solve for all solutions of sin(3x)cos(6x)-cos(3x)sin(6x) = sqrt(2)/2​

The combination of goods that would maximize the consumer's Utility per dollar leads to a utility equal to 14 utils.

To determine the combination of goods that would maximize the consumer's utility per dollar, we need to use the concept of marginal utility. Assuming that the consumer has $14 to spend and chips cost $1 while soda costs $2, the consumer could purchase either 14 chips, 7 sodas or a combination of the two.

Let's say the consumer decides to buy x chips and y sodas. The total expenditure would be 1x + 2y, and the total utility would be the sum of the marginal utility of each good multiplied by the quantity purchased.

Assuming the utility function is U(x, y) = x^(1/2) * y^(1/2), the marginal utility of chips would be (1/2)x^(-1/2)*y^(1/2) and the marginal utility of soda would be (1/2)x^(1/2)*y^(-1/2).

The consumer's goal is to maximize the utility per dollar, which is calculated as the total utility divided by the total expenditure. Therefore, the utility per dollar function would be U(x, y)/(1x + 2y).

Using the Lagrange multiplier method, we can find the combination of x and y that maximizes the utility per dollar. The first-order conditions give us the following system of equations:

(1/2)x^(-1/2)*y^(1/2) = λ

(1/2)x^(1/2)*y^(-1/2) = 2λ

x^(1/2)*y^(1/2) = 14

Solving this system of equations, we get x = 4 and y = 49/4. Therefore, the consumer should buy 4 chips and 12.25 sodas, which costs $29.50.

The utility function evaluated at this point is U(4, 49/4) = (4^(1/2))*(49/4)^(1/2) = 14. The utility per dollar function evaluated at this point is U(4, 49/4)/(1(4) + 2(49/4)) = 14/29.50 = 0.4746.

Therefore, the combination of goods that would maximize the consumer's utility per dollar leads to a utility equal to 14 utils.

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Answer:

Step-by-step explanation:

Answer:

The perimeter of the polygon is 59ft.

Step-by-step explanation:

PERIMETER = Sum of all sides of any polygon.

Let p = Perimeter

p = 18 + 17 + 15 + 9

p = 59ft.

If it helps pls like and mark as brainliest!

Answer:

27499.975

Step-by-Step Reasoning:

1. Start with the given number: 27500.00

2. Subtract .025 from 27500.00

3. 27500.00 - .025 = 27499.975

Answer:

Step-by-step 27500.075

For question 12, if you put the $629.63 toward reducing your credit card debt of $2,856.74 at a 14.75% APR, then you would save in interest in 1 full month is  $7.75

what is  interest ?

Interest is the cost of borrowing money. It is the amount that a lender charges a borrower for the use of their money or assets. When you borrow money, you typically agree to pay back the amount you borrowed plus interest over a set period of time.

In the given question,

Yes, your calculation is correct. If your realized income is $3,167.89 per month and your fixed expenses are $954.32 every 2 weeks (so $1,908.64 per month), then your monthly discretionary income is:

$3,167.89 - $1,908.64 = $1,259.25

If you save 50% of this discretionary income each month, you would save:

50% x $1,259.25 = $629.63

For question 12, if you put the $629.63 toward reducing your credit card debt of $2,856.74 at a 14.75% APR, then you would save in interest in 1 full month:

$629.63 x (14.75%/12) = $7.75

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Answer:

-12.6

Step-by-step explanation:

-4.5 - 8.1

= -4.5 + -8.1

= -12.6

Using intersecting secant theorem, the value of x is approximately 6

The value of the variable is solved using intersection secant theorem

The theorem gives the formula applied as follows

(6 + 6) * 6 = (5 + x) * x

Starting with the left side of the equation:

(6 + 6) * 6 = 12 * 6 = 72

Now, let's simplify the right side of the equation:

(5 + x) * x = 5x + x^2

Substituting these simplified expressions back into the original equation, we get:

72 = 5x + x^2

This is now a quadratic equation, which we can solve by rearranging and factoring:

x^2 + 5x - 72 = 0

Using calculator the solution is x = 6.35 or -11.35

Taking the positive value we have that x = 6.35

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By answering the presented question, we may conclude that As a result, surface area the prism's surface area is 640 square feet.

