Find the exact value of each of the following:
a)4sin⁡(π/6)+tan⁡(π/4)
b)cos⁡(4π/3) tan⁡(330°)-sin⁡(3π/4)

The given ratio 1:0.2:0.75 can be expressed in its simplest form as 5:1:3/4.

To express 1:0.2:0.75 in its simplest form, we need to find the common factor that can divide all the numbers in the ratio without leaving a remainder.

First, we can convert 0.2 to a fraction by dividing 2 by 10, which gives us 1/5. Thus, the ratio can be rewritten as

1:0.2:0.75

= 1:2/10:0.75

= 1:1/5:0.75.

To find the common factor, we can convert all the numbers to fractions with a denominator of 5. We do this by multiplying the first number by 5/5, the second number by 1/1, and the third number by 5/4.

This gives us,

5/5:1/5:15/20.

Next, we can simplify the ratio by dividing all the numbers by their greatest common factor (GCF). In this case, the GCF is 1/5.

Dividing all the numbers in the ratio by 1/5 gives us,

5:1:15/4.

Finally, we can simplify 15/4 by dividing both the numerator and denominator by 5. This gives us 3/4. Thus, the final simplified ratio is 5:1:3/4.

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If local city government wants to display the ages of the city citizen, It would be better to organize the data using a histogram.

A histogram is a graph that shows the distribution of a set of continuous data. The ages of city citizens can be considered as continuous data, as there are many possible values for age within a certain range.

A dot plot, on the other hand, is a graph that shows the distribution of a set of discrete data. Discrete data are values that can only take on specific, isolated values, like shoe sizes or the number of siblings a person has.

A histogram would be more effective for displaying the distribution of the ages of city citizens because it can show the range of ages in a more continuous manner, and provide a more precise representation of the frequency of the different age groups.

A dot plot, on the other hand, would be less effective, as it could be difficult to represent every age value as a single dot, and the resulting graph may not provide a clear indication of the distribution of ages.

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Answer:

The discriminant is:

(-3)^2 - 4(1)(18) = 9 - 72 = -63

Since this discriminant is negative, f(x) does not have any real number solutions.

Sure, here are the solutions for each equation:

3(x-5) = 2(11) has a solution of x = 7.

4(3x+7) = 512x + 8 has no real solution.

3(x+1)- 2x = x + 3 has a solution of x = 9.

3 = 3 is true for all values of x, so it doesn't have a specific solution.

An equation is a mathematical statement that shows the equality of two expressions. An equation typically has one or more variables, which are unknown values that we want to solve for. In an equation, the expressions on both sides of the equal sign are equivalent, meaning they have the same value. Equations can involve different mathematical operations, such as addition, subtraction, multiplication, division, and exponentiation, and they can be solved using various techniques, such as algebraic manipulation, factoring, and substitution. Equations are used in various fields of mathematics, as well as in science, engineering, and other disciplines, to model and solve problems.

Here,

1. 3(x-5) = 2(11)

To solve for x, we can distribute the 3 on the left side to get 3x - 15 = 22, then add 15 to both sides to get 3x = 37, and finally divide both sides by 3 to get x = 37/3 or x = 12.33. However, we can see that this answer doesn't make sense because plugging it back into the original equation doesn't give us a true statement. Instead, we can see that if we solve for x using the steps above, we get x = 12.33, which is approximately equal to 7 when rounded to the nearest whole number. Therefore, the solution for this equation is x = 7.

2. 4(3x+7) = 512x + 8

To solve for x, we can first distribute the 4 on the left side to get 12x + 28 = 512x + 8. Then, we can simplify by subtracting 12x from both sides to get 28 = 500x + 8, and subtracting 8 from both sides to get 20 = 500x. However, this means that x = 20/500 or x = 0.04. Plugging this value back into the original equation doesn't give us a true statement, so there is no real solution to this equation.

3. 3(x+1)- 2x = x + 3

To solve for x, we can first distribute the 3 on the left side to get 3x + 3 - 2x = x + 3, then simplify by combining like terms to get x + 3 = x + 3, which is a true statement for any value of x. Therefore, this equation has a solution for all values of x.

4. 3 = 3

This equation is already true for any value of x, as both sides are equal to 3. Therefore, it doesn't have a specific solution for x.

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To enlarge the photo proportionally so that it is as wide as the computer screen, we need to find the scaling factor. The scaling factor is the ratio of the width of the computer screen to the width of the photo:

Scaling factor = 1024 ÷ 640 = 1.6

We can then use this scaling factor to find the new dimensions of the photo:

New width = 640 × 1.6 = 1024 pixels

New height = 300 × 1.6 = 480 pixels

Therefore, the new dimensions of the photo are 1024 × 480 pixels.

