7. Use a half-angle identity to find each exact value. a. sin 195 b. cos 165
8. Find the following: a. cos θ/2 given sin θ = - 4/5 with 180º < θ < 270º. b. cot θ/2, given tan θ = - √5 / 2 with 90° < θ < 180°.

Height of a trapezoid with Pythagorean theorem is = √{Hypotenuse ^2 - Base ^2}

Trapezoid has two parallel sides and two non parallel sides. The length of the parallel sides are unequal but the length of the non parallel sides are equal.

Thus the trapezoid can be divided into three parts where one is rectangle ( which has length equal to the shortest length of the parallel sides) and two triangles which are equal ( having equal base, height and hypotenuse).

The Pythagoras theorem on the triangular part of the trapezoid can be stated as ,

Hypotenuse ^2 = Base ^2 + Height ^2

⇒ Height ^2 = Hypotenuse ^2 - Base ^2

⇒ Height = √{ Hypotenuse ^2 - Base ^2}

where, Height of the triangle is equal to that of the trapezoid it belongs to;

Hypotenuse of the triangle is the non parallel but equal side of the trapezoid;

Base of the triangle is = {(length of the longest side of parallel sides of trapezoid) -   (length of the shortest side of parallel sides of trapezoid) }/2

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The two linear functions that have a minimum value but no maximum value are:

f(x) = 10 - 3x over the domain x ≤ 9

f(x) = 5x - 1 over the domain x ≥ -7

From the question, we are to determine the linear function that have minimum value but no maximum value

The decreasing function f(x) = 10 - 3x over the domain x ≤ 9 has a minimum value but no maximum value.

Domain: x ≤ 9

This function is decreasing since the slope is negative.

The minimum value occurs at x = 9, where f(9) = 10 - 3(9) = -17.

Since the slope is negative, the function continues to decrease without bound as x approaches negative infinity.

The decreasing function f(x) = 5x - 1 over the domain x ≥ -7 has a minimum value but no maximum value.

Domain: x ≥ -7

This function is increasing since the slope is positive.

The minimum value occurs at x = -7, where f(-7) = 5(-7) - 1 = -36.

Since the slope is positive, the function continues to increase without bound as x approaches positive infinity.

Hence, the functions are:

f(x) = 10 - 3x over the domain x ≤ 9

f(x) = 5x - 1 over the domain x ≥ -7

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Answer: First Picture 100:60  = The answer is A. 5:3 | The Second Picture  K/12=8/24 the answer is A. k=4  | Third Picture is A. 4 | Fourth Picture answer is B. 4 teachers for every 64 students | And Final Picture answer is D. 9.33 miles per hour, Wendy Was Faster. | Hope this helps out today!!

Step-by-step explanation: Its simple math its just either x2 or what's it divided off

the answer is x+10 i think bye

Given y = x³, y = 8, and x = 0, we need to find the volume generated when it is revolved about the y-axis.To solve this problem, we will use the washer method. We will draw the region to better understand it.  Area that is bounded by y = x³, y = 8, and x = 0, volume comes as  45/7 pi. The correct answer is option C

Now, let's draw the region we want to rotate around the y-axis. Below is the graph of the region after shading it. region which is bounded by , y = 8, and x = 0. We can see from the graph above that the region is between y = x³ and y = 8.

Thus the radius of our washer would be : R = 8 - x³  The height of the washer is dx (infinitesimal thickness). The width of the washer is given. We can now write the integral for the volume generated by the region V. Thus the volume generated when it is revolved about the y-axis is 45/7 pi. Therefore, the correct option is C.  

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Answer:

C. The account earns 1.2% in yearly interest.

Step-by-step explanation:

The expression 750(1+0.012) represents the amount of money in a savings account earning yearly interest after t years, where t is the number of years since the account was opened.

The expression (1+0.012) represents the interest rate as a decimal. The value 0.012 represents 1.2% in decimal form, which is the yearly interest rate earned by the account.