The surface area of an object indicates the overall space occupied by its surface. The surface area of a three-dimensional form is the entire amount of space that surrounds it. The surface area of a three-dimensional form refers to its full surface area. By summing the areas of each face, the surface area of a cuboid with six rectangular faces may be computed. As an alternative, you may use the following formula to name the box's dimensions: 2lh + 2lw + 2hw = surface (SA). Surface area is a measurement of the total amount of space occupied by the surface of a three-dimensional form (a three-dimensional shape is a shape that has height, width, and depth).

To get the surface area of the prism, sum the areas of all six faces.

The top and bottom rectangular faces have the same size and area:

10 feet x 8 feet = 80 square feet

The dimensions of the other four faces are the same:

12 feet tall x 10 feet wide = 120 square feet

As a result, the total surface area is:

2 (80 square feet) + 4 (120 square feet) = 160 square feet + 480 square feet = 640 square feet

As a result, the prism's surface area is 640 square feet.

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In the tangent line , Value of [tex]m\angle{VKT}[/tex] is A)54°.

A circle's tangent is a single-pοinted line that intersects the circle. The "pοint οf tangency" is the lοcatiοn where the circle and the tangent crοss. The radius οf the circle, with which it crοsses, is perpendicular tο the tangent. All curved shapes can be thοught οf as tangent. As a tangent is a line, it has an equatiοn as well.

The line that οnly tοuches the circumference οf a circle οnce is said tο be tangent tο it. Fοr each circled pοint, there can οnly be οne tangent. The tangent's pοint οf tangency is where it jοins the circle.

Here the given circle arc length mVPT = 234°. Then

arc length οf mVT = 360°-234° = 126°

Nοw using tangent line fοrmula in circle then,

=> [tex]m\angle{VKT}=\frac{mVPT-mVT}{2}[/tex]

=> [tex]m\angle{VKT} =\frac{234\textdegree-126\textdegree}{2}[/tex]

=> [tex]m\angle{VKT} =\frac{108\textdegree}{2} =54\textdegree[/tex].

Hence the correct option is A)54°.

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The equation for the volume of the rectangular prism is V = x³ + 2x² - 8x.

A three-dimensional solid structure called a rectangular prism contains six rectangular faces, each of which is perpendicular to the faces around it. The term "rectangular parallelepiped" is another name for it. Rectangular prisms are frequently employed in geometry because they may be used to represent a variety of real-world objects, including boxes, buildings, and rooms. A rectangular prism's surface area is equal to the total of the areas of its six faces, whereas its volume is determined by the product of its length, breadth, and height.

The volume V of a rectangular prism is given by the formula:

V = lwh

Substituting the values:

V = (x + 4)(x - 2)(x)

= (x² + 2x - 8)(x)

= x³ + 2x² - 8x

Hence, the equation for the volume of the rectangular prism is V = x³ + 2x² - 8x.

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we use the first four terms, we get: equation.

[tex]f(x) =x^2 + x^3 + x^4/2! + x^5/3![/tex]

And so on.

A Maclaurin series, named after the Scottish mathematician Colin Maclaurin, is a special type of power series expansion of a function that is centered at zero. In other words, it is a way to represent a function as an infinite sum of terms that involve the function's derivatives evaluated at zero.

The general form of a Maclaurin series for a function f(x) is given by:

[tex]f(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + ...[/tex]

To evaluate the function f(x) = x^2 * [tex]e^x[/tex] using Maclaurin series, we can first find the Maclaurin series for [tex]e^x[/tex] and then multiply it with x^2 to obtain the Maclaurin series for f(x).

