation:

r^2 - 2r - 3 = 0

Factoring, we get:

(r - 3)(r + 1) = 0

So the roots are r = 3 and r = -1.

The general solution to the homogeneous equation y'' - 2y' - 3y = 0 is:

y_h = c1e^3x + c2e^(-x)

To find the particular solution, we use the method of undetermined coefficients. Since e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = Ae^4x

Taking the first and second derivatives of y_p, we get:

y_p' = 4Ae^4x

y_p'' = 16Ae^4x

Substituting these into the original differential equation, we get:

16Ae^4x - 8Ae^4x - 3Ae^4x = e^4x

Simplifying, we get:

5Ae^4x = e^4x

So:

A = 1/5

Therefore, the particular solution is:

y_p = (1/5)*e^4x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^3x + c2e^(-x) + (1/5)*e^4x

b) To solve the differential equation y'' + y' - 2y = 3xe^x, we first find the characteristic equation:

r^2 + r - 2 = 0

Factoring, we get:

(r + 2)(r - 1) = 0

So the roots are r = -2 and r = 1.

The general solution to the homogeneous equation y'' + y' - 2y = 0 is:

y_h = c1e^(-2x) + c2e^x

To find the particular solution, we use the method of undetermined coefficients. Since 3xe^x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax + B)e^x

Taking the first and second derivatives of y_p, we get:

y_p' = Ae^x + (Ax + B)e^x

y_p'' = 2Ae^x + (Ax + B)e^x

Substituting these into the original differential equation, we get:

2Ae^x + (Ax + B)e^x + Ae^x + (Ax + B)e^x - 2(Ax + B)e^x = 3xe^x

Simplifying, we get:

3Ae^x = 3xe^x

So:

A = 1

Therefore, the particular solution is:

y_p = (x + B)e^x

Taking the derivative of y_p, we get:

y_p' = (x + 2 + B)e^x

Substituting back into the original differential equation, we get:

(x + 2 + B)e^x + (x + B)e^x - 2(x + B)e^x = 3xe^x

Simplifying, we get:

-xe^x - Be^x = 0

So:

B = -x

Therefore, the particular solution is:

y_p = xe^x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^(-2x) + c2e^x + xe^x

c) To solve the differential equation y" - 9y' + 20y = x^2*e^4x, we first find the characteristic equation:

r^2 - 9r + 20 = 0

Factoring, we get:

(r - 5)(r - 4) = 0

So the roots are r = 5 and r = 4.

The general solution to the homogeneous equation y" - 9y' + 20y = 0 is:

y_h = c1e^4x + c2e^5x

To find the particular solution, we use the method of undetermined coefficients. Since x^2*e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax^2 + Bx + C)e^4x

Taking the first and second derivatives of y_p, we get:

y_p' = (2Ax + B)e^4x + 4Axe^4x

y_p'' = 2Ae^4x +

Briana and her mother will need to measure and cut the fabric for the quilt. They will need to decide on a pattern and color scheme for the quilt. They will need to sew the pieces of fabric together to create the quilt top. They will need to layer the quilt top with batting and backing fabric and then quilt the layers together. Finally, they will need to bind the edges of the quilt.

The statement "The product of a number and ten decreased by eight" is represented by the correct equation D. 10n - 8.

A number's product is the outcome of multiplying it by another number.

Addition, subtraction, multiplication, division, exponents, and square roots are some other examples of mathematical expressions.

A mathematical action known as an exponent symbolizes the repeated multiplication of the same integer. It is represented by a tiny number above and to the right of a base number.

For instance, 23 signifies that the number 2 has been multiplied three times, for a result of 8.

The sentence "The product of a number and ten decreased by eight" is correctly expressed as D. 10n - 8.

With this formula, a number is multiplied by 10 first, and the result is then reduced by 8.

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Answer:

x = 58.03°

Step-by-step explanation:

Given:

Solve for x:

1. [tex]x=cos^{-1}(\frac{9}{17})=58.03[/tex]

Answer:

So, the measure of angle x is equal to 58.03.

Answer:

length = 17;  breadth = 9

Step-by-step explanation:

Let x = area of rectangle; a = length; b = breadth.

