Solve the rational equation 5 over X plus X +5 over X +2 equals -11 over X to the 2nd+ 2X

The exact value of the following trig expression tan(11pi/12) is (18sqrt(3) + 18sqrt(2-sqrt(2)))/(8-9sqrt(3)*sqrt(2-sqrt(2))).

The exact value of the following trig expression tan(11pi/12) can be found using trig identities and the special points on the unit circle.

First, we can use the double angle formula for tangent:

tan(2x) = 2tan(x)/(1-tan^2(x))

We can then use this formula to rewrite the given expression:

tan(11pi/12) = tan(2*11pi/24) = 2tan(11pi/24)/(1-tan^2(11pi/24))

Next, we can use the sum formula for tangent to rewrite the expression in the numerator:

tan(11pi/24) = tan(7pi/24 + 4pi/24) = (tan(7pi/24) + tan(4pi/24))/(1-tan(7pi/24)*tan(4pi/24))

We can then use the special points on the unit circle to find the exact values of these expressions:

tan(7pi/24) = tan(pi/3 + pi/8) = (tan(pi/3) + tan(pi/8))/(1-tan(pi/3)*tan(pi/8)) = (sqrt(3) + sqrt(2-sqrt(2)))/(1-sqrt(3)*sqrt(2-sqrt(2)))

tan(4pi/24) = tan(pi/6) = sqrt(3)/3

Plugging these values back into the original expression, we get:

tan(11pi/12) = 2(sqrt(3) + sqrt(2-sqrt(2)))/(1-sqrt(3)*sqrt(2-sqrt(2)))/(1-(sqrt(3)/3)^2) = 2(sqrt(3) + sqrt(2-sqrt(2)))/(1-sqrt(3)*sqrt(2-sqrt(2)))/(8/9) = (2(sqrt(3) + sqrt(2-sqrt(2))))*(9/(8-9sqrt(3)*sqrt(2-sqrt(2))))

Simplifying further, we get:

tan(11pi/12) = (18sqrt(3) + 18sqrt(2-sqrt(2)))/(8-9sqrt(3)*sqrt(2-sqrt(2)))

Therefore, the exact value of the following trig expression tan(11pi/12) is (18sqrt(3) + 18sqrt(2-sqrt(2)))/(8-9sqrt(3)*sqrt(2-sqrt(2))).

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From the equation of the least squares regression line, we can say that the height of the flower compared to the soil line, in the beginning, is 3.16 units and it grows by 0.56 units each day.

A mathematical equation is a formula that uses the equals symbol (=) to connect two expressions and express their equality. Two expressions joined by an equal sign form a mathematical statement known as an equation. The expression on the left and the expression on the right is shown to be equal in relation to one another.

LHS = RHS (left-hand side = right-hand side) appears in all mathematical equations. You can solve equations to determine an unknown variable's value, which corresponds to an unknown quantity. It is not an equation if there is no "equal to" symbol in the statement. It will be taken into account as an expression.

Given,

The predicted height = 0.56

Days in soil = 3.16

The y-axis of the plot is the height of the line compared to the soil line.

The x-axis gives the number of days.

from the equation of the least squares regression line, we get the above predicted height and the days in the soil.

y  = mx+b is the equation of a line.

Then, we can say that,

m = slope = 0.56

b = y-intercept = 3.16

Therefore from the equation of the least squares regression line, we can say that the height of the flower compared to the soil line, in the beginning, is 3.16 units and it grows by 0.56 units each day.

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Using algebra we can conclude that Mrs. Ong brought a total of 12 fruits.

Algebra is a discipline of mathematics that studies symbols and the mathematical operations that can be applied to them.

These symbols are referred to as variables because they don't have predetermined values.

These symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations in order to ascertain the values.

Now, let the total fruits be x.