The Maclaurin series for [tex]e^x[/tex]is given by:

[tex]e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ...[/tex]

Multiplying both sides by x^2, we get:

[tex]x^2 * e^x = x^2 + x^3 + x^4/2! + x^5/3! + x^6/4! + ...[/tex]

Therefore, the Maclaurin series for f(x) is:

[tex]f(x) = x^2 * e^x = x^2 + x^3 + x^4/2! + x^5/3! + x^6/4! + ...[/tex]

To approximate the value of f(x) using the Maclaurin series, we can truncate the series after a certain number of terms. The more terms we include, the more accurate the approximation will be. For example, if we use the first three terms of the series, we get:

[tex]f(x) = x^2 + x^3 + x^4/2![/tex]

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Answer:

⅙

Step-by-step explanation:

I just know the answer cuz I am better

Answer:

C

Step-by-step explanation:

The percentage rate of change per day is approximately 0.1932%.

A function has an input and an output.

A function can be one-to-one or onto one.

It simply indicated the relationships between the input and the output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

To represent the number of users on the website after t years, we can use the exponential growth formula:

[tex]N_t = N_0 \times e^{rt}[/tex]

where [tex]N_0[/tex] is the initial number of users, r is the growth rate, and t is the time in years.

Given that [tex]N_0[/tex] = 6300 and the growth rate is 73%, or 0.73, per year,

The function is:

[tex]N_t = 6300 \times e^{0.73t}[/tex]

To find the percentage rate of change per day, we can use the formula:

[tex]r = (e^{0.73/365} - 1) \times 100%[/tex]

where 0.73/365 is the daily growth rate.

Solving for r, we get:

[tex]r = (e^{0.73/365} - 1)[/tex] x 100%

r ≈ 0.1932%

Therefore,

The percentage rate of change per day is approximately 0.1932%.

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The prism A have lateral surface area = 685 and total surface area = 1021, while the prism B have lateral surface area = 328 and total surface area = 448.

The triangular prisms A and B have two same size triangle faces and two different size rectangle face with a rectangle bottom face.

For 6 and 7;

area of one triangle face = 1/2 × 21 × 9

area of one triangle face = 94.5

area of the two triangle faces = 2(94.5) = 189

area of one smaller rectangle face = 16 × 14 = 224

area of one bigger rectangle face = 17 × 16 = 272

Area of lateral surface area = 188 + 224 + 272 = 685

Area of bottom rectangle face = 21 × 16 = 336

Total surface area of prism A = 685 + 336 = 1021

For 8 and 9;

area of one triangle face = 1/2 × 24 ×

area of one triangle face = 84

area of the two triangle faces = 2(84) = 168

area of one smaller rectangle face = 7 × 5 = 35

area of one bigger rectangle face = 25 × 5 = 125

Area of lateral surface area = 168 + 35 + 125 = 328

Area of bottom rectangle face = 24 × 5 = 120

Total surface area of prism A = 328 + 120 = 448

Therefore, the prism A have lateral surface area = 685 and total surface area = 1021, while the prism B have lateral surface area = 328 and total surface area = 448.

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To solve this expression, we need to apply the order of operations (also known as PEMDAS): 8 15/22 - 3/4 x 2 divided by 12 = 351/44.

What is PEMDAS?

1.  First, we need to simplify any expressions inside parentheses, but there are no parentheses in this expression.

2. Next, we need to perform any multiplication or division, working from left to right. So, we need to calculate 3/4 x 2 = 3/2.

3. Then, we can rewrite the expression with the new result: 8 15/22 - 3/2 divided by 12.

4. Next, we need to perform the division, which means we need to divide 3/2 by 12. To do this, we can rewrite 12 as a fraction with a denominator of 2: 12/1 x 1/2 = 6/1. Then, we can simplify 3/2 divided by 6/1 by multiplying by the reciprocal of 6/1: (3/2) x (1/6) = 1/4.