+)       a × b = x (1)

+)      (a - 5) × (b + 3) = x - 9 => x = ab + 3a - 5b - 15 + 9 (2)

+)      (a + 3) × (b + 2) = x + 67 => x = ab + 2a + 3b + 6 - 67 (3)

Replace (1) into (2) & (3):

[tex]\left \{ {{3a - 5b=6} \atop {2a+3b=61}} \right. = > \left \{ {{a=17} \atop {b=9}} \right.[/tex]

The intersection points in the function is (-4, 61)

To find the intersection points of two functions, we can set them equal to each other and solve for x. Substituting x = -4, we have:

4(-4)² + 3(-4) + 3 = (-4)³ + 7(-4)² - 3(-4) + d

Simplifying, we get:

64 - 12 + 3 = -64 + 112 + 12 + d

d = -77

So the two functions intersect at x = -4 and y = 61. Therefore, the only intersection point is (-4, 61).

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The exponential function is y=5.[tex]2^x[/tex].

What is exponential function?

A mathematical function called an exponential function is employed frequently in everyday life. It is mostly used to compute investments, model populations, determine exponential decline or exponential growth, and so forth.

Here the exponential function is [tex]y=ab^x[/tex]

Since (-3,5/8) is on the graph, -[tex]\frac{5}{8}[/tex]=[tex]ab^{-3}[/tex] -----> 1

Since (3, 40) is on the graph, 40=[tex]ab^3[/tex] ------> 2

So, [tex]\frac{ab^3}{ab^{-3}}=\frac{40}{\frac{-5}{8}}[/tex]

=> [tex]b^{3+3}=8\times8[/tex]

=> [tex]b^6=2^6[/tex]

=> b = 2

put b=2 into 2 then,

=> 40= [tex]a\times2^3[/tex]

=> 8a=40

=> a =5

Then the exponential function is y=5.[tex]2^x[/tex].

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Answer:

The answer is 15:6 or 15 to 6

Step-by-step explanation:

white=15

black=6

white ratio black

15:6

The equation that shows how to find a percentage is:

part/whole = percent/100

This equation can be used to solve for any of the three variables, given the values of the other two.

The correct answer is C. 0.3, which is equivalent to 3/10.

What are fractions?

A fraction is a mathematical representation of a quantity that expresses a part of a whole or a ratio between two quantities.

To find the total fraction of the diagram that is shaded, we need to add the fractions representing the shaded areas of the two squares.

Given that one square is shaded 2/12 and the other shaded square is 2/15, we can add these fractions:

2/12 + 2/15

To add fractions, we need to find a common denominator. The least common multiple (LCM) of 12 and 15 is 60, so we can use 60 as the common denominator for both fractions:

(2/12) + (2/15) = (2/12) * (5/5) + (2/15) * (4/4)

= 10/60 + 8/60

Now we can add the numerators:

= (10 + 8)/60

= 18/60

= 3/10

So, the total fraction of the diagram that is shaded is 3/10.

Hence, the correct answer is C. 0.3, which is equivalent to 3/10.

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Answer:

14.4

Step-by-step explanation:

percentage for corn =30

total no. of acres of the land=48

=30 × 48

100

=14.4

Answer: d=10

Step-by-step explanation:

Answer: To find the derivative of the function h(x), we can use the quotient rule, which states that if f(x) = u(x) / v(x), then f'(x) = [v(x)u'(x) - u(x)v'(x)] / [v(x)]^2.

Using this rule, we can find the derivative of h(x) as follows:

h(x) = (x + 2) / (x - 2)

h'(x) = [(x - 2)(1) - (x + 2)(1)] / (x - 2)^2 // apply the quotient rule and differentiate numerator and denominator

h'(x) = (-2 - 2x) / (x - 2)^2

Therefore, the derivative of h(x) is h'(x) = (-2 - 2x) / (x - 2)^2.

Step-by-step explanation:

Answer: the length of TR is 24.5.

Step-by-step explanation: 9x + 1 = 13x - 11

22 = 4x

x = 5.5

Presently that we know x, ready to discover the values of PR and QS:

PR = 9x + 1 = 9(5.5) + 1 = 49

QS = 13x - 11 = 13(5.5) - 11 = 56

Since TR is half of the PR bisector, ready to discover it by separating PR by 2:

TR = PR/2 = 49/2 = 24.5

Answer:

Step-by-step explanation:

a lot (90,000) (i think)

The width of the rectangle is 4 units.

What is rectangle?

Rectangle is a four sided polygon or specifically a particular type of parallelogram having two opposite sides are equal and one angle is right angle that is 90°. It has four vertices and two diagonals intersect each other.

Given that,

The length of a rectangle is 2 units more than the width.

Let, the width of the rectangle is w units.

Then the length of the rectangle is w+2 units.

Area of any rectangle is length × width

                                        = (w+2)×w square units

Given that,

The area of the rectangle is   24 square units.

Equating both  the values we get,

                    w(w+2)= 24

We have to solve the equation for w.

    Multiplying the bracket term with w we get,

w²+ 2w = 24

⇒ w² + 2w- 24=0

⇒ (w+6)(w-4)=0

so either w= -6 or w=4

As width cannot be negative so w= 4.

Hence, the width of the rectangle is 4 units.

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The distance between city A and city B is approximately 442.3 km.

What is the law of cosine?

The Law of Cosines, also known as the Cosine Rule, is a mathematical formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. Specifically, it states that:

c² = a² + b² - 2ab cos(C).

We can use the Law of Cosines to find the distance between City A and City B. Let's call this distance d.

From the information given, we know that:

The distance between the satellite and city A is 450 km.

The distance between the satellite and city B is 340 km.

The angle between city A, the satellite, and city B is 1.5 degrees.

Using the Law of Cosines, we have:

d² = 450² + 340² - 2(450)(340)cos(1.5)

d² = 202500 + 115600 - 2(450)(340)cos(1.5)

d² = 318100 - 122328.8

d² = 195771.2

d = √195771.2

d ≈ 442.3

Therefore, the distance between city A and city B is approximately 442.3 km (rounded to the nearest tenth).

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We can use the procedures below to calculate what proportion of the original principal the overall amount of interest represents. This total amount of interest received corresponds to about 5.34% of the initial investment.

Divide the principal even by rate of interest, the time period, and other factors to arrive at simple interest. Simple return Equal principal + interests + hours is the marketing formula.The most typical technique to figure out interest is to use a portion of the principal sum. He would only pay his share of the 100% interest, for example, if somebody borrows $100 from the a partner and pledges to repay the loan with 5% interest. $x (0.05) = $5. When, you must pay interest.when you lend money after borrowing it and adding interest. Interest is often determined as an indicator of the overall of the loan total. The interest rate of the loan is the name given to this percentage.

Determine the total interest that was earned in Step 1.

It states that $33.75 was earned in interest overall.

Step 2: Determine the interest generated before to taxes.

The sum of the interest earned after tax can be determined by dividing the entire sum of the interest by (1 + rate of taxation), where the rate of tax is given as a decimal. Daniela was subject to a tax of 15% on the investment earnings. The tax rate in this instance is 15%, which really is equal to 0.15.

Interest gained before taxes is equal as $33.75 / (1 Plus 0.15), that results in a value of $29.35.

3. Determine the initial principal.

The $550 that Daniela first put into the CD is referred to as the original primary.

Compute the proportion of the initial principle in step four.

By dividing the sum of interest generated after tax by the original principal and multiplying the result by 100 as express it as a percentage, one can determine what proportion of the original principal the entire amount of interest represents.

(Interest paid before taxation / Original principal) / 100 equals the percentage of the original principal.

= ($possess / $550) w x 100

≈ 5.34%

Hence, the total interest earned is equivalent to roughly 5.34% of the initial capital.

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The length of line segment AB is equal to 8 units.

The magnitude of m∠BAQ is equal to 58°.

In Mathematics and Geometry, a perpendicular bisector can be defined as a line that bisects or divides a line segment exactly into two (2) equal halves and forms an angle that has a magnitude of 90 degrees at the point of intersection.

This ultimately implies that, the length of line segment AB can be calculated by using the following mathematical equation;

RP = 2AB

AB = RP/2

AB = 16/2

AB = 8 units.

Since A and B are the midpoints of PQ and RQ respectively, we have the following angles;

m∠P + m∠Q + m∠R = 180° (sum of all interior angles of ∆PQR)

58° + 38° + m∠R = 180°

m∠R = 180° - (58° + 38°)

m∠R = 84°

Since PR || AB, we have;

m∠R = m∠ABQ = 84° (corresponding angles).

m∠Q + m∠ABQ + m∠BAQ = 180° (sum of all interior angles of ∆ABQ).

m∠BAQ = 180° - (84° + 38°)

m∠BAQ = 180° - 122°

m∠BAQ = 58°.

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Complete Question:

In the ΔPQR, A and B are the midpoints of PQ and RQ respectively. If RP = 16, m∠P = 58°, and m∠Q = 38°, obtain AB and m∠BAQ.

The given information and the transitive property of inequalities, we can prove that [tex]AC[/tex] is greater than [tex]EF[/tex] .

Statement Reason

[tex]BC = EF[/tex] Given

Betweenness Given

[tex]AC > BC[/tex] Given

[tex]AC > EF[/tex] Transitive property [tex](3, 1)[/tex]

Explanation:

[tex]BC = EF[/tex] Given: Given statement that BC is equal to EF.

Betweenness Given: Given statement that states the concept of betweenness, where BC is between AC and EF.

AC > BC Given: Given statement that  [tex]AC[/tex] is greater than BC.

[tex]AC > EF[/tex] Transitive property: Using the transitive property, we can conclude that  [tex]AC[/tex] is greater than EF (based on statement 3 and 1).

Therefore, using the given information and the transitive property of inequalities, we can prove that AC is greater than [tex]EF[/tex]  .

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We can label the point (75, 50) as the optimal solution for the banquet, as it represents the maximum number of guests that can be invited while staying within the constraints.

What is banquet?

A banquet is a large formal meal that usually involves multiple courses and is served to a group of people on special occasions such as weddings, awards ceremonies, or fundraising events. Banquets often include speeches, presentations, and entertainment, and are typically held in a large venue such as a hotel ballroom, banquet hall, or conference center. Banquets can be hosted for a variety of purposes, such as to honor a special guest, celebrate an achievement, or raise money for a charitable cause.

To create a graph showing the solution region of the system of inequalities representing the constraints of the situation, we can use custom relationships to define the variables and constraints.

Let's define the variables:

Let x be the number of adult guests.

Let y be the number of student guests.

Now, let's write the system of inequalities representing the constraints of the situation:

The total number of guests cannot exceed 125: x + y ≤ 125

The cost of hosting the banquet cannot exceed $3375: 45x + 15y ≤ 3375

To graph this system of inequalities, we can plot the boundary lines of each inequality and shade the region that satisfies all the constraints.

The boundary lines of each inequality are:

x + y = 125 (the line that connects the points (0, 125) and (125, 0))

45x + 15y = 3375 (the line that connects the points (0, 225) and (75, 0))

To find the viable combinations of guests that satisfy all the constraints, we need to shade the region that is below the line x + y = 125 and to the left of the line 45x + 15y = 3375.

The resulting graph should look like this:

The point where the two lines intersect, (75, 50), represents the maximum number of adult guests (75) and the maximum number of student guests (50) that can be invited to the banquet while staying within the budget and venue capacity. Any point within the shaded region represents a viable combination of guests.

We can label the point (75, 50) as the optimal solution for the banquet, as it represents the maximum number of guests that can be invited while staying within the constraints.

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Answer: y=6x-1

Step-by-step explanation:

y=6x-1

x represents the time luke has been riding(hours) and the y represents the distance he has travelled (miles)

To calculate the variance of complaints received per week, we first need to find the mean or expected value of the number of complaints. This can be calculated by multiplying the number of complaints in each category by its corresponding probability, and then adding up the results.

Expected value = (0 x 0.13) + (1 x 0.21) + (2 x 0.44) + (3 x 0.07) + (4 x 0.06) + (5 x 0.09) = 2.54

Next, we can use the following formula to calculate the variance:

Variance = Σ[(x - µ)²P(x)]

where Σ = sum, x = number of complaints, µ = expected value, and P(x) = probability.

Using this formula and the probabilities given in the table, we get:

Variance = (0 - 2.54)²(0.13) + (1 - 2.54)²(0.21) + (2 - 2.54)²(0.44) + (3 - 2.54)²(0.07) + (4 - 2.54)²(0.06) + (5 - 2.54)²(0.09) = 1.4856

Rounding this answer to the nearest hundredth, we get a variance of 1.49.