Oranges = 1/2 x - 3

Pears = [x - 1/2 (1/2x -3 ) ]- 2= x - 1/4x + 3/2 - 2 = 3x /4 - 1/2

Apples = x - 1/2 (3x /4 - 1/2) - 1

           = x - 3/8 x + 1/4 - 1 = 5x /8 - 3/4

Mangoes = 5

x - 5x/ 8 + 3/4 = 5

3x /8 + 3/4 = 5    

3x + 6 = 40

3x = 34

x = 11.333

Therefore, using algebra we can conclude that Mrs. Ong brought a total of 12 fruits.

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Complete question:

Mrs. Ong bought some fruits. 3 fewer than 1/2 of the fruits were oranges. 2 fewer than 1/2 of the remaining fruits were pears. 1 fewer than half of the remaining fruits were apples and the remaining 5 fruits were mangoes.

How many fruits did Mrs Ong buy altogether?

Answer: $15

Step-by-step explanation:

Remember percent makes the decimal move two places

The cost of Frank's shirt is 12 + 12(0.25)

12 + 3 = $15

Hope this helps!

The stopping distance for a car traveling at 50 mph is 208.3 feet.

We can set up a proportion using the given information and the formula for stopping distance:

D = k s²

where D is the stopping distance, s is the speed of the car, and k is a constant of proportionality.

Using the given information, we can solve for k:

75 = k * 30²

k = 75 / (30²) = 0.0833

Now we can use this value of k to find the stopping distance for a speed of 50 mph:

D = 0.0833 * 50² = 208.3 feet

Therefore, the stopping distance for a car traveling at 50 mph is 208.3 feet.

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The equation of total time taken by the commuter in travelling is 14x + 11, thus the correct option (b).

An algebraic equation can be defined as a mathematical statement in which two expressions are equivalent. Algebraic equations usually consist of variables, coefficients and constants .

Algebraic equations are also called polynomials because they contain polynomials on both sides of the equality sign. Algebraic equations consist of variables, coefficients, constants, and algebraic operations such as addition, subtraction, multiplication, division, and exponentiation.

According to the question:

(4x + 13) + (10x - 2) = 4x + 10x + 13 - 2

                               = 14x + 11

Thus, the equation for the total time taken is 14x + 11.

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Solving the given radical equation [tex]\sqrt{(4x+5)} = x-4[/tex] ,  the solution set is {11}

To solve the radical equation, we need to isolate the radical on one side of the equation and then square both sides to eliminate the radical. We can then solve for x and check our proposed solutions. Here are the steps:

Step 1: Isolate the radical on one side of the equation:
[tex]\sqrt{(4x+5)} = x-4[/tex]

Step 2: Square both sides of the equation to eliminate the radical:
[tex](\sqrt{(4x+5))^2} = (x-4)^2\\4x+5 = x^2-8x+16[/tex]

Step 3: Rearrange the equation and solve for x:
[tex]x^2-12x+11 = 0\\(x-11)(x-1) = 0[/tex]

Step 4: Solve for x:
[tex]x-11 = 0 , x-1 = 0\\x = 11 , x = 1[/tex]

Step 5: Check the proposed solutions by plugging them back into the original equation:
[tex]\sqrt{(4(11)+5)} = 11-4\\\sqrt{49 } = 7[/tex]
7 = 7 (True)

[tex]\sqrt{(4(1)+5) } = 1-4\\\sqrt{9} = -3[/tex]
3 = -3 (False)

Therefore, the solution set is {11}.

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The present value of A = $35,000, r = 7% compounded monthly, t = 6 years is $24,036.89.

The present value is the amount of money that is needed today in order to produce a specified future value at a specified interest rate and time period. It is calculated using the formula: PV = A / (1 + r/n)^(n*t), where A is the future value, r is the interest rate, n is the number of compounding periods per year, and t is the time period in years.

Given that A = $35,000, r = 7%, n = 12 (since the interest is compounded monthly), and t = 6 years, we can plug these values into the formula to calculate the present value:

PV = $35,000 / (1 + 0.07/12)⁷²
PV = $35,000 / (1.00583)⁷²
PV = $35,000 / 1.45558
PV = $24,036.89

Therefore, the present value is $24,036.89.