5. Now we can substitute this value back into the expression: 8 15/22 - 1/4.

6. Finally, we need to convert the mixed number 8 15/22 to an improper fraction: 8 x 22/22 + 15/22 = 181/22.

7. Now we can combine like terms: 181/22 - 1/4 = (181 x 4)/(22 x 4) - 1/4 = 724/88 - 22/88 = 702/88.

8. We can simplify the fraction by dividing both the numerator and denominator by the greatest common factor (GCF), which is 2: 702/88 = 351/44.

Therefore, 8 15/22 - 3/4 x 2 divided by 12 is equal to 351/44.

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a) The length of the secant segment [tex]HC = 31.5[/tex].

b) The length of the secant segment [tex]XY = 4.8[/tex]

In geometry, a secant is a line that intersects a circle at two distinct points. A secant segment is part of a secant that lies between its two points of intersection with the circle.

In geometry, a segment is part of a line that is bounded by two distinct endpoints. A segment can be straight or curved. Straight segments are also called line segments, and curved segments are also called arcs.

According to the given information

a) In the figure, HG is a tangent and HD is a secant to the circle, and HC is the length of the secant segment.

We know that the product of the lengths of the two segments of a secant from an exterior point to a circle is equal to the product of the lengths of the entire secant and its external segment. That is,

[tex]HD*HE =HG^{2}[/tex]

where [tex]HE[/tex] is the length of the external segment of the secant.

Substituting the given values, we get:

[tex]31.5*HE =21^{2}[/tex]

[tex]HE = 21^{2} /31.5 = 14[/tex]

Now, we can use the same property to find [tex]HC[/tex]. That is,

[tex]HC *HE =HG^{2}[/tex]

Substituting the values we have found, we get:

[tex]HC*14 = 21^{2} \\HC = 21^{2} /14 = 31.5[/tex]

Therefore, the length of the secant segment [tex]HC = 31.5[/tex].

b) [tex]VX*VZ= VY*VW[/tex]

Substituting the given values, we get:

[tex]14.4*12= VY *36\\VY = (14.4*12)/36 = 4.8[/tex]

Therefore, the length of [tex]VY, XY = 4.8[/tex]

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The second difference is constant in quadratic number patterns.  The correct option is C

Quadratic number patterns are sequences of numbers that follow a quadratic or second-degree pattern or rule. This means that the sequence is generated by adding a constant difference between terms, which is itself increasing or decreasing linearly.

A quadratic sequence can be defined by a formula of the form :

an = a + bn + cn^2

Where

Therefore, the correct option is C

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Answer:

Step-by-step explanation:

To find [fo(hog)] (1), we need to first evaluate the composition hog at x = 1, and then plug the result into f.

Let's start by evaluating hog at x = 1:

hog(x) = h(g(x)) = h(x + 4) = (x + 4)^2 - 1

So, hog(1) = (1 + 4)^2 - 1 = 24

Now we can evaluate f at hog(1):

f(hog(1)) = f(24) = 3(24) = 72

Therefore, [fo(hog)] (1) = 72.

Answer:

The given expression is,

-1/6+2/3(9-3/4)-1/2

According to BODMAS rule, we first do the operation of bracket ( as B comes first in the BODMAS).

∴ 9-3/4= (36-3)/4=33/4

Putting 33/4 in place of (9-3/4), the expression becomes

-1/6+2/3×33/4-1/2

Now, the priority of M in BODMAS is more than A and S, so we do the operation of multiplication.

∴2/3×33/4=11/2 ( as 33÷3=11 and 4÷2=2)

Replacing 2/3×33/4 by 11/2, the expression becomes,

-1/6+11/2-1/2

= -1/6+(11-1)/2

= -1/6+5

= (-1+30)/6

=29/6

Hence the value of the given expression is 29/6.

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the given pairs of functions are inverse of each other.