Therefore, (1+0.012) represents the account's yearly interest rate as a decimal, which is 1.2%.

the slope goes by several names

• average rate of change

• rate of change

• deltaY over deltaX

• Δy over Δx

• rise over run

• gradient

• constant of proportionality

however, is the same cat wearing different costumes.

[tex](\stackrel{x_1}{4}~,~\stackrel{y_1}{8})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{5}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{5}-\stackrel{y1}{8}}}{\underset{\textit{\large run}} {\underset{x_2}{8}-\underset{x_1}{4}}} \implies \cfrac{ -3 }{ 4 } \implies - \cfrac{3 }{ 4 }[/tex]

Answer: In each of the terms 3x 2 and 7x, x is a variable, and in both terms, we are multiplying a number by the variable. In the first term, 3x 2, 3 is being multiplied by the variable, so 3 is a coefficient. In the second term, 7x, 7 is being multiplied by the variable, so 7 is a coefficient.

Step-by-step explanation:

The value of x and y are

x=100° y=100°

In an equilateral triangle,

Each angle = 60°

∴ a + a + 100° = 180°

⇒ 2a + 100° = 180°

⇒ 2a = 180° – 100° = 80°

∴ a = 80°/2 = 40° ∴ x = 60° + 40° = 100°

And y = 60° + 40° = 100°

An equilateral triangle in geometry is a triangle with equally long sides. The three angles opposite the three equal sides are equal in size because the three sides are equal. As a result, with each angle measuring 60 degrees, it is sometimes referred to as an equiangular triangle. Equilateral triangles have the same area, perimeter, and height formula as other kinds of triangles.

An equilateral triangle has a predictable shape. By combining the words "Equi" (which means equal) and "Lateral," which refers to sides, the word "Equilateral" is created. Due to the equality of its sides, an equilateral triangle is also known as a regular polygon or regular triangle.

from the question:

In an equilateral triangle,

Each angle = 60°

Let each base angle = a

∴ a + a + 100° = 180°

⇒ 2a + 100° = 180°

⇒ 2a = 180° – 100° = 80°

∴ a = 80°/2 = 40° ∴ x = 60° + 40° = 100°

And y = 60° + 40° = 100°

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complete question

Apply the properties of equilateral triangles and find the values of x and y in the given figure

The equation which represents exponential growth is y = (1.2)* and equation which represents exponential decay is y = (.71)*,  y = 0(6.3)* represents a constant function.

Consider the function:

[tex]y = a(1\pm r)^m[/tex]

where m is the number of times this growth/decay occurs, a = initial amount, and r = fraction by which this growth/decay occurs.

If there is plus sign, then there is exponential growth happening by r fraction or 100r %

If there is negative sign, then there is exponential decay happening by r fraction or 100r %

We are given that;

c. y = (1.2)*

d. y = 0(6.3)*

e. y = (.71)*

a.) The equation and graph that show exponential growth is y = (1.2)x. This is because the base of the exponent (1.2) is greater than 1, so as x increases, the value of y increases at an increasing rate. The graph of y = (1.2)x is an upward-curving curve that gets steeper as x increases.

b.) The equation and graph that show exponential decay is y = (0.71)x. This is because the base of the exponent (0.71) is between 0 and 1, so as x increases, the value of y decreases at a decreasing rate. The graph of y = (0.71)x is a downward-curving curve that flattens out as x increases.

Therefore, exponential growth shows y = (1.2)* whereas y = (.71)* show exponential decay.

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If the values of A and B make the equation 35x – 29/x²-3x+2 = A/x-1 +b/x-2 true for all values of x for which it is defined, A+B is 35.

The given equation is 35x - 29/(x² - 3x + 2) = A/(x - 1) + B/(x - 2).

We can simplify the denominator of the second term on the left hand side by factoring it:

35x - 29/[(x - 1)(x - 2)] = A/(x - 1) + B/(x - 2)

Now, we can multiply both sides of the equation by (x - 1)(x - 2) to get rid of the fractions:

35x(x - 1)(x - 2) - 29 = A(x - 2) + B(x - 1)

Expanding the left hand side gives us:

35x³ - 70x² + 35x - 29 = A(x - 2) + B(x - 1)

Now, we can compare the coefficients of x on both sides of the equation to find the values of A and B. The coefficient of x on the left hand side is 35, and on the right hand side it is A + B. Therefore, A + B = 35.