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You will have 4500L of water in the tank after it rains 90mL.

If it rains 90mL and you have a tank with a roof area of 50m^2, you can calculate the amount of water collected in the tank using the following formula:

Amount of water collected = (Rainfall in mm) x (Roof area in m^2)

In this case, the amount of water collected would be:

Amount of water collected = (90mm) x (50m^2)

Since 1mm of rain is equivalent to 1L of water per square meter, we can convert the rainfall from mm to L:

Amount of water collected = (90L/m^2) x (50m^2)

Simplifying the equation gives us:

Amount of water collected = 4500L

Therefore, you will have 4500L of water in the tank after it rains 90mL.

I hope this helps! Let me know if you have any further questions.

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The volume of the prism would be h(60 - hp/2) cubic feet.

Volume is a collection of two - dimensional points enclosed by a single dimensional line. Mathematically, we can write -

V = ∫∫∫ F(x, y, z) dx dy dz

Given is that the surface area of the right prism is 120 feet.

The formula for finding the surface area for the right prism is -

A = 2b + hP

2b + hp = 120

We can write the base area as -

2b = 120 - hp

b = 60 - hp/2

The volume of the prism would be -

V = base area x height

V = (60 - hp/2) x (h)

V = h(60 - hp/2)

Therefore, the volume of the prism would be h(60 - hp/2) cubic feet.

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2x should be added to 3x to get 5x.

To find out what should be added to 3x to get 5x, we can use simple algebra. We can set up an equation as follows:

3x + ? = 5x

Next, we can isolate the unknown variable on one side of the equation by subtracting 3x from both sides:

? = 5x - 3x

Simplifying the right side of the equation gives us:

? = 2x

Therefore, the answer is 2x. This means that 2x should be added to 3x to get 5x.

In conclusion, 2x should be added to 3x to get 5x.

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The solutions to the equation are w = (5 + i√(5))/(10) and w = (5 - i√(5))/(10). These are the fully simplified answers.

To solve the equation 10w(w-1)+3=0 using the Quadratic Formula, we first need to rearrange the equation to the standard form of a quadratic equation, which is ax²+bx+c=0.

In this case, we can distribute the 10w to get 10w²-10w+3=0. This gives us the values of a=10, b=-10, and c=3.

Now, we can plug these values into the Quadratic Formula, which is:

w = (-b ± √(b²-4ac))/(2a)

w = (-(-10) ± √((-10)²-4(10)(3)))/(2(10))

w = (10 ± √(100-120))/(20)

w = (10 ± √(-20))/(20)

w = (10 ± √(-1)(20))/(20)

w = (10 ± √(-1)√(20))/(20)

w = (10 ± i√(20))/(20)

w = (10 ± i√(4)√(5))/(20)

w = (10 ± 2i√(5))/(20)

w = (5 ± i√(5))/(10)

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The conditional density function ????(????|???? = y) = 1/y. This is a valid density function because it is non-negative and integrates to 1.

So the answer is ????(????|???? = y) = 1/y.

The question asks us to find the conditional density function ????(????|???? = y) given the joint density function ????(x, y) = 6y(1 − y), 0 ≤ x ≤ y ≤ 1.

To find the conditional density function, we need to divide the joint density function by the marginal density function of y.

The marginal density function of y can be found by integrating the joint density function with respect to x:

????(y) = ∫????(x, y) dx = ∫6y(1 − y) dx = 6y(1 − y) ∫dx = 6y(1 − y) (x) |0≤x≤y = 6y(1 − y) (y - 0) = 6y^2(1 − y)

Now we can find the conditional density function by dividing the joint density function by the marginal density function:

????(????|???? = y) = ????(x, y)/????(y) = 6y(1 − y)/6y^2(1 − y) = 1/y

Therefore, the conditional density function ????(????|???? = y) = 1/y. This is a valid density function because it is non-negative and integrates to 1.

So the answer is ????(????|???? = y) = 1/y.

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The proportion that could be used to solve for x is 32/x = 12/15. The correct option is 3.