A function that can reverse into another function is known as an inverse function or anti-function. In other words, the inverse of a function "f" will take y to x if any function "f" takes x to y. When a function is written as "f" or "F," its inverse is written as "f-1" or "F-1." Here, (-1) should not be confused with an exponent or a reciprocal.

We have two function

f(x) = √(x+4)+9

let y=√(x+4)+9

y-9=√(x+4)

(y-9)²=x+4

x=(y-9)²-4

inverse of f(x) is (x-9)²-4

Now for g(x)= (x-9)²-4

The inverse of g(x)
will be f(x) = √(x+4)+9

Hence the given pairs of function is inverse of each other

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Any expression that involves only integers and basic Arithmetic operations can simplify to a rational answer, while expressions involving irrational numbers cannot.

There are several expressions that simplify to a rational answer. In mathematics, a rational number is any number that can be expressed as the ratio of two integers, where the denominator is not zero. Therefore, any expression that involves only integers, addition, subtraction, multiplication, and division can simplify to a rational answer.

For example, if you have an expression like (3 + 2)/(4 - 1), this can be simplified to 5/3, which is a rational number. Similarly, expressions like 6/3 or 7 - 2 also simplify to rational answers.

However, expressions that involve irrational numbers such as pi or the square root of 2 cannot simplify to rational answers. For example, the expression sqrt(2) + 3 cannot be simplified to a rational number.

In summary, any expression that involves only integers and basic arithmetic operations can simplify to a rational answer, while expressions involving irrational numbers cannot.

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The original dosage will therefore decrease to 5mg at the eleventh hour (from the time of decay's beginning).

A logical statement that two forms are equal is known as an equation. It comprises of two side, left and right, which are separated by the equals sign (=). It may be necessary to solve one or maybe more unknown variables in an equation in order for it to be true.

As an illustration, the expression 3x + 5 = 14 has one unknown quantity, x. In order to solve for x, we must rewrite the equation using operations that preserve equality, such as deducting 5 both from sides of the equation:

given

We may change A = 5 into the equation above and solve for t to get the time at which there are 5mg left: 5 = 20(1/2)(t/5)

20 divided by both sides:

Using base 2, take the logarithm of both sides:

log2(1/4) = t/5 -2 = t/5

t = -10 after multiplying both sides by 5.

As time cannot be negative, we can infer that after 10 hours, the starting amount will have decreased to 5mg.

The original dosage will therefore decrease to 5mg at the eleventh hour (from the time of decay's beginning).

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Answer:

Step-by-step explanation:

The probability that a randomly selected student from the set has a height less than 115 cm is approximately 0.04, or 4%

A z-score (or standard score) is a measure of how many standard deviations a data point is from the mean of its distribution. It is a way to standardize data and compare observations from different distributions.

To find the probability that a randomly selected student from the set has a height less than 115 cm, we need to standardize the value using the standard normal distribution.

First, we calculate the z-score:

z = (115 - 150) / 20

  = -1.75

Then, we look up the area under the standard normal distribution curve to the left of z = -1.75 using a table or a calculator. The table or calculator gives us the value 0.0401.

Therefore, the probability that a randomly selected student from the set has a height less than 115 cm is approximately 0.04, or 4%, rounded to two decimal places:

P(X < 115) = 0.04

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A set of middle school student heights are normally distributed with a mean of 150150150 centimeters and a standard deviation of 202020 centimeters. Let XXX represent the height of a randomly selected student from this set. Find P(X<115)P(X<115)P, left parenthesis, X, is less than, 115, right parenthesis. You may round your answer to two decimal places.

yes. use u substitution to get the answer

Answer:Let's assume that the cost price for both products A and B is "C", and the selling price for both products A and B is "S". We can use this information to set up two equations, one for Tom and one for John, that relate the costs and profits for the two products:

Tom: 5C - 2(5S) = P1

John: 3(5C) - 13C = P2

where P1 and P2 are the profits made by Tom and John, respectively.

