Find Tan A and Tan B. write each answer as a fraction and as a decimal rounded into four places.

a. The scalar product of vectors A = A₁i + A2₂j + A₃k and B = B₁i + B₂j + B₃k. is A ∙ B = A₁B₁ + A₂B₂ + A₃B₃.

b. The angle between the vectors A = 2i +2j - k and B = 7i + 24k is 97.67°

The scalar product of two vectors a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k is given by

a.b = abcosФ = a₁b₁ + a₂b₂ + a₃b₃ where

6a) Suppose A = A₁i + A2₂j + A₃k and B = B₁i + B₂j + B₃k. Prove that A ∙ B = A₁B₁ + A₂B₂ + A₃B₃.

We proceed as follows

We know that

A ∙ B = (A₁i + A₂j + A₃k). (B₁i + B₂j + B₃k)

= A₁i.(B₁i + B₂j + B₃k) + A₂j.(B₁i + B₂j + B₃k) + A₃k.(B₁i + B₂j + B₃k)

Expanding the brackets, we have

= A₁i.B₁i + A₁i.B₂j + A₁i.B₃k + A₂j.B₁i + A₂j.B₂j + A₂j.B₃k + A₃k.B₁i + A₃k.B₂j + A₃k.B₃k

= A₁B₁i.i + A₁B₂i.j + A₁B₃i.k + A₂B₁j.i + A₂B₂j.j + A₂B₃j.k + A₃B₁k.i + A₃B₂k.j + A₃B₃k.k

We know that

So, we have that

= A₁B₁i.i + A₁B₂i.j + A₁B₃i.k + A₂B₁j.i + A₂B₂j.j + A₂B₃j.k + A₃B₁k.i + A₃B₂k.j + A₃B₃k.k

= A₁B₁(1) + A₁B₂(0) + A₁B₃(0) + A₂B₁(0) + A₂B₂(1) + A₂B₃(0) + A₃B₁(0) + A₃B₂(0) + A₃B₃(1)

= A₁B₁ + 0 + 0 + 0 + A₂B₂ + 0 + 0 + 0 + A₃B₃

= A₁B₁ + A₂B₂ + A₃B₃

So, A ∙ B = A₁B₁ + A₂B₂ + A₃B₃.

6b) To find the angle between the vectors A = 2i +2j - k and B = 7i + 24k, we proceed as follows.

We know that the angle between two vectors is given by

Ф = cos⁻¹[(A.B)/AB​] where

Now, A.B =  (2i +2j - k).(7i + 24k) = (2 × 7 + 2 × 0 + (-1) × 24)

= (14 + 0 - 24)

= -10

Now, the magnitude of a vector C = C₁i + C₂j + C₃k is C = √(C₁² + C₂² + C₃²)

Since vector A = 2i +2j - k its magniude A = √(2² + 2² + (-1)²)

= √(4 + 4 + 1)

= √9

= 3

Also since vector B = 7i + 24k,  its magniude A = √(7² + 0² + 24²)

= √(49 + 0 + 576)

= √625

= 25

So, substituting the values of the variables into the equation, we have that

Ф = cos⁻¹[(A.B)/AB​]

Ф = cos⁻¹[(-10)/(3 × 25)]

Ф = cos⁻¹[(-2)/(3 × 5)]

Ф = cos⁻¹[-2/15]

Ф = -0.13333

Ф = 97.67°

So, the angle betwen the vectors is 97.67°

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Answer:

19

Step-by-step explanation:

tan35=x/20

20tan35=x

x=14.0041507641942

x+5=19.0041507641942

The nearest foot is 19

Answer: The average is a descriptive statistic that provides an idea of the central or typical value in a dataset

Step-by-step explanation:

The average of the numbers is 4

The given parameters are;

The numbers; 2, 5, 8, and 1

The required parameter

To find the average;

Solution:

The average is given by the sum of the data divided by the size or count of the data points

Therefore, the average of the given numbers is given as follows;

Average, x= 2+5+8+1/4=4

The average of the numbers,  = 4

Answer:

50% ate uncle joe and 50% ate robert so, uncle joe make 14 cookies

Answer:

1. Michael ate 50% of the cookies, so the remaining cookies are 50% of the total. Let's denote the total number of cookies as T. So, after Michael ate his share, 0.5 x T cookies remained.

2. Robert then ate 50% of what was left, leaving only 50% of the cookies that were left after Michael ate. So, after Robert ate his share, 0.5 x 0.5 x T = 0.25 x T cookies remained.

3. We know that 14 cookies remained after Robert ate his share.

So, we can set up the equation 0.25 x T = 14 and solve for T:

T = 14/0.25 = 56

So, Uncle Joe made 56 chocolate chip cookies.

Step-by-step explanation:

Hope it helps!!!

Answer:

Step-by-step explanation:

Answer: 0.04

Step-by-step explanation:

The area [tex]A_1[/tex] of the square target with side length [tex]s_1=20 cm[/tex] is:

[tex]A_1=s_1^{2}=20^{2}=400[/tex]

The area [tex]A_2[/tex] of the shaded square with side length [tex]s_2=4cm[/tex] is:

[tex]A_2=s_2^{2}=4^{2}=16[/tex]

So, the probability that the dart will land in the shaded square on the target is:

[tex]\frac{A_{2}}{A_{1}}=\frac{16}{400}=0.04[/tex]

The fisherman can expect to catch 120 unhealthy salmon during the next month.

We can use the proportion of unhealthy to healthy salmon that was seen in the fisherman's catch of 15 salmon (12 unhealthy, 3 healthy) and multiply that proportion by the total amount of salmon he plans to catch (150) to find our answer.

Unhealthy = 150 × (12/15)

Unhealthy = 120

Therefore, the fisherman can expect to catch 120 unhealthy salmon during the next month.

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The length of the arc RT is 11.72 units.

An arc is defined as a portion of the boundary of a circle or a curve.

To calculate the length of the arc RT, we use the formula below

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

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Given statement solution is :- The volume of the resulting part is approximately 267 cubic inches.

To find the volume of the resulting part, we need to calculate the volume of the larger cone and subtract the volume of the smaller cone.

The formula for the volume of a cone is given by:

V = (1/3) * π * [tex]r^2[/tex] * h

where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the radius of the cone's base, and h is the height of the cone.

For the larger cone:

Radius (r) = 5 inches

Height (h) = 12 inches

V_large_cone = (1/3) * π * [tex](5^2)[/tex] * 12

V_large_cone = (1/3) * π * 25 * 12

V_large_cone = (1/3) * π * 300

V_large_cone = 100π

For the smaller cone:

Radius (r) = 3 inches

Height (h) = 5 inches

V_small_cone = (1/3) * π * [tex](3^2)[/tex] * 5

V_small_cone = (1/3) * π * 9 * 5

V_small_cone = (1/3) * π * 45

V_small_cone = 15π

Therefore, the volume of the resulting part is:

V_resulting_part = V_large_cone - V_small_cone

V_resulting_part = 100π - 15π

V_resulting_part = 85π

Now, we can approximate the value of π as 3.14159:

V_resulting_part ≈ 85 * 3.14159

V_resulting_part ≈ 267.10715

Rounded to the nearest whole number, the volume of the resulting part is approximately 267 cubic inches.

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Answer:

Step-by-step explanation:

9x-3y=-10 ...............(1)

3x-4y=1...............(2)

multiplying equation (2) by 3

9x-12y=3...................(3)

Using elimination method, then

9x-3y=-10 ...............(1)

9x-12y=3...................(3

9y= -13

y= -13/9

substituting y= -13/9 in equation (1) then

9x-3(-13/9)= -10

9x+13/3= -10

multiplying throughout by 3

27x+13= -30

27x= -30-13

27x= -43

x= -43/27

since x and y values are known, then

12x-7y = 12(-43/27) - 7(-13/9)

12x-7y = -516/27 + 91/9

12x-7y = -9

Answer:

40

Step-by-step explanation:

plugging in a and b will get us:

[tex]2^{2} + 6^{2} = c^{2}[/tex]

4 + 36 = [tex]c^{2}[/tex]

40 = [tex]c^{2}[/tex]

c= [tex]\sqrt{40\\[/tex]

The output value of f(-1) include the following: -13.

In Mathematics and Geometry, a piecewise-defined function simply refers to a type of function that is defined by two (2) or more mathematical expressions over a specific domain.

Generally speaking, the domain of any piecewise-defined function simply refers to the union of all of its sub-domains.

Since the value of x is -1, the output value of this piecewise-defined function can be calculated by using the first function as follows;

f(x) = 3x - 10

f(-1) = 3(-1) - 10

f(-1) = -3 - 10

f(-1) = -13

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The completely factorized forms are:

1.1) 2x + 4y - 6z (no further factorization)

[tex]1.2) 2pq(5p^5q - 2p^2 + 1)[/tex]

1.3) (m + n)(m + n - 5p)

1.4) 7c(4 - a) + d(-4 + a)

We have,

Let's factorize each expression completely:

1.1)

2x + 4y - 6z

There are no common factors among the terms, so we can't factorize it further.

1.2)

[tex]10p^6q^2 - 4p^3q^2 + 2pq[/tex]

The common factor among the terms is 2pq, so we can factor it out:

[tex]2pq (5p^5q - 2p^2 + 1)[/tex]

1.3)

(m + n)² - 5p(m + n)

Using the distributive property, we can expand the squared term:

(m + n)(m + n) - 5p(m + n)

Now we can factor out the common factor (m + n):

(m + n)(m + n - 5p)

1.4)

4(7c - d) + a(d - 7c)

Using the distributive property, we can expand the terms:

28c - 4d + ad - 7ac

Rearranging the terms:

(28c - 7ac) + (-4d + ad)

Factoring out common factors:

7c(4 - a) + d(-4 + a)

Thus,

The completely factorized forms are:

1.1) 2x + 4y - 6z (no further factorization)

[tex]1.2) 2pq(5p^5q - 2p^2 + 1)[/tex]

1.3) (m + n)(m + n - 5p)

1.4) 7c(4 - a) + d(-4 + a)

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Answer:

6.6

Step-by-step explanation:

I’ve done this before. The original answer I did was 6.63 but it says round to the nearest tenth so yep…

The correct statement is:

C. The function has a y-intercept at (0, -2).

Let's analyze each statement:

A. As the value of x increases, the value of f(x) moves toward a constant.

This statement is false.

The function f(x) = (1/5)ˣ - 2 is an exponential function with a base of 1/5. As x increases, the exponential term (1/5)ˣ becomes smaller and approaches zero, causing f(x) to approach -2.

However, it does not approach a constant value.

B. The domain of the function is (-2, ∞).

This statement is false.

The function f(x) = (1/5)ˣ - 2 is defined for all real numbers.

There are no restrictions on the domain, so the correct statement is that the domain is (-∞, ∞).

C. The function has a y-intercept at (0, -2).

This statement is true.

To find the y-intercept, we set x = 0 in the function and solve for f(x):

f(0) = (1/5)⁰ - 2

= 1 - 2

= -1

Therefore, the y-intercept of the function is (0, -2).

D. The function is increasing.

This statement is true.

An exponential function with a positive base, such as (1/5)ˣ, is always decreasing.

However, when we subtract 2 from the exponential term, it shifts the graph vertically downward by 2 units.

This transformation makes the function f(x) = (1/5)ˣ - 2 increasing.

So, the correct statement is:

C. The function has a y-intercept at (0, -2).

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The required sample size to estimate the population proportion with 98% confidence and a margin of error of 1.5% is approximately 4,847.60.

To find the sample size required to estimate a population proportion with a 98% level of confidence, we will use the formula:

n = (z² * p * q) / E²

where:

n = sample size

z = z-value associated with the desired confidence level

p = population proportion - estimated

q = 1 - p (complement of the estimated proportion)

E = desired margin of error

We, first of all, will calculate the required sample size:

Given:

z = z-value corresponding to a 98% confidence level

E = 1.5% = 0.015

p = 28% = 0.28

q = 1 - p = 1 - 0.28 = 0.72

Then, we use a standard normal distribution table or a calculator to find the z-value. For a 98% confidence level, the z-value is approximately 2.326.

Putting the values into the formula:

n = (2.326² * 0.28 * 0.72) / 0.015²

n = (5.410 * 0.2016) / 0.000225

n≈ 1.091/0.000225

n ≈ 4,847.60

Hence, to estimate the population proportion with a 98% confidence level and a margin of error of 1.5%, a sample size of approx. 4,848.60 is required.

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Answer:

When [tex]x=0[/tex], the value of f(x) is 1.

Each time x increases by 1, f(x) is multiplied by 4.

Equation of function: [tex]f(x)=1\cdot 4^x[/tex]

Step-by-step explanation:

The detailed explanation is attached below.

The value of the unknown side to the nearest hundredth is 6.78

Cosine rule states that; if a, b,c are the sides of a triangle and A,B,C are the opposite angles of the sides, then

c² = a² + b² -2ab CosC

Cosine rule is used when all the sides or two of the sides are given. The angle opposite to side c is angle C.

c² = 4² + 10² - 2(4)(10)cos 29

c² = 16 +100 - 80cos29

c² = 116 -69.97

c² = 46.03

c = √ 46.03

c = 6.78 ( nearest hundredth).

Therefore the value of c is 6.78

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The frequency distribution table would be:

Class ______ Freq ___ Midpoint __R/Freq__ C/Freq

0 - 9 _______ 4 ______ 4.5 ______ 0.2 _____ 4

10-19 _______ 5 ______ 14.5 _____0.25 _____9

20-29 ______ 4 ______ 24.5_____ 0.2______ 13

30-39_______3 ______ 34.5 _____0.15______ 16

40-49_______4 ______ 44.5 _____ 0.2 _____ 20

The data ranges from 0 - 49 . we had to use a class interval of 10 since we were interested in having just 5 classes.

The midpoint is the average value of the upper and lower class.

Relative frequency is the ratio of the class frequency to the total number of data in the distribution.

Cumulative Frequency is the running sum of frequencies in the distribution table.

Therefore, the frequency distribution captures all the information required.

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Using the division operation , Natalie can do 64 loads of laundry with her bottle of detergent.

From the line plot , the detergent with 4 bottles has 8 cups in it.

Fraction of cup used per laundry = 1/8

Total cups of detergent in bottle = 8

Number of laundry Natalie can do :

(Total cups of detergent in bottle / Fraction of cup used per laundry )

Number of loads of laundry that can be done :

= 8 / (1/8)

= 8 × 8/1

= 64

Therefore, Natalie can do 64 loads of laundry with her bottle of detergent.

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If u =(1+i√3) and v=(1-i√3), product uv is: D. 4.

To find the product of u and v let us simply multiply them together:

u = 1 + i√3

v = 1 - i√3

uv = (1 + i√3)(1 - i√3)

Using the difference of squares formula (a² - b² = (a + b)(a - b)) we can simplify the expression:

uv = (1 + i√3)(1 - i√3)

uv= 1² - (i√3)²

uv= 1 - (-3)

uv= 1 + 3

uv= 4

Therefore the product uv is equal to 4.

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Answer:

5(5y - 3z)

Step-by-step explanation:

25y - 15z ← factor out common factor of 5 from each term

= 5(5y - 3z)

Answer: [tex]x=2\sqrt{5}[/tex]

Step-by-step explanation:

Detailed explanation is attached below.

34. I haven't seen Tom since his wedding day.

36. Mary hasn't spoken French in years.

37. I haven't had a good night's sleep in ten weeks.

38. He hasn't paid taxes since 1970.

40. The windows haven't been cleaned for 3 months.

41. It's been years since I took photographs.

The perfect tense is a grammatical construction used to express actions that are completed or have happened before the present moment. It indicates that an action was performed and finished in the past but has relevance or impact on the present.

There are three main forms of the perfect tense: present perfect, past perfect, and future perfect.

1. Present Perfect Tense: This tense is used to describe an action or event that started in the past and has a connection to the present. It emphasizes the result or completion of the action.

2. Past Perfect Tense: This tense is used to describe an action or event that occurred before another action or a specific point in the past. It expresses an action completed before a specified time in the past.

3. Future Perfect Tense: This tense is used to describe an action that will be completed before a specified time or point in the future.

Each form of the perfect tense can be used with different time references to indicate the relationship between the completed action and the present or other points in time.

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Topic Sentence: In a business setting, concepts from this course can be applied when optimizing customer service operations and improving customer satisfaction.

Supporting Detail: One concept from the course that can be applied is the use of queuing theory to manage customer wait times and reduce service bottlenecks. By analyzing arrival rates, service rates, and queue lengths, businesses can optimize staffing levels and allocate resources efficiently.

Worked Example: One practical application of queuing theory is in call centers. With the aid of this theory, the center can analyze the number of customer service reps required in different periods of the day. The objective is to keep waiting times at their lowest and deliver timely service to customers.

Supporting Detail: Another concept that can be applied is the concept of customer lifetime value (CLV). By understanding CLV, businesses can identify their most valuable customers, prioritize their needs, and tailor personalized marketing strategies to enhance customer retention.

Worked Example: An e-commerce business may use information on customer buying patterns, typical order amounts, and the rate of customer loss to determine the overall lifetime value of a customer. By utilizing this data, the organization has the opportunity to execute tailored loyalty initiatives and customized suggestions, ultimately enhancing customer contentment and optimizing overall revenue for the long haul.

Conclusion:

The application of course concepts in a business environment can result in better customer service, shorter wait times, higher customer approval, and increased customer allegiance, thereby fueling the expansion and profitability of a business.

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The remainder of the polynomial division [tex]\frac{\left(y^4-y^3+2y^2+y-1\right)}{\left(y^3+1\right)}[/tex] is 2y²

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

[tex]\frac{\left(y^4-y^3+2y^2+y-1\right)}{\left(y^3+1\right)}[/tex]

When the numerator is expanded, we have

[tex]\frac{\left(y^4-y^3+2y^2+y-1\right)}{\left(y^3+1\right)} = \frac{(y - 1)(y^3 + 1) + 2y^2}{y^3 + 1}[/tex]

Split the expanded expression

[tex]\frac{\left(y^4-y^3+2y^2+y-1\right)}{\left(y^3+1\right)} = \frac{(y - 1)(y^3 + 1)}{y^3 + 1} + \frac{2y^2}{y^3 + 1}[/tex]

Evaluate

[tex]\frac{\left(y^4-y^3+2y^2+y-1\right)}{\left(y^3+1\right)} = y - 1 + \frac{2y^2}{y^3 + 1}[/tex]

From the above, we have

Quotient = y - 1

Remainder = 2y²

Hence, the remainder of the polynomial division is 2y²

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Proved that, sin⁴x-cos⁴x/ sin²x-cos²x = 1​.

Here, we have,

given that,

the LHS is:

sin⁴x-cos⁴x/ sin²x-cos²x

now, we know that,

sin²x+cos²x = 1

and, we know that,

a² - b² = (a+b) (a-b)

so, we get,

sin⁴x-cos⁴x/ sin²x-cos²x

= (sin²x+cos²x) (sin²x-cos²x ) / sin²x-cos²x

=(sin²x+cos²x)

=1

= RHS

Hence, Proved that, sin⁴x-cos⁴x/ sin²x-cos²x = 1​

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The length x in the similar triangles is given as follows:

x = 15 cm.

Two triangles are defined as similar triangles when they share these two features listed as follows:

The proportional relationship for the side lengths in this triangle is given as follows:

x/25 = 9/15 = 18/30

Hence the value of x is obtained as follows:

x/25 = 9/15

15x = 25 x 9

x = 25 x 9/15

x = 15 cm.

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Converting the temperature from Celsius to Fahrenheit:

Subtract the given temperature by 32 and divide it by 1.8°C.

°C = (F - 32) / 1.8°C,   or use the equation °F= 1.8 °C+32

(°F-32) / 1.8°C

a) The given temperature is 60°F

substitute the given values in the above equation:

(60 - 32) / 1.8

28 / 1.8

= 15.5555 approx 16°C

60°F = 16°C

b) The given temperature is 10°F

°C = (F - 32) / 1.8°C

substitute the given values in the above equation:

(10 -32) / 1.8

-22 / 1.8

-12.222 approx -12°C

10°F = -12°C

correct question:

What is the temperature in °C, if the temperature is

a) 60F

b) 10F

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The number of students taking all three languages would be 1 student. The number of students taking none of the languages are 32.

The total number of students taking each language is given so the number of people taking these languages are:

15 + 14 - X + X = 30

X = 1

So, there is 1 student taking all three languages.

For the students taking none of these languages:

Total students - students taking at least one language = students taking none of these languages.

= 100 (total students) - (53 ( taking only one language ) + 14 ( taking Arabic and Bulgarian ) + X (taking all three languages))

= 100 - 53 - 14 - 1

= 32 students

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