There are two primary methods for resolving proportional problems: cross-products and cross-multiplications. Using proportions and ratios.

By cross-multiplying, you multiply the denominator of the second ratio by the numerator of the first ratio.

Then you set that equal to the first ratio's denominator multiplied by the second ratio's numerator. By cross-multiplying the above proportion as an example, we may find the value of x.

24/9 = 32/12 = x/15

32/12 = x /15

32/x = 12/15

Thus, the correct option is 3) 32/x = 12/15.

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The question is incomplete. The missing options are written below:

The accompanying diagram shows two similar triangles. Which proportion could be used to solve for x?

1) x/ 24 = 9/ 15

2) 24/9 = 15/x

3) 32/x = 12/15

4) 32/12 = 15/x

The dosage of Jack and Jill is obtained using geometric series formula which is 500 mg and 225 mg respectively.

What is geometric series?

A geometric series is a collection of terms in mathematics that have an unlimited number and a fixed ratio between each term. A geometric series is typically expressed as a + ar + ar² + ar³ + ..., where r is the common ratio between neighbouring terms and a is the coefficient of each term.

Let's start by finding the amount of medication that Jack takes in total.

Each hour, Jack takes a dosage that is 80% of the previous dosage (since it drops by 20%).

So his dosages form a geometric sequence with first term 100 and common ratio 0.8 -

100, 80, 64, 51.2, 40.96, ...

The sum of an infinite geometric series with first term a and common ratio r (where |r| < 1) is given by -

sum = a / (1 - r)

In this case, a = 100 and r = 0.8, so the sum is -

sum = 100 / (1 - 0.8) = 500

Therefore, Jack takes a total of 500 mg of medication.

Now let's find the original dosage for Jill.

If Jill takes half the amount of medication that Jack eventually took, then she takes -

500 / 2 = 250 mg

Let's call Jill's original dosage x.

Then her dosages form a geometric sequence with first term x and common ratio 0.7 (since it drops by 30%).

The sum of this sequence must be 250.

x + 0.7x + 0.49x + 0.343x + ... = 250

This is an infinite geometric series with first term x and common ratio 0.7, so we can use the formula for the sum of an infinite geometric series -

sum = x / (1 - 0.7) = x / 0.3

Simplifying the equation from before -

x / 0.3 × (1 + 0.7 + 0.49 + 0.343 + ...) = 250

x / 0.3 × (1 / (1 - 0.7)) = 250

x / 0.3 × 3.333... = 250

x = 250 / 1.111... = 225

Therefore, Jill's original dosage was 225 mg.

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Answer: 3

Step-by-step explanation:

To find the x value, we must plug 6 in for y

6 = 15 - 3x

-9 = -3x

3 = x

Hope this helps!

The mass of suitcase D is 2x -2 kg.

What is Algebra?

A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them. Variables are the names given to these symbols because they lack fixed values.

As per the given data:

mass of A = x kg

B has a mass 6 kg less than A.

C has a mass twice of B

total mass = 6x - 20 kg

Let's consider mass of D as M.

mass of B = x - 6

mass of C = 2(x - 6)

For the total mass:

x + x - 6 + 2(x - 6) + M = 6x - 20

2x - 6 + 2x - 12 + M = 6x - 20

4x - 18 + M = 6x - 20

M = 2x - 2 kg

Hence, mass of suitcase D is 2x -2 kg.

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The expression can be expanded as a difference of two logarithms:

[tex]ln(\sqrt\frac{x^2}{y^5})[/tex] = [tex]2 ln(x) - (5/2) ln(y)[/tex]

What is the logarithms?

Logarithms are mathematical functions that help to simplify the representation of very large or very small numbers. They are the inverse functions of exponential functions.

We can use the properties of logarithms to expand the expression as follows:

[tex]ln(\sqrt\frac{x^2}{y^5})[/tex] = [tex]ln(x^2) - ln(y^5/2)[/tex]

Next, we can simplify each logarithm using the property that log(a^b) = b log(a):

[tex]ln(x^2) - ln(y^5/2)[/tex] = [tex]2 ln(x) - (5/2) ln(y)[/tex]

Hence, the expression can be expanded as a difference of two logarithms:

[tex]ln(\sqrt\frac{x^2}{y^5})[/tex] = [tex]2 ln(x) - (5/2) ln(y)[/tex]

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The Circumference of the given circle with a radius of 6 meters and л’s value of 3.14 is 37.68 meters.

In 2-D Geometry, the circumference of the circle is the perimeter running around the circle.

The Circumference of a circle is given by the following formula:

C=2 лr…..(i),

Where,

C = Circumference of the circle,

Л = 3.14 (given value)

r= Radius of the circle = 6 meters (given).

Substituting the values of each variable in equation (i), we get;

C = 2 лr = 2x3.14x6 meters,

Or, C = 37.68 meters

Therefore, the circumference of the given circle is 37.68 meters

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Answer:

21.5

Step-by-step explanation:

The total number of words which starts and end with the letter N formed from the letters of the word NIPUN is 4

To form words that start and end with the letter N, we need to first fix the letter N at the beginning and end of the word. This leaves us with the letters I, P, and U to arrange in the middle.

There are 3! (3 factorial) ways to arrange these 3 letters, which is equal to 3 × 2 × 1 = 6.

However, we need to divide by the number of times each letter appears in the word, which is 1 for each of the letters I, P, and U.

So the total number of words that can be formed is 6/1/1/1 = 6.

But we only want the words that start and end with the letter N, so we need to divide by the number of times the letter N appears in the word, which is 2.

So the final answer is 6/2 = 3.

Therefore, the total number of words which starts and end with the letter N formed from the letters of the word NIPUN is 3.

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Answer:

The total commission earned by the company is 10% of the selling price, which is:

0.10 x $180,000 = $18,000

Marge receives 1/4 of the company's commission, which is:

(1/4) x $18,000 = $4,500

Therefore, Marge receives $4,500 in commission.

The difference in inches between 7 3/4 and 8 whole numbers is 1/4 inch.

What is Subtraction?

One of the four arithmetic operations, along with addition, multiplication, and division, is subtraction, which is denoted by the minus sign. The operation of subtraction represents the removal of items from a collection. In the adjacent image, for instance, there are 5 2 peaches, which means 5 peaches with 2 subtracted, making a total of 3 peaches.

As a result, 5 minus 2 equals 3, or the difference between 5 and 2. Although subtraction is most often used in mathematics with natural numbers, it can also be used with other types of objects, such as negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices, to remove or decrease physical and abstract quantities.

The difference in inches between 7 3/4 and 8 whole numbers is 1/4 inch.

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If Allister’s father measures 180 cm, then the height of Allister would be 150 cm

Let us assume that 'h' represents the height of Allister and 'm' represents the height of Allister’s father.

Here, Allister’s father is 120% of Allister’s height.

This means that m is 120 percent of 'h'

Using the formula of percentage,

m = 120% of h

m = 120/100 × h

m = 6h/5

But Allister’s father actually measures 180 cm

This means m = 180

so , 180 = 6h/5

We solve this equation to find the value of h.

⇒ h = 180 × 5/6

⇒ h = 150 cm

Therefore, Allister's height = 150 cm.

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The complete question is:

Allister’s father is 120% of Allister’s height. If his father measures 180 cm, how tall is Allister?

The percent of powdered sugar in mixture B is 25.82%.

To solve this problem, we can use a table to organize the information and find the percent of powdered sugar in mixture B.

MixturePoundsPercent Powdered SugarPounds of Powdered SugarA520%1B11x%0.11xTotal1624%3.84

From the table, we can create an equation to find the percent of powdered sugar in mixture B:

1 + 0.11x = 3.84

Solving for x:

0.11x = 2.84

x = 25.82%

Therefore, the percent of powdered sugar in mixture B is 25.82%.

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We get the cartesian equation of the plane:

-48x - 28y + 20z = -240

To find the cartesian equation of a plane using the given vectors u and v, we need to find a normal vector to the plane. A normal vector is a vector that is perpendicular to the plane. We can find a normal vector by taking the cross product of u and v. The cross product of two vectors u and v is given by:

n = u x v = <(u2*v3 - u3*v2), (u3*v1 - u1*v3), (u1*v2 - u2*v1)>

Plugging in the values for u and v, we get:

n = <-6,8,4> x <2,-6,-9> = <(8*-9 - 4*-6), (4*2 - -6*-6), (-6*-6 - 8*2)>

n = <-72+24, 8-36, 36-16> = <-48, -28, 20>

Now that we have a normal vector to the plane, we can use it to find the cartesian equation of the plane. The cartesian equation of a plane is given by:

n1(x-x0) + n2(y-y0) + n3(z-z0) = 0

where n =  is the normal vector and (x0, y0, z0) is a point on the plane. We can use the vector u as a point on the plane, so (x0, y0, z0) = (-6, 8, 4). Plugging in the values for n and (x0, y0, z0), we get:

-48(x+6) - 28(y-8) + 20(z-4) = 0

Simplifying, we get the cartesian equation of the plane:

-48x - 28y + 20z = -240

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Class B has higher percentage of girls than class A that is 26.6 % more.

A percentage is a number or ratio that can be expressed as a fraction of 100. If you need to calculate the percentage of a number, divide the number by the whole number and multiply by 100. Percentages therefore mean 1 in 100.

Given,

Total number of students in class A = 60

Number of girls student in class A = 28

percentage of girls in class A

= (28/60)×100

= 0.467 × 100

= 46.7%

Total number of students in class B = 60

Number of girls student in class B = 44

percentage of girls in class A

= (44/60)×100

= 0.733 × 100

= 73.3 %

Difference in percentage of girls in class A and class B.

= 73.3% - 46.7%

= 26.6%

Hence, class B has 26.6 percent more girls than class A.

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So, the solution to the system of equations is:

x = -26/7
y = -25/7
z = -37/7

Solving a system of linear equations by the Matrix Inversion method involves finding the inverse of the coefficient matrix, multiplying it by the constant matrix, and solving for the variables.

First, we need to write the system of equations in matrix form:

[ 2 6 2 ] [ x ] [ 8 ]
[ 6 6 3 ] [ y ] = [ -3 ]
[ 2 3 1 ] [ z ] [ 3 ]

Next, we need to find the inverse of the coefficient matrix:

A = [ 2 6 2 ]
[ 6 6 3 ]
[ 2 3 1 ]

The determinant of A is:

det(A) = 2(6-9) - 6(2-6) + 2(18-6) = -6 + 24 + 24 = 42

The adjoint of A is:

adj(A) = [ (6-3) (-2+6) (-18+6) ]
[ (-6+3) (2-4) (12-6) ]
[ (6-18) (-2+12) (2-18) ]

= [ 3 4 -12 ]
[ -3 -2 6 ]
[ -12 10 -16 ]

The inverse of A is:

A' = (1/det(A))adj(A) = (1/42)[ 3 4 -12 ]
[ -3 -2 6 ]
[ -12 10 -16 ]

= [ 1/14 2/21 -4/7 ]
[ -1/14 -1/21 2/7 ]
[ -4/7 5/21 -8/21 ]

Now, we can multiply the inverse of A by the constant matrix to find the solution:

[ x ] [ 1/14 2/21 -4/7 ] [ 8 ]
[ y ] = [ -1/14 -1/21 2/7 ] [ -3 ]
[ z ] [ -4/7 5/21 -8/21 ] [ 3 ]

= [ (8/14) + (16/21) + (-32/7) ]
[ (-8/14) + (-3/21) + (-6/7) ]
[ (-32/7) + (15/21) + (-24/21) ]

= [ -26/7 ]
[ -25/7 ]
[ -37/7 ]

So, the solution to the system of equations is:

x = -26/7
y = -25/7
z = -37/7

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