In conclusion, the variance of complaints received per week is 1.49, meaning that the number of complaints is quite spread out around the mean, which is 2.54. This could indicate that the department store has some variability in its service quality, as the number of complaints can vary widely from week to week.

None of the given numbers make the equation 8/y² + 2 true

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas. It involves the study of variables, expressions, equations, and functions.

To solve this problem, we can substitute each of the given numbers (0, 1, 2, 3, 4) for y in the equation 8/y² + 2 and see if the equation is true.

Substituting y=0 would make the denominator of the fraction zero, which is undefined, so y=0 is not a valid choice.

Substituting y=1 would give us:

8/1² + 2 = 8 + 2 = 10

So, 1 is not the answer.

Substituting y=2 would give us:

8/2² + 2 = 8/4 + 2 = 2 + 2 = 4

So, 2 is not the answer.

Substituting y=3 would give us:

8/3² + 2 = 8/9 + 2 = 0.888 + 2 = 2.888

So, 3 is not the answer.

Substituting y=4 would give us:

8/4² + 2 = 8/16 + 2 = 0.5 + 2 = 2.5

So, 4 is not the answer.

Therefore, none of the given numbers make the equation 8/y² + 2 true.

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0 is the value of (p o q) (0) in function .

What are functions in mathematics?

Function, in mathematics, a formula, rule, or law that defines the relationship between one variable (the independent variable) and another (the dependent variable).

                                  Functions are ubiquitous in mathematics and essential in the natural sciences for formulating physical relationships. A function is a relationship between inputs, with each input associated with exactly one output. Each function has a domain and a codomain or scope. 

P(0) = 0

Q(0) = - 1

(P o q)(0)  =  P(0) * Q(0)

         = 0 * - 1

          = 0

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The radius of the circle is 12, and the equation of the circle is:

(x + 2)² + (y - 8)² = 144

What is a circle?

It is the center of an equidistant point drawn from the center. The radius of a circle is the distance between the center and the circumference.

To find the equation of a circle that is tangent to a horizontal line at a given point, we can use the standard form of the equation of a circle:

(x - h)² + (y - k)² = r²

where (h, k) is the center of the circle and r is the radius.

In this case, the center of the circle is (-2, 8), which gives us:

(x + 2)² + (y - 8)² = r²

To find the radius, we need to know where the circle intersects the y = -4 line. Since the circle is tangent to the line, the distance from the center of the circle to the line is equal to the radius.

The distance between a point (x, y) and a horizontal line y = k is given by |y - k|.

In this case, the distance between the center (-2, 8) and the line y = -4 is |8 - (-4)| = 12.

Therefore, the radius of the circle is 12, and the equation of the circle is:

(x + 2)² + (y - 8)² = 144

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Complete question:

What is the center at (-2, 8), tangent to the line y = -4?

The two middle terms, -11w and +11w, cancel each other out, leaving only the first and last terms. This is why there is no middle term with a variable in the factorization of the difference of squares pattern.

When expanding a binomial expression in the form of (a + b)ⁿ, the middle term can be found using the following formula:

Middle term coefficient = nC(k), where k = (n+1)/2 if n is odd, and k = n/2 or (n/2 + 1) if n is even. The middle term coefficient is then multiplied by the product of a raised to the power of (n-k) and b raised to the power of k.

The difference of squares pattern is a special algebraic pattern that arises when we factor a polynomial that is the difference between two perfect squares. For example,

x² - y² = (x+y)(x-y)

When we apply this pattern to the expression w - 121, we can rewrite it as:

w² - 11²

And, we can use the difference of squares pattern to factor it as:

(w + 11)(w - 11)

Notice that there is no middle term with a variable in this factorization. This is because when we multiply (w + 11)(w - 11), the middle term cancels out.

To see why this happens, let's expand the product:

(w + 11)(w - 11) = w² - 11w + 11w - 121

The two middle terms, -11w and +11w, cancel each other out, leaving only the first and last terms. This is why there is no middle term with a variable in the factorization of the difference of squares pattern.

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Answer: 30 and 5

Step-by-step explanation: