Prove : cscx - sinx = cosxcotx

The solution of the equations ​2a-b=-1 and a-2b=4 is b = -3 and a = -2. The solution of the equations 3m-n=8 and m+n=4​ is m = 3 and n = 1.

In the elimination method, the two equations are solved by eliminating one variable and by substituting the value of one variable in the equation to get the value of another variable.

The two equations can be solved as below,

2a-b=-1

a-2b=4

Multiply the second equation by two and subtract from the first equation,

2a-b=-1

2a - 4b = -8

________

       3b = - 9

         b = -3

2a + 3 = -1

2a = -4

a = -2

For the other two equations,

3m-n=8

m+n= 4

_______

4m = 12

m = 3

m + n = 4

n = 4 - 3

n = 1

Therefore, the solution of the equations ​2a-b=-1 and a-2b=4 is b = -3 and a = -2. The solution of the equations 3m-n=8 and m+n=4​ is m = 3 and n = 1.

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[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$113\\ r=rate\to 6\%\to \frac{6}{100}\dotfill &0.06\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{weekly, thus 52} \end{array}\dotfill &52\\ t=years\dotfill &4 \end{cases}[/tex]

[tex]A = 113\left(1+\frac{0.06}{52}\right)^{52\cdot 4} \implies \boxed{A \approx 143.63}\hspace{5em}\underset{ earned~interest }{\stackrel{ 143.63~~ - ~~113 }{\boxed{\approx 30.63}}}[/tex]

The height of the helium tank is 49 inches.

Volume refers to the amount of space occupied by a three-dimensional object, measured in cubic units.

The volume V of a cylinder is given by the formula:

V = πr²h

where r is the radius of the base and h is the height of the cylinder.

Since the diameter of the helium tank is given as 14 inches, the radius r is half of the diameter, which is 7 inches.

We also know that the volume inside the tank is 2,401π cubic inches. Therefore, we can write:

2,401π = π(7)²h

Simplifying the right-hand side, we get:

2,401π = 49πh

Dividing both sides by 49π, we get:

h = 2,401π / 49π

Simplifying, we get:

h = 49

Therefore, the height of the helium tank is 49 inches.

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Answer:

49

Step-by-step explanation:

I took the test and got it right :)

You should invest $26,100 in Stock Boll and $12,900 in Stock Coff to maximize your annual interest earned, and the maximum annual interest is $2,337.

To maximize your annual interest earned, you should invest as much money as possible in the stock with the higher interest rate, which is Stock Coff. However, there are two constraints that you need to consider: the buying limit on Stock Coff and the requirement to spend at least twice as much money on Stock Boll as Stock Coff.

Let's use algebra to solve this problem. Let x be the amount of money you invest in Stock Boll and y be the amount of money you invest in Stock Coff. Then we have the following equations:

x + y = $39,000 (the total amount of money you have to invest)
y <= $12,900 (the buying limit on Stock Coff)
x >= 2y (the requirement to spend at least twice as much money on Stock Boll as Stock Coff)

To maximize your annual interest earned, you want to maximize the expression 0.05x + 0.08y (the sum of the interest earned from Stock Boll and Stock Coff).

If you invest the maximum amount of money in Stock Coff ($12,900), then you have $39,000 - $12,900 = $26,100 left to invest in Stock Boll. This satisfies the requirement to spend at least twice as much money on Stock Boll as Stock Coff ($26,100 >= 2 * $12,900).

So the optimal solution is to invest $26,100 in Stock Boll and $12,900 in Stock Coff. The maximum annual interest earned is 0.05 * $26,100 + 0.08 * $12,900 = $1,305 + $1,032 = $2,337.

Therefore, you should invest $26,100 in Stock Boll and $12,900 in Stock Coff to maximize your annual interest earned, and the maximum annual interest is $2,337.

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The 7th term of the geometric sequence is 5,103.

To find the 7th term of a geometric sequence with a first term of 7 and a common ratio of -3, we can use the formula:

an = a1 × [tex]r^(n-1)[/tex]

where an is the nth term of the sequence, a1 is the first term of the sequence, r is the common ratio, and n is the number of the term we want to find.

Substituting the given values, we get:

a7 = 7 × [tex](-3)^(7-1)[/tex]

Simplifying the exponent, we get:

a7 = 7 × [tex](-3)^6[/tex]

Evaluating the exponent, we get:

a7 = 7 × 729

Multiplying, we get:

a7 = 5,103

Therefore, the 7th term of the geometric sequence is 5,103.

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The graph of the functions are added as an attachment

The graphs are sinusoidal functions such that they are mathematical functions that exhibit a repetitive pattern over time or space

From the question, we have the following parameters that can be used in our computation:

y = 4cot(4x)

y = -1/2cot(2x - π/4)

To plot y = 4cot(4x) using two periods, we make use of the domain

{0 < x < π/2}

To plot y = 4cot(4x) using two periods, we make use of the domain

{0 < x < π}

See attachement for the graphs of the functions

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The length of MN, to the nearest tenth of a unit is equal to: C. 14.4 units.

In order to determine the missing side length of a geometric figure with the adjacent and hypotenuse side lengths given, you should apply the law of cosine:

C² = A² + B² - 2(A)(B)cosθ

Where:

A, B, and C is the length of side of a given triangle.

By substituting the given side lengths and angle into the law of cosine formula, we have the following;

MN² = LM² + LN² - 2(LM)(LN)cosθ

MN² = 12² + 10² - 2(12)(10)cos(52)

MN² = 144 + 100 - 147.76

MN = √96.24

MN = 9.8 units.

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The set of vectors {v1, v2, v3} is linearly dependent for all real number h ≠ 4 and all real number h ≠ -4.

The set of vectors {v1, v2, v3} is linearly dependent if there exists a set of real numbers a, b, and c such that a*v1 + b*v2 + c*v3 = 0 and at least one of a, b, or c is not equal to zero. In this case, we can write the equation as:

a*1 + b*2 + c*(-1) = 0

We can rearrange this equation to solve for h:

h = -a - 2*b + c

Now we can plug in the values for v1, v2, and v3:

h = -1 - 2*2 + (-1)*(-1)

h = -4

Therefore, the set of vectors {v1, v2, v3} is linearly dependent when h = -4.

Alternatively, we can also use the determinant of the matrix formed by the vectors to determine when the set is linearly dependent. The determinant of the matrix is given by:

| 1 2 -1 |
| -1 1 3 | = (1)(1)(h) + (2)(3)(-1) + (-1)(-1)(2) - (-1)(1)(-1) - (2)(-1)(1) - (1)(3)(2)
| 2 1 h |

Simplifying this equation gives:

h - 6 - 2 + 1 + 2 - 6 = 0

h - 11 = 0

h = 11

Therefore, the set of vectors {v1, v2, v3} is also linearly dependent when h = 11.

So the set of vectors {v1, v2, v3} is linearly dependent for all real number h ≠ 4 and all real number h ≠ -4.

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Answer:

21.6 feet?

Step-by-step explanation:

8 percent of 3 is 0.24

0.24 x 90 is 21.6

Answer:

[tex]\frac{y + x}{y\y - x}[/tex]

Step-by-step explanation:

Helping in Jesus' name.

The results of the matrix operations are:

To compute each matrix sum or product, we first need to determine if the operation is defined. Then, we can use the rules of matrix arithmetic to find the result.

-2A is defined, since we can multiply a matrix by a scalar. To find the result, we simply multiply each element of A by -2:

-2A = [-2(2) -2(0) -2(-1); -2(4) -2(-3) -2(2)] = [-4 0 2; -8 6 -4]

B - 2A is defined, since both matrices have the same dimensions. To find the result, we first compute -2A as shown above, then subtract each corresponding element of B and -2A:

B - 2A = [7 -5 1; 1 -4 -3] - [-4 0 2; -8 6 -4] = [7-(-4) -5-0 1-2; 1-(-8) -4-6 -3-(-4)] = [11 -5 -1; 9 -10 1]

AC is undefined, since the number of columns in A (3) does not match the number of rows in C (2).

CD is defined, since the number of columns in C (2) matches the number of rows in D (2). To find the result, we multiply each row of C by each column of D and sum the products:

CD = [(1)(3)+(2)(-1) (1)(5)+(2)(4); (-2)(3)+(1)(-1) (-2)(5)+(1)(4)] = [1 13; -7 -6]

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Answer:

I believe the answer you are looking for is [tex]6b^{4}[/tex] -  [tex]8b^{2}[/tex]

Step-by-step explanation:

Hope it helps

Sorry if I'm wrong

Answer:

θ = 78.69°

Step-by-step explanation:

As this is a right angle triangle we use trigonometric ratios/

The perpendicular i.e the height of the mast = 50 ft

The base i.e the shadow = 10 ft

tanθ = perpendicular / base

tanθ = 50/10

θ = [tex]tan^{-1}[/tex] (50/10)

θ = 78.69°

The average amount of trash collected is found to be 3 pounds.

Let the average amount of trash collected be 'x'.

As per the given conditions, equation form:

80*(1.5 + x ) = 360

Applying the Distributive Property:

120 + 80x = 360

Simplifying:

80x = 240

x = 240/80

x = 3

Thus, the average amount of trash collected is found to be 3 pounds.

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The complete question is attached.

For (a): 52 + 35/10m, rationalizing the denominator gives the result (25√2 + 6√5)/10.
For (b): 33x + 3y/3xy, rationalizing the denominator gives the result √3(x+y)/xy.

To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression is the same expression with the sign between the terms changed. For example, the conjugate of (a+b) is (a-b) and the conjugate of (a-b) is (a+b). By multiplying by the conjugate, we eliminate any radicals in the denominator.

(a) 5/√2 + 3/√5

  = (5/√2)(√2/√2) + (3/√5)(√5/√5)

  = (5√2)/2 + (3√5)/5

  = (25√2 + 6√5)/10

(b) (3√3x + 3√y)/(3√xy)

  = (3√3x + 3√y)(3√xy)/(3√xy)(3√xy)

  = (9x√3xy + 9y√3xy)/(9xy)

  = (9√3xy(x+y))/(9xy)

   = √3(x+y)/xy

Therefore, the rationalized denominators are (a) (25√2 + 6√5)/10 and (b) √3(x+y)/xy.

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The solutions to this problem are y = 5 and y = -1/3.

Sure! The quadratic formula is a useful tool for solving equations of the form ax^2 + bx + c = 0. In this problem, we need to rearrange the equation so that it fits this form before we can use the quadratic formula.

First, let's multiply both sides of the equation by y to get rid of the fraction:

y^2 + 1 = 13y/6

Next, let's rearrange the equation so that it equals zero:

y^2 - 13y/6 - 1 = 0

Now we can use the quadratic formula to solve for y. The quadratic formula is:

y = (-b ± √(b^2 - 4ac))/(2a)

In this problem, a = 1, b = -13/6, and c = -1. Plugging these values into the quadratic formula gives us:

y = (-(-13/6) ± √((-13/6)^2 - 4(1)(-1)))/(2(1))

Simplifying the equation gives us:

y = (13/6 ± √(169/36 + 4))/2

Finally, we can solve for y by simplifying the square root and dividing by 2:

y = (13/6 ± √(289/36))/2

y = (13/6 ± 17/6)/2

y = (30/6)/2 or y = (-4/6)/2

y = 5 or y = -1/3

So the solutions to this problem are y = 5 and y = -1/3. I hope this helps! Let me know if you have any further questions.

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The polynomial equation in the complete 30x^(4)+83x^(3)-52x^(2)+4x+1=0. The solutions are: x = -1 and x = -1/7, 1/7, -5/7, 5/7.

The question asks you to solve the polynomial equation: 30x4 + 83x3 - 52x2 + 4x + 1 = 0.

To solve this equation, you will need to factor it into the form (ax + b)(cx + d) = 0. The solutions of this equation will be when either ax + b or cx + d are equal to 0.

First, factor out the Greatest Common Factor (GCF) of the polynomial, which is 4x. This will leave us with:

4x(7x3 + 20x2 - 13x + 1) = 0.

Now, factor this expression into two binomials. One of the binomials is already in the form ax + b, and the other will need to be factored further.

4x(7x2 + 5x - 1)(x + 1) = 0.

Therefore, the solutions of this equation are when

or

.

The solutions are: x = -1 and x = -1/7, 1/7, -5/7, 5/7.

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Option (c) has the same equation as above after simplification, so the answer is (c):y-8=-0.2(x+10).

Simplification involves reducing an expression, equation, or problem to a simpler form. This can involve performing operations such as combining like terms, factoring, or canceling out common factors.

Point-slope form is a way of writing the equation of a straight line in two-dimensional space, using a given point on the line and the line's slope.

The point-slope form equation for a line passing through a point (x1, y1) with slope m is:y - y1 = m(x - x1).

In the given question,

To write the equation of a line in point-slope form, we need to use the formula:

y - y1 = m(x - x1)

where m is the slope of the line and (x1, y1) is any point on the line.

We are given two points on the line: (-10, 8) and (10, 4). We can use these points to find the slope of the line:

m = (y2 - y1) / (x2 - x1)

= (4 - 8) / (10 - (-10))

= -0.15

Now we can use the point-slope form with the point (-10, 8) and slope -0.15:

y - 8 = -0.15(x - (-10))

Simplifying this equation, we get:

y - 8 = -0.15(x + 10)

Option (c) has the same equation as above after simplification, so the answer is (c):

y-8=-0.2(x+10).

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Answer:

a/sin(A) = b/sin(B) = c/sin(C)

where a, b, and c are the side lengths of the triangle, and A, B, and C are the opposite angles, respectively.

We are given A, C, and c, so we can use the Law of Sines to find B, a, and b.

a/sin(A) = c/sin(C)

a/sin(22) = 2.69/sin(51.4)

a = sin(22) * (2.69/sin(51.4))

a = 1.00

b/sin(B) = c/sin(C)

b/sin(B) = 2.69/sin(51.4)

b = sin(B) * (2.69/sin(51.4))

We can use the fact that the sum of the angles in a triangle is 180 degrees to find B:

B = 180 - A - C

B = 180 - 22 - 51.4

B = 106.6

Now we can substitute the values we have found into the Law of Sines to find b:

a/sin(A) = b/sin(B)

1/sin(22) = b/sin(106.6)

b = sin(106.6) * (1/sin(22))

b = 2.69

Therefore, B is approximately 106.6 degrees, a is approximately 1.00, and b is approximately 2.69.

Answer:

33361

Step-by-step explanation:

The equation appears to be
75 ( 1 + 0.84 )^x

Since it increases by 84% every 2 days, you'd devide 20 by 2, making 10

So to do the equation you'd shove that into a calculator since no one wants to hand do that sort of equation.

75 ( 1 + 0.84 )^10 = 33361.0332844 = 33361

This is a very small number, as indicated by the negative exponent. Specifically, it is 207.7 divided by 1,000,000.

An algebraic expression is a combination of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, and division) that can be simplified and evaluated.

It may contain one or more variables, which are symbols that represent unknown quantities or values that can vary. Algebraic expressions can be written using letters, symbols, or a combination of both, and they can be used to model real-world situations, solve problems, and express mathematical relationships.

For example, "3x + 5" is an algebraic expression that represents a linear equation with one variable "x".

2.077×10⁻⁴ means "2.077 times 10 to the power of -4", which can be rewritten as 0.0002077. This is a very small number, as indicated by the negative exponent. Specifically, it is 207.7 divided by 1,000,000.

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The equation that represents the proportional relationship between the initial height and the rebound height of the bouncy ball is D . r = 0.6 h

The proportional relationship can be found by the formula :

= Change in y / Change in x

= Change in rebound height / Change in initial height

= ( 0. 50 - 0. 30 ) / ( 0. 3 / 0 . 18 )

= 0.6 meters

The proportional relationship is :

= 0. 6 x initial height

= 0. 6 h

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All the real solutions of the equation are x=3, x=1, and x=-2. The rational root theorem states that any rational root of the equation x³-2x²-5x+6=0 can be written in the form p/q, where p is a factor of the constant term (6) and q is a factor of the leading coefficient (1). Therefore, the possible rational roots of the equation are: ±1, ±2, ±3, ±6

We can use synthetic division to test each of these possible roots until we find one that gives a remainder of 0.

1 | 1 -2 -5 6
 |   1 -1 4
 |_____________
 | 1 -1 -6 10

The remainder is not 0, so 1 is not a root of the equation.

-1 | 1 -2 -5 6
  |  -1  3 2
  |_____________
  | 1 -3 -2 8

The remainder is not 0, so -1 is not a root of the equation.

2 | 1 -2 -5 6
 |   2  0 10
 |_____________
 | 1  0 -5 16

The remainder is not 0, so 2 is not a root of the equation.

-2 | 1 -2 -5 6
  |  -2  8 14
  |_____________
  | 1 -4  3 20

The remainder is not 0, so -2 is not a root of the equation.

3 | 1 -2 -5 6
 |   3  3 0
 |_____________
 | 1  1 -2 6

The remainder is 0, so 3 is a root of the equation. We can use the quotient (1x²+1x-2) to find the remaining roots of the equation.

1x²+1x-2=0

Using the quadratic formula, we can find the remaining roots:

x = (-1 ± √(1²-4(1)(-2)))/(2(1))

x = (-1 ± √(1+8))/2

x = (-1 ± √9)/2

x = (-1 ± 3)/2

x = 1 or x = -2

Therefore, the real solutions using rational root theorem and quadratic formula are of the equation are x=3, x=1, and x=-2.

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The probability of rolling a sum greater than 5 is 13/18.

Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.

We know that, probability of an event = Number of favourable outcomes/Total number of outcomes

There are six different possible outcomes for a dice, the set (S) of all the outcomes can be listed as follows:

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

a) The probability of rolling a sum of even number

Number of favorable outcomes = 9

Total number of outcomes = 36

Now, probability = 9/36

= 1/4

b) The probability of rolling a sum of odd numbers

Number of favorable outcomes = 9

Total number of outcomes = 36

Now, probability = 9/36

= 1/4

c) The probability of rolling a sum of greater than 5

Number of favorable outcomes = 26

Total number of outcomes = 36

Now, probability = 26/36

= 13/18

Therefore, the probability of rolling a sum greater than 5 is 13/18.

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Gail should pick a Beautiful bank if she plans to have her money in the account for 15 years.

To determine which bank would be better for Gail, we need to calculate the future value of her investment after 15 years for each bank and compare the results.

For Neighborhood Bank, we can use the formula:

FV = P(1 + r/n)^(n*t)

where FV is the future value,

P is the principal (the initial amount invested),

r is the annual interest rate (as a decimal),

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

and t is the number of years.

Plugging in the values, we get:

FV = 5,500(1 + 0.012/1)^(1*15)

FV = 5,500(1.012)^15

FV = 7,130.34

For Beautiful Day Bank, we can use the formula:

FV = P(1 + r/n)^(n*t)

Plugging in the values, we get:

FV = 5,500(1 + 0.012/365)^(365*15)

FV = 5,500(1.000032876)^5475

FV = 7,145.25

Therefore, Beautiful Day Bank would be the better choice for Gail, as she would earn a higher amount of interest and end up with a higher future value for her investment after 15 years.

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There will be 22 blue signs at the end of the event.

The probability of an event can be determined by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of rolling a six on a standard six-sided die is 1/6, as there is only one favorable outcome (rolling a six) out of six possible outcomes (rolling a one, two, three, four, five, or six).

To solve this problem, we need to consider the factors that determine whether a house's sign will be flipped or not. A sign will only be flipped if the number of residents can divide the house number. For example, the sign on house number 12 will be flipped by the residents of that house because 12 is divisible by 2, 3, 4, and 6 (the factors of 12).

At the beginning, all signs are red, which means that none of the houses have had their sign flipped yet. Let's start with house number 1 and consider which houses will flip its sign.

For house number 1, its sign will not be flipped because 1 is only divisible by 1.

For house number 2, its sign will be flipped because 2 is divisible by 2.

For house number 3, its sign will not be flipped because 3 is only divisible by 1 and 3.

For house number 4, its sign will be flipped because 4 is divisible by 2 and 4.

For house number 5, its sign will not be flipped because 5 is only divisible by 1 and 5.

And so on, continuing this process for all 500 houses.

We can see that the signs on the houses with numbers that have an odd number of factors will end up blue, and the signs on the houses with numbers that have an even number of factors will end up red. This is because a factor always comes in pairs, except when the number is a perfect square, in which case the factor pair is identical (e.g. 4 has factors 1, 2, and 4, which is an odd number of factors).

So we need to count the number of integers from 1 to 500 that have an odd number of factors. These are the perfect squares, which are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, and 484.

Therefore, there will be 22 blue signs at the end of the event.

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If A varies directly as B and inversely as C then The value of A when B is 12 and C is 3 is 16.

To find the value of A, we can use the formula for direct and inverse variation of the given :
A = k*B/C

We are given that A is 12 when B is 6 and C is 2. We can plug these values into the formula to find the value of k:
12 = k*6/2
12 = 3k
k = 4

Now that we know the value of k, we can plug in the new values for B and C to find the value of A:
A = 4*12/3
A = 48/3
A = 16

Therefore, the value of A when B is 12 and C is 3 is 16. The correct answer is O 16.

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[tex] \: [/tex]

_________

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \small{\star \boxed{ \tt \color{blue} volume \: of \: cone \: = \frac{1}{3}h\pi {r}^{2} }}[/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━━━

hope it helps! :)

The corner point (8, 4) and is equal to 550

The minimum value of P = 50x + 75y exists at the corner point (3, 0) and is equal to 150. The maximum value of P = 50x + 75y exists at the corner point (8, 4) and is equal to 550.

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Answer:

277 milliliters

Step-by-step explanation:

do 945-668

this gives you 277

Answer:

The answer to your problem is, 277 milliliters.

Step-by-step explanation:

The reason I say that is because our problem that you are is a subtraction problem, so.. :

945 - 668 = 277.

Then we just add milliliters.

Thus the answer to your problem is, 277 milliliters.