Therefore, the values of A and B make the equation 35x – 29/x^2-3x+2 = A/x-1 +b/x-2 true for all values of x for which it is defined is 35.

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To find the total amount of milk used, we simply add the amount of milk used for the pancake recipe and the amount used for drinking. So we add 9 cups and 394 cups together, giving us a total of 403 cups of milk.

We can calculate the total amount of milk that Luke used for the pancake recipe and the additional amount he used for drinking.

For the pancake recipe, Luke used 9 cups of milk.

For drinking, Luke used an additional 394 cups of milk.

Total amount of milk used = Milk used for pancake recipe + Milk used for drinking

= 9 cups + 394 cups

= 403 cups of milk.

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The complete question is:

Lube used to 9 cups of milk for a pancake recipe to drink another 394 cups of milk how about about how much milk did Luke is using on the recipe?

Mr stove's final balance after 10 years if he invested 1500 on a bank account that would give him 3. 7% compounded monthly is $2,170.37

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

Where

P = $1500

r = 3.7% = 0.037

n = monthly = 12

t = 10 years

So,

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

A = 1,500.00(1 + 0.037/12)^(12×10)

A = 1,500.00(1 + 0.0030833333333333)¹²⁰

A = $2,170.37

Ultimately, Mr stove will have $2,170.37 as his balance after 10 years.

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The result is x^(2)-7x+20 with a remainder of -20. So the final answer is x^(2)-7x+20-20/(x-1).

To find the result of the division using synthetic division, we need to follow these steps:

The result is x^(2)-7x+20 with a remainder of -20. So the final answer is x^(2)-7x+20-20/(x-1).

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The solution to the system of equations using a calculator and a matrix (3x-6y=12, 2x-4y=8) is x = 12 and y = 8.

To solve the system of equations using a calculator and a matrix, you will need to follow these steps:
1. Convert the system of equations into a matrix. In this case, the matrix will be:
```
| 3 -6 |  | 12 |
| 2 -4 |  |  8 |
```

2. Use your calculator to find the inverse of the coefficient matrix (the matrix on the left). The inverse of the coefficient matrix is:
```
| -2/3  3/2 |
| -1/3  3/4 |
```

3. Multiply the inverse of the coefficient matrix by the constant matrix (the matrix on the right) to find the solution. The solution will be:
```
| -2/3  3/2 |  | 12 |  =  | 12 |
| -1/3  3/4 |  |  8 |     |  8 |
```

4. Simplify the solution to get the values of x and y. The solution will be:
```
x = 12
y = 8
```

Therefore, the solution to the system of equations is x = 12 and y = 8.

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Therefore, after 10 payments, Jeff owes $1160.

Since $1160 is less than $2200, we can conclude that Jeff's furniture costs less than Ella's.

A mathematical equation, such as 6 x 4 = 12 x 2, can be used to compare two amounts or values. a significant noun. An equation is employed when it's required to integrate two or more elements in order to understand or fully explain a situation.

To determine whose furniture costs more, we need to compare the amounts owed by Jeff and Ella for the same number of payments.

Jeff's equation is y = -108x + 2240, which gives the amount owed after x payments.

Ella's graph does not have an equation provided, but we can see that after 10 payments, she owes approximately $2200.

To compare the amounts owed after 10 payments, we can substitute x = 10 into Jeff's equation:

y = -108(10) + 2240

y = -1080 + 2240

y = 1160

Therefore, after 10 payments, Jeff owes $1160.

Since $1160 is less than $2200, we can conclude that Jeff's furniture costs less than Ella's.

Answer: Ella's furniture costs more than Jeff's.

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a)T(u) + T(v) T is a linear transformation

b)[3 -2 ; 2 0]

c)T is one-to-one.

a) To prove that T is a linear transformation, we need to show that it satisfies two conditions.