We know that at the end of the business day, John banks Ksh 110,000/- while Tom banks Ksh

230,000. So we can write:

P1 = 230,000 - 5C

P2 = 110,000 - 18C

Substituting these values into the equations for Tom and John, we get:

5C - 2(5S) = 230.000 - 5C

Simplifying these equations, we get:

10C - 10S = 230,000

2C = 110,000

Solving for C, we get:

C = 55,000

Substituting this value back into the first equation, we can solve for S:

10(55,000) - 10S = 230,000

Simplifying this equation, we get:

S = 32,000

Therefore, the price for product A is Ksh 55,000 and the price for product B is Ksh 32,000.

Step-by-step explanation:

Approximately 75 students voted for the eagle.

A fraction is a numerical quantity that represents a part of a whole or a quotient of two numbers. It consists of two numbers separated by a horizontal or diagonal line called a fraction bar or a division bar.

The falcon won with 5/8 of the votes, which means that 3/8 of the votes were for the other candidates (since 5/8 + 3/8 = 1).

Let's represent the total number of votes as "x".

Then, the number of votes for the falcon is (5/8)x and the number of votes for the other candidates is (3/8)x.

Out of these other candidates, the eagle received 1/3 of the votes. So, the number of votes for the eagle is:

(1/3)(3/8)x = (1/8)x

We know that the total number of votes is 536, so we can set up the equation:

(5/8)x + (3/8)x + (1/8)x = 536

Simplifying the equation, we get:

(9/8)x = 536

Multiplying both sides by 8/9, we get:

x = 596

Therefore, the number of votes for the eagle is:

(1/8)x = (1/8)(596) = 74.5

Therefore, approximately 75 students voted for the eagle.

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A) The Volume of wooden box is 72 ft³ while the surface area is 124 ft²

B) The measure of Surface area helped the carpenter know how much wood to buy.

C) The measure of Volume will be used to find the amount of sand the box will hold.

Vοlume is a measure οf the amοunt οf space οccupied by an οbject οr substance. In mathematics, the vοlume οf a three-dimensiοnal οbject οr space is calculated by measuring the amοunt οf cubic units that the οbject οr space can hοld.

Fοr example, the vοlume οf a cube is calculated by multiplying the length, width, and height οf the cube. The units used tο measure vοlume depend οn the type οf οbject οr substance being measured, but cοmmοnly used units include cubic meters (m), cubic centimeters (cm), and liters (L).

A) Given dimensions: L = 9 ft, B = 4 ft and H = 2 ft

Volume = L × B × H

Volume = 9 × 4 × 2

Volume = 72 ft³

Surface area = 2LB + 2LH + 2HB

Surface area = 2(9 × 4) + 2(9 × 2 ) + 2(4 × 2)

Surface area = 2(9 × 4) + 2(9 × 2 ) + 2(4 × 2)

Surface area = 72 + 36 + 16

Surface area = 124 ft²

B) The measure of Surface area helped the carpenter know how much wood to buy.

C) The measure of Volume will be used to find the amount of sand the box will hold.

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Complete question:

Pedro earns $5.1 per hour than Oliver according to the given graph.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side. It demonstrates the equality of the relationship between the expressions printed on the left and right sides. We have LHS = RHS (left hand side = right hand side) in every mathematical equation. To determine the value of an unknown variable that represents an unknown quantity, equations can be solved. A statement is not an equation if it has no "equal to" sign. That will be regarded as a phrase.

Given that, Pedro earned $521.40 in 22 hours.

The per hour rate is thus,

521.40 / 22 = 23.7 per hour.

For Olive we have:

372 - 186 / 20- 10 = 186 / 10 = 18.6

The difference between their earnings is:

23.7 - 18.6 = 5.1

Hence, Pedro earns $5.1 per hour than Oliver.

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The complete question is: