Find (v) x (w) if ||v||=8, ||w||=12, and the angle between v and w is 60 degrees

The height of the can is approximately 1.26 cm.

The formula for the surface area of a cylinder is:

SA = 2πr² + 2πrh

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

Since the diameter of the can is 7 cm, the radius is 7/2 = 3.5 cm.

We are given that the surface area of the can is 180 square centimeters, so we can set up an equation:

180 = 2π(3.5)² + 2π(3.5)h

Simplifying:

180 = 24.5π + 7πh

180 = 45.5πh

h = 180 / (45.5π)

h = 1.25924790139

h ≈ 1.26 cm (rounded to the nearest hundredth)

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The sum of the first 5 terms of the sequence is 183.

We have,

The first term of the sequence is 3 and the common ratio is -3.

We can use the formula for the sum of the first n terms of a geometric sequence:

Sn = a(1 - r^n) / (1 - r)

where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.

Plugging in the values we have:

S5 = 3(1 - (-3)^5) / (1 - (-3))

Simplifying:

S(5) = 3(1 + 243) / 4

S(5) = 3(244) / 4

S(5) = 183

Therefore,

The sum of the first 5 terms of the sequence is 183.

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Answer:

[tex]\frac{-11}{4} (x + 2)^2 + 6.[/tex]

Step-by-step explanation:

As we know that

The standard equation of a parabola with vertex at point (h,k) is given by:

[tex]y = a(x-h)^2 + k[/tex]

where (h,k) is the vertex

and "a" is a coefficient that determines the direction and shape of the parabola.

We know that the vertex of our parabola is (-2, 6),

so we can substitute those values into the equation to get:

[tex]y = a(x + 2)^2 + 6[/tex]

Now finding a.

Since the parabola passes through point (-4, -5). Substituting these coordinates into the equation, we get:

-5 = a(-4 + 2)^2 + 6

-5 =a(-2)^2+6

-5=4a +6

-5-6=4a

4a=-11

a =- 11/4

Substituting the value of an equation second, we get

[tex]\frac{-11}{4} (x + 2)^2 + 6.[/tex]

Therefore, the equation of the parabola with a vertex at (-2, 6) and passing through (-4, -5) is y =

[tex] \frac{-11}{4} (x + 2)^2 + 6[/tex]

Answer:

[tex]y=-\dfrac{11}{4}(x+2)^2+6[/tex]

Step-by-step explanation:

The vertex form of a parabola is:

[tex]\boxed{y=a(x-h)^2+k}[/tex]

where:

Given the vertex is at (-2, 6), substitute h = -2 and k = 6 into the equation:

[tex]y=a(x-(-2))^2+6[/tex]

[tex]y=a(x+2)^2+6[/tex]

To find the value of a, substitute the point (-4, -5) into the equation and solve for a:

[tex]\begin{aligned}-5&=a(-4+2)^2+6\\-5&=a(-2)^2+6\\-5&=4a+6\\-5-6&=4a+6-6\\-11&=4a\\\dfrac{-11}{4}&=\dfrac{4a}{4}\\-\dfrac{11}{4}&=a\\a&=-\dfrac{11}{4}\end{aligned}[/tex]

Substitute the found value of a back into the equation:

[tex]y=-\dfrac{11}{4}(x+2)^2+6[/tex]

Therefore, the equation of a parabola with a vertex at (-2, 6) and that passes through the point (-4, -5) is:

[tex]\boxed{y=-\dfrac{11}{4}(x+2)^2+6}[/tex]

The value of x in the triangle is 1/√3

We know that tan function is the ratio of opposite side and adjacent side

In triangle, for angle 60 degrees the opposite side is 1 unit

Adjacent side is x units

tan 60 = opposite side/adjacent side

√3= 1/x

We need to find value of x

Apply cross multiplication

x=1/√3

Hence, the value of x in the triangle is 1/√3

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Answer:  the best answer is option A: the data is symmetric with an approximate range of 20. The lowest value of 6 might mean that most students did not participate in activities.

Step-by-step explanation: From the line plot, we can see that the data is not symmetric because the dots are not equally distributed on both sides of the plot. Therefore, we can eliminate options A and C.

Option D suggests that the data is bimodal, meaning that there are two peaks in the distribution. However, there is no clear indication of two peaks in the plot. Therefore, we can eliminate option D.

Option B suggests that the data is skewed, and the most frequent number of activities is 16 or more. However, this is not the case as the highest number of dots is above 14 and 16. Therefore, we can eliminate option B.

The volume of the prism will be 88 cubic units. Then the correct option is B.

Given that:

Length, L = 4 units

Width, W = 4 units

Height, H = 5.5 units

The volume of the prism is given as,

V = L x W x H

The volume of the prism is calculated as,

V = 4 x 4 x 5.5

V = 88 cubic units

The volume of the prism will be 88 cubic units. Then the correct option is B.

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The complete question is given below.

The maximum of 1,159 m/s occurs when t = 7, and the minimum of 16 m/s occurs when t = 6.

It should be noted that upon deriving v(t), an equation of 12t² - 96t + 144 is produced. Subsequently, v'(t) must be set to zero, resulting in a quadratic equation of 12t² - 96t + 144 = 0, requiring division of both sides by 12 to produce t² - 8t + 12 = 0.

The quadratic can now be factored to obtain (t - 2)(t - 6) = 0, indicating that critical points lie on the values t = 2 and t = 6.

After calculating these values— v(1) = 100 m/s, v(2) = 64 m/s, v(6) = 16 m/s and v(7) = 1159 m/s—it is clear that the maximum of 1,159 m/s occurs when t = 7, and the minimum of 16 m/s occurs when t = 6.

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The length of AB in the right triangle is 8 units.

The sides of a right angle triangle can be found as follows:

The right angle triangle has one of its angles as 30 degrees. The side AB can be found using trigonometric ratios.

Hence,

sin 30 = opposite / hypotenuse

where

opposite side = AB

hypotenuse side = 16 units

Therefore,

sin 30 = AB / 16

cross multiply

AB = 16 sin 30

AB = 16 × 0.5

AB = 8 units

Therefore,

AB = 8 units

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Answer:

A. 45(x+3)=60x

Step-by-step explanation:

The value of m in terms of other variables of the right angle triangle is: m = √(p² - n²)

Pythagoras Theorem is the way in which you can find the missing length of a right angled triangle.

The triangle has three sides, the hypotenuse (which is always the longest), Opposite (which doesn't touch the hypotenuse) and the adjacent (which is between the opposite and the hypotenuse).

Pythagoras is in the form of;

a² + b² = c²

However, it can also be written in the form of c² = a² + b²

From the given triangle, we can write as:

p² = m² + n²

m² = p² - n²

m = √(p² - n²)

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Answer:

f(- 4) = 44 , g(5) = - 21

Step-by-step explanation:

to evaluate f(- 4) substitute x = - 4 into f(x)

f(- 4) = 2(- 4)² - 3(- 4) = 2(16) + 12 = 32 + 12 = 44

to evaluate g(5) substitute x = 5 into g(x)

g(5) = - 4(5) - 1 = - 20 - 1 = - 21

The percentage error for each of the given situations are:

1) 23%

2) 21%

3) 11%

4) 16.67%

5) 43%

The formula for percent error is:

Percentage Error = ((Actual Number – Estimated Number)/ Estimated number) * 100%

1) Estimated Number = 2310

Actual Number = 3000

Thus:

Percent Error = (3000 - 2310)/3000 * 100%

Percent error = 23%

2) Estimated Number = 17600

Actual Number = 21300

Thus:

Percent Error = (21300 - 17600)/17600 * 100%

= 21%

3) Estimated Number = 4.5

Actual Number = 4

Thus:

Percent Error = (4 - 4.5)/4.5 * 100%

= 11%

4) Estimated Number = 15000

Actual Number = 12500

Thus:

Percent Error = (12500 - 15000)/15000 * 100%

= 16.67%

5) Estimated Number = 350

Actual Number = 200

Thus:

Percent Error = (200 - 350)/350 * 100%

= 43%

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Answer:

Step-by-step explanation:

You want to compare the future values of two accounts after 10, 20, 50, and 85 years:

The formula for account value with compounding monthly is ...

  A = P(1 +r/12)^(12t) . . . . . . principal P invested at rate r for t years

  A = 4500(1 +0.075/12)^(12·t) . . . . . using given values for P and r

The first line of calculator output shown in the attachment gives the value after 10, 20, 50, and 85 years.

The formula for account value with continuous compounding is ...

  A = P·e^(rt)

  A = 5500·e^(0.0675t) . . . . . using given values for P and r

The second line of calculator output shown in the attachment gives the value after 10, 20, 50, and 85 years.

You will notice this (second) account has higher value up to 20 years, then lower value after that.

The account with the higher interest rate will make more money after the "break-even" period of about 20.6 years.

__

Additional comment

We have rounded the account values to the nearest dollar. For the purpose of value comparison, three significant figures would be sufficient.

The second attachment shows the different account values over the long term. You can see the higher interest rate really makes a difference.

The third attachment shows the shorter-term account values, and the "break-even" time period of about 20.6 years, when the values in both accounts are about $22,063.

The derivative of the function cos(xº) in radians is y' = -sin(xπ/180)

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

cos(xº)

Changing the degree into radian, we have

cos(xπ/180)

Express as a function

So, we have

y = cos(xπ/180)

When the cosine function is differentiated, we have

y' = -sin(xπ/180)

Hence, the differentiated function is y' = -sin(xπ/180)

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Answer: The slope is -5/4

m=-5/4

Miriana rolled the sequence of 3 and then 6 two more times than the theoretical probability would have predicted.

The theoretical probability of rolling a three-on-one number cube is 1/6, and the same is true for rolling a six. Since the two rolls are independent,

The probability of rolling a three and then a six in the sequence is the product of the probabilities: (1/6) x (1/6) = 1/36.

To find the expected number of times this sequence would occur in 180 experiments, we multiply the probability by the number of experiments: (1/36) x 180 = 5.

So, if Miriana had rolled the number of cubes according to the theoretical probability, we would have expected her to roll the sequence of 3 and then 6 five times in 180 experiments.

However, she actually rolled this sequence 7 times, so she rolled it 2 more times than the theoretical probability would have predicted.

Therefore, Miriana rolled the sequence of 3 and then 6 two more times than the theoretical probability would have predicted.

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The numeric value of the function f(x) at x = 5 is given as follows:

f(5) = 74.75.

The second derivative of the function f(x) is given as follows:

f''(x) = 2x + 6sin(x).

Integrating the second derivative, the first derivative of the function is given as follows:

f'(x) = x² - 6cos(x) + K.

When x = 0, y' = 2, hence the constant K is given as follows:

2 = -6 + K

K = 8.

Hence:

f'(x) = x² - 6cos(x) + 8.

Integrating the first derivative, the function is given as follows:

f(x) = x³/3 - 6sin(x) + 8x + K.

When x = 0, y = 4, hence the constant K is given as follows:

K = 4.

Then:

f(x) = x³/3 - 6sin(x) + 8x + 4.

The numeric value at x = 5 is given as follows:

f(5) = 5³/5 - 6sin(5) + 8(5) + 4

f(5) = 74.75.

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The balance of the account after 20 years is given as follows:

A(20) = $13,268.58.

The amount of money earned, in compound interest, after t years, is given by:

[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]

In which:

The parameters for this problem are given as follows:

P = 6500, r = 0.036, n = 2, t = 20.

Hence the balance of the account after 20 years is given as follows:

A(20) = 6500 x (1 + 0.036/2)^(2 x 20)

A(20) = $13,268.58.

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The task here is to determined where the table given is a function. Thus we can state that the above table is NOT a function. This is called Non-Function Relation.

The table given above is suggestive of the relationship between the values of x and y given.

It may be inferred that x values are the input values while the y values are the output.

So the criteria for the values on the table to constitute a function is that each input value (x) must be associated with only one value of y.

From the table, it is clear to see that this rule if violated. You have the x value (-1) being associated with the y values of 2 and 4 respectively, hence, it is proper to conclude that the table is NOT  a function.

This is called a Non-Function Relation.

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The condition that must be true in order to get a correct answer will be that the number found in the columns of the first matrix must be equal to the numbers in the rows of the second matrix.

A matrix is a rectangular arrangement of numbers that can be used to demonstrate mathematical properties. Often, two or three matrices can be evaluated to determine an outcome.

For the given matrices, a × b  and c x d the only way to arrive at a correct answer will be if the figures found in the columns of the first matrix equal the figures found in the rows of the second matrix.

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The bank employs a total of 32 bankers, and the answer is not listed among the options provided. The correct option is B.

Let's denote the number of bankers who manage only personal loans as P and the number of bankers who manage only business loans as B. Then, we can use the following information to form equations:

There are 20 bankers who manage personal loans, which means P + 3 bankers in total manage personal loans.

There are 25 bankers who manage business loans, which means B + 3 bankers in total manage business loans.

The bank has a total of X bankers.

Using these equations, we can solve for X as follows:

P + 3 = 20 (equation 1)

B + 3 = 25 (equation 2)

P + B + 3 = X (equation 3)

Solving equation 1 for P, we get P = 17.

Solving equation 2 for B, we get B = 12.

Substituting P = 17 and B = 22 into equation 3, we get:

17 + 12 + 3 = X

Simplifying, we get X = 32.

Therefore, the bank employs a total of 32 bankers, and the answer is not listed among the options provided.

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Answer:

180

Step-by-step explanation:

The word HORSES has 6 distinct letters, so we can arrange them in 6! = 720 ways if all the letters were unique.

However, in this case, the letters E and S are repeated twice. Therefore, we must divide by the number of arrangements of the repeated letters to avoid counting the same arrangement more than once.

The letter E appears twice, so there are 2! = 2 ways to arrange the E's within the word. Similarly, the letter S appears twice, so there are 2! = 2 ways to arrange the S's within the word.

Thus, the total number of unique arrangements of the letters in the word HORSES is:

6! / (2! × 2!) = 720 / 4 = 180

Therefore, there are 180 unique arrangements of the letters in the word HORSES.

1.) The scale factor = 4

The perimeter ratio = 1:3

The area ratio = 1:16

To calculate the scale factor of the given shapes, the formula that should be used would be ;

Scale factor = Bigger dimensions/ Smaller dimensions

Length of the bigger rectangle (B) = 16

Length of the smaller rectangle (A) = 4

The scale factor = 16/4 = 4

The perimeter = 2(l+w)

length B = 16

width B = 8

perimeter B = 2(16+8)

= 2(24) = 48

Length A = 4

width A = 2

perimeter A =2(4+2)

= 16

The perimeter ratio = A:B = 16:48 = 1:3

Area = length×width

length A = 4

width A = 2

Area A = 4×2 = 8

length B = 16

width B = 8

area B = 8×16= 128

Area ratio of A:B = 8:128 = 1:16

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The solution to the system of equations is (d) More than 1 solution

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

3y-12x=18

2y-8x=12

Using the graphing function on a calculator, we have the solution to be

Infinite many solutions

This is so because the equations 3y-12x=18 and 2y-8x=12 are equivalent equations

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If each tile cost $2 then the amount spend by  Nicole’s mother on tiles is $130

Given that Nicole‘s mother put tiles on a section of the kitchen floor.

The section included 5 rows with 13 tiles in each row

Each tile cost $2.

To find the total amount spend by Nicolas mother on tiles we have to find the number of tiles

Number of tiles =5×13

=65 tiles

Now let us find the amount

Total amount on tiles = 2×65

=$130

Hence, if each tile cost $2 then the amount spend by  Nicole’s mother on tiles is $130

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Answer:28

Step-by-step explanation:

•C= 2(3.14 x r)

•C= 2(3.14 x4.5)

•C=2(14.13)

•C=28.26

The best response to such statement will be "Let me explain how my questions are related to our discussion" which is option (A)

The statement the customer made shows that he is disappointed with whatever the situation is.

Customer dissatisfaction refers to the feeling of displeasure, frustration, or disappointment that a customer experiences when their expectations are not met by a product or service they have purchased or received. It can arise from various factors, such as poor quality, ineffective communication, slow response times, inadequate support, or a mismatch between customer needs and what is being offered.

To prevent customer dissatisfaction, businesses must focus on providing high-quality products and services, being responsive and attentive to customer needs, and proactively addressing any issues or concerns that arise.

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No, the relation is not a function.

Given the relation in the question:

x: -2, -4, -1, -4, 5, 3

y: 1, -3, 2, 8, 0, -1

A function is simply a relationship that maps one input to one output.

To determine whether the relation is a function, we need to check whether each x-value in the relation is associated with a unique y-value.

Looking at the x-values, we see that there are repeated values of -4, which means that there are multiple y-values ( -3 and 8 ) associated with x = -4.

The mapping of x to y is not well-defined for all values of x, since one value of x is associated with multiple values of y, hence, it is not a function.

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The numeric value of the function f(x) at x = 5 is given as follows:

f(5) = 74.75.

The second derivative of the function f(x) is given as follows:

f''(x) = 2x + 6sin(x).

Integrating the second derivative, the first derivative of the function is given as follows:

f'(x) = x² - 6cos(x) + K.

When x = 0, y' = 2, hence the constant K is given as follows:

2 = -6 + K

K = 8.

Hence:

f'(x) = x² - 6cos(x) + 8.

Integrating the first derivative, the function is given as follows:

f(x) = x³/3 - 6sin(x) + 8x + K.

When x = 0, y = 4, hence the constant K is given as follows:

K = 4.

Then:

f(x) = x³/3 - 6sin(x) + 8x + 4.

The numeric value at x = 5 is given as follows:

f(5) = 5³/5 - 6sin(5) + 8(5) + 4

f(5) = 74.75.

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1. The lateral surface area of the prism is 200 units²

2. The lateral surface area of the cylinder is 125.6 cm²

The lateral surface area is the area of the lateral surface.

The lateral surface area of a prism is expressed as;

LSA = (a+b+c)h

or perimeter of the base × height

LSA = (6+12+7)8

= 25 × 8

= 200 units²

The lateral surface area of a cylinder

= 2πrh

r = 4cm and h = 5cm

LSA = 2 × 3.14 × 4 × 5

LSA = 40 × 3.14

= 125.6cm²

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