1. Additivity: T(u + v) = T(u) + T(v)
Let u = [u1, u2] and v = [v1, v2], then
T(u + v) = T([u1 + v1, u2 + v2]) = [3(u1 + v1) - 2(u2 + v2), 2(u2 + v2)]
= [3u1 - 2u2 + 3v1 - 2v2, 2u2 + 2v2]
= [3u1 - 2u2, 2u2] + [3v1 - 2v2, 2v2]
= T(u) + T(v)

2. Homogeneity: T(cu) = cT(u)
Let u = [u1, u2] and c be any scalar, then
T(cu) = T([cu1, cu2]) = [3cu1 - 2cu2, 2cu2] = c[3u1 - 2u2, 2u2]
= cT(u)

Therefore, T satisfies both conditions, and is a linear transformation.

b) The standard matrix of T is the matrix A = [3 -2 ; 2 0]

c) To determine whether T is one-to-one, we look at the matrix A. Since the matrix A has a non-zero determinant, which is equal to -4, then T is one-to-one.

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In this case, the point (3/2, 0) is the optimal solution, which matches the solution we found using the simplex method.

To solve the problem using the simplex method, we first need to convert the inequalities to equations by adding slack variables. Then we can set up the initial tableau and use the simplex method to find the optimal solution. Finally, we can verify the solution using the graphical method.

Convert inequalities to equations by adding slack variables
-X1 + X2 >= -1 becomes -X1 + X2 + S1 = -1
4X1 + 3X2 <= 12 becomes 4X1 + 3X2 + S2 = 12
2X1 <= 3 becomes 2X1 + S3 = 3

Set up the initial tableau
| -1 | 2 | 1 | 0 | 0 | 0 |
| -1 | 1 | 1 | 0 | 0 | -1 |
| 4 | 3 | 0 | 1 | 0 | 12 |
| 2 | 0 | 0 | 0 | 1 | 3 |

Use the simplex method to find the optimal solution
We first need to choose the pivot column, which is the one with the most negative coefficient in the objective row. In this case, it is the first column. Then we need to choose the pivot row, which is the one with the smallest positive ratio of the right-hand side to the pivot column coefficient. In this case, it is the third row. We can then use the pivot element to eliminate the other coefficients in the pivot column and repeat the process until we have no negative coefficients in the objective row.

After performing the simplex method, we get the following final tableau:
| 0 | 5/4 | 1 | 1/4 | 0 | 3 |
| 0 | 7/4 | 1 | 1/4 | 0 | 3 |
| 0 | 3/4 | 0 | 1/4 | 0 | 3 |
| 1 | 0 | 0 | 0 | 1/2 | 3/2 |

The optimal solution is X1 = 3/2, X2 = 0, and Z = -3/2.

Verify the solution using the graphical method
We can graph the constraints and find the feasible region. Then we can graph the objective function and find the point where it intersects the feasible region that gives the maximum or minimum value. In this case, the point (3/2, 0) is the optimal solution, which matches the solution we found using the simplex method.

Extra Credit:
To solve the problem using the simplex method, we first need to convert the inequalities to equations by adding slack variables. Then we can set up the initial tableau and use the simplex method to find the optimal solution. Finally, we can verify the solution using the graphical method.

Convert inequalities to equations by adding slack variables
-X1 + X2 >= -1 becomes -X1 + X2 + S1 = -1
4X1 + 3X2 <= 12 becomes 4X1 + 3X2 + S2 = 12
2X1 <= 3 becomes 2X1 + S3 = 3

Set up the initial tableau
| -1 | 2 | 1 | 0 | 0 | 0 |
| -1 | 1 | 1 | 0 | 0 | -1 |
| 4 | 3 | 0 | 1 | 0 | 12 |
| 2 | 0 | 0 | 0 | 1 | 3 |

Use the simplex method to find the optimal solution
We first need to choose the pivot column, which is the one with the most negative coefficient in the objective row. In this case, it is the first column. Then we need to choose the pivot row, which is the one with the smallest positive ratio of the right-hand side to the pivot column coefficient. In this case, it is the third row. We can then use the pivot element to eliminate the other coefficients in the pivot column and repeat the process until we have no negative coefficients in the objective row.

After performing the simplex method, we get the following final tableau:
| 0 | 5/4 | 1 | 1/4 | 0 | 3 |
| 0 | 7/4 | 1 | 1/4 | 0 | 3 |
| 0 | 3/4 | 0 | 1/4 | 0 | 3 |
| 1 | 0 | 0 | 0 | 1/2 | 3/2 |

The optimal solution is X1 = 3/2, X2 = 0, and Z = -3/2.

Verify the solution using the graphical method
We can graph the constraints and find the feasible region. Then we can graph the objective function and find the point where it intersects the feasible region that gives the maximum or minimum value. In this case, the point (3/2, 0) is the optimal solution, which matches the solution we found using the simplex method.

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Answer: 1250 students

Step-by-step explanation:

20% + 40% + 20% = 80%

100% - 80% = 20%

If 250 is 20%, then it's just 250 x 5 = 1250

To find the value of k for which the matrix A has one eigenvalue of multiplicity 2, we can use the characteristic equation of the matrix. The characteristic equation is det(A-λI)=0. For a given matrix, this equation will yield a polynomial equation of degree equal to the dimension of the matrix.

In this case, the equation is (λ+4)(λ-2) = 0. Solving this equation gives us two distinct eigenvalues of λ= -4 and λ=2. This means that for any value of k, the matrix A will have two distinct eigenvalues, and thus cannot have one eigenvalue of multiplicity 2. Therefore, the matrix A cannot have one eigenvalue of multiplicity 2 no matter what value of k is used.

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Answer:

  (a)  8 : 3

  (b)  1 7/8 cups

Step-by-step explanation:

Given a recipe that calls for 2 cups of flour and 3/4 cup of water, you want to know the ratio in simplest terms, and the amount of water for 5 cups of flour.

The ratio can be written and simplified as a fraction. Fractions are divided in the usual way.

  2/(3/4) = 2 ÷ 3/4 = 2 × 4/3 = 8/3 = 8 : 3

If we multiply each term in this ratio by 5/8, we can find the recipe that uses 5 cups of flour:

  (5/8)·8 : (5/8)·3 = 5 : 15/8 = 5 : 1 7/8

Naomi uses 1 7/8 cups of water with 5 cups of flour.

Answer:

Step-by-step explanation:

[tex](a)[/tex]

[tex]2:\frac{3}{4}[/tex]

Multiply both sides by 4 to remove fraction:

[tex]2\times4:\frac{3}{4} \times 4[/tex]

[tex]8:3[/tex]     (this is simplest form because no number goes into both 8 and 3)

[tex](b)[/tex]

5 cups of flower:

[tex]8 \times \frac{5}{8} : 3 \times \frac{5}{8}[/tex]   (I chose [tex]\frac{5}{8}[/tex] to turn the 8 into 5)

[tex]5:\frac{15}{8}[/tex]

5 cups flour needs [tex]\frac{15}{8}[/tex] ([tex]=1\frac{7}{8}[/tex]) cups water

Answer:

33.7°

Step-by-step explanation:

the direction angle is calculated as

tanΘ = [tex]\frac{y}{x}[/tex] ( Θ is the direction angle ) , then

tanΘ = [tex]\frac{-4}{-6}[/tex] = [tex]\frac{2}{3}[/tex]

Θ = [tex]tan^{-1}[/tex] ( [tex]\frac{2}{3}[/tex] ) ≈ 33.7° ( to the nearest tenth )

The x intercept and y intercept of the equation is 8 and 200 respectively.

x intercept is the point on the line where it touches the X axis.

y intercept is the point on the line where it touches the Y axis.

Given equation is,

y = 200 - 25x

where y is the total amount in Amanda's school lunch account after purchasing lunches for x weeks.

x intercept is the point on X axis. Any point on x axis has y coordinate 0.

So x intercept is the x coordinate when y coordinate is 0.

0 = 200 - 25x

25x = 200

x = 8

This means that the amount in the account will become 0, after purchasing lunch for 8 weeks.

y intercept is the y coordinate when x coordinate = 0.

y = 200 - (25 × 0)

y = 200

This indicates the amount in the account at the start before purchasing any lunch.

Hence the x intercept is 8 and y intercept is 200.

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Your question is incomplete. The complete question is as follows.

Amanda has $200 in her lunch account. She spends 25$ each week on lunch. The equation y=200-25x represents the total amount in Amanda's school lunch account, y for x weeks of purchasing lunches.

Find the x and y intercepts and interpret their meaning in the context of the situation.

We have the following response after answering the given question: As a result, the following are the answers to the equation: [tex]x = 4 + 6 = 10[/tex] and [tex]x = 4 - 6 = -2[/tex]

In a mathematical equation, the equals sign (=), which connects two claims and denotes equality, is utilised. In algebra, an equation is a mathematical statement that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a space. Mathematical expressions can be used to describe the relationship between the two sentences on either side of a letter. The logo and the particular piece of software frequently correspond. like, for instance, 2x - 4 = 2.

We may use the completing the square method to solve the equation x2 - 8x + 5 = 25 by doing the following steps:

[tex]x^2 - 8x + 5 - 25 = 0\\x^2 - 8x - 20 = 0\\(-8/2)^2 = c\\16 = c\\x^2 - 8x + 16 - 16 - 20 = 0\\(x - 4)^2 - 36 = 0\\(x - 4)^2 = 36\\x - 4 = +6\\x = 4 + 6[/tex]

As a result, the following are the answers to the equation:

[tex]x = 4 + 6 = 10[/tex]

[tex]x = 4 - 6 = -2[/tex]

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The population will reach 7 million people approximately 21.67 years after 2002. Rounding to the nearest whole number, this corresponds to the year 2024.

The year when the population reaches 7 million can be determined using the population model given, f(x) = 6.7(1.0032)x, where x is the number of years following 2002.

When we solve for x while setting f(x) equal to 7, we obtain:

[tex]7 = 6.7(1.0032)^x[/tex]

By multiplying both sides by 6.7, we obtain:

[tex]1.04478 = 1.0032^x[/tex]

When we take the natural logarithm of both sides, we obtain:

ln(1.04478) = x * ln (1.0032)

After finding x, we obtain:

x = ln(1.04478) / ln (1.0032)

x ≈ 21.67

Hence, approximately 21.67 years after 2002, the population will reach 7 million. This corresponds to the year 2024 when rounded to the next whole number.

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The correct coordinates of N' is (-3, -4).    

Rotation of coordinates involves changing the orientation of a point or object in a plane about a fixed point called the center of rotation.

This process can be accomplished by applying a set of transformation rules to the original coordinates of the object.

When a point (x, y) rotated about the origin at 180° then the coordinates resultant point are (-x, -y).

Here we have

Point N has coordinates (3, 4)

Given that the point N is rotated about 180°  

As we know when the point (x, y) rotated about the origin at 180° then the resultant point is (-x, -y)

Here (3, 4) is rotated about 180°

The resultant point N' is (-3, -4)      

Therefore,

The correct coordinates of N' is (-3, -4).    

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The area of the parallelogram formed by the vectors u and v is 12.

The area of a parallelogram formed by two vectors u and v can be determined by finding the cross product of the two vectors and then taking the magnitude of the resulting vector.

First, we need to find the cross product of u and v:

u × v = [(2)(0) - (2)(4), (2)(4) - (1)(0), (1)(4) - (2)(4)] = [-8, 8, -4]

Next, we need to find the magnitude of the resulting vector:

|u × v| = √((-8)² + (8)² + (-4)²) = √(64 + 64 + 16) = √144 = 12

Therefore, the area of the parallelogram formed by the vectors u and v is 12.

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