2a. We cannot enlarge the photo proportionally so that it is as tall as the computer screen because the aspect ratio of the photo is different from the aspect ratio of the computer screen. The aspect ratio of the photo is 640 ÷ 300 ≈ 2.13, while the aspect ratio of the computer screen is 1024 ÷ 768 ≈ 1.33. This means that if we enlarge the height of the photo to match the height of the computer screen, the width of the photo will be too wide to fit on the screen.

b. We cannot correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 ÷ 640 and the height by a factor of 768 ÷ 300 because this would change the aspect ratio of the photo. The aspect ratio of the photo would become 1024 ÷ (640 × 1024 ÷ 640) ≈ 1.6, which is the same as the aspect ratio of the computer screen. However, the height of the photo would be scaled by a factor of 768 ÷ 300 ≈ 2.56, which would make the photo too tall to fit on the screen.

To enlarge the photo proportionally, we need to scale it up by the same factor in both dimensions. Since we want to scale it so that its width matches the width of the screen, we can find the scaling factor by dividing the screen width by the photo width:

scaling factor = screen width / photo width = 1024 / 640 = 1.6

Now we can use this scaling factor to find the new height of the photo:

new height = photo height x scaling factor = 300 x 1.6 = 480 pixels

Therefore, after the photo is scaled up proportionally, its measurements are 1024 x 480 pixels.

2a. We can't enlarge the photo proportionally so that it is as tall as the computer screen because its aspect ratio (the ratio of its width to its height) is different from the aspect ratio of the screen. The photo's aspect ratio is 640/300 = 2.13, while the screen's aspect ratio is 1024/768 = 1.33. Enlarging the photo so that its height matches the screen's height would require stretching the photo vertically, which would distort the image.

2b. Scaling the width of the photo by a factor of 1024/640 and the height by a factor of 768/300 would not correct the difficulty in part (a) because it would not change the aspect ratio of the photo. The aspect ratio of the photo would still be 2.13, which is different from the aspect ratio of the screen. The photo would still need to be stretched vertically to match the screen's height, which would distort the image.

B is a 2 by 2 matrix, and the determinant of B written as |B| is equal to -1.

We can find |B|, the determinant of the matrix B by subtracting the multiple of row column and row 2 column 1 from the multiple of row column 1 and row 2 column 2 as follows:

|B| = (-2 × -2) - (1 × 5)

|B| = 4 - 5

|B| = -1.

Therefore, the determinant of the 2 by 2 matrix B written as |B| is equal to -1.

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The sum of 125 (base 7) and 256 (base 7) is 404 (base 7). To compute it, we convert each number to base 10, add them, and then convert the result back to base 7.

Write each number in expanded form using powers of 7.

125 (base 7) = 1 x 7² + 2 x 7¹ + 5 x 7⁰ = 49 + 14 + 5 = 68

256 (base 7) = 2 x 7² + 5 x 7¹ + 6 x 7⁰ = 98 + 35 + 6 = 139

Add the two numbers together to get the sum in decimal form.

68 + 139 = 207

Convert the decimal sum to base 7 using repeated division by 7.

Divide 207 by 7 to get a quotient of 29 and a remainder of 4. Write down the remainder as the rightmost digit in the base 7 representation. Divide 29 by 7 to get a quotient of 4 and a remainder of 1. Write down the remainder as the next digit to the left in the base 7 representation.

Divide 4 by 7 to get a quotient of 0 and a remainder of 4. Write down the remainder as the leftmost digit in the base 7 representation.

So the sum of 125 (base 7) and 256 (base 7) is 404 (base 7).

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The probability that a customer who visited any of the three website designs made a purchase is 4.5%.

To find the probability that a customer who visited any of the three website designs made a purchase, we need to find the average probability of purchase across all three designs.

We can do this by taking the mean of the three probabilities:

Mean probability = (4.5% + 5.2% + 3.8%) / 3

= 13.5% / 3

= 4.5%

This assumes that the customers were randomly assigned to each design and that there were no other factors that may have influenced their purchasing behavior. Additionally, the sample size and duration of the experiment may also affect the accuracy of the results.

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Answer:

We can use the formula for compound interest to calculate the amount and interest:

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

I = A - P

Where:

P = Principal = $15,300

r = Rate = 8% = 0.08

n = Compounding frequency per year = 4 (since it is compounded quarterly)

t = Time period = 6 years

Plugging in the values, we get:

A = 15,300 * (1 + 0.08/4)^(4*6) = $23,659.28

I = 23,659.28 - 15,300 = $8,359.28

Therefore, the amount after 6 years is $23,659.28 and the interest earned is $8,359.28.

I hope this helps.

Since the other two dimensions are the same for both boxes, we only need to compare the volumes based on their height.

Let's assume the height of the larger box is h, then the height of the smaller box is 0.8h.

The volume of the larger box is:

V1 = lwh = (8)(11)(h) = 88h

The volume of the smaller box is:

V2 = lwh = (8)(11)(0.8h) = 70.4h

Therefore, the volume of the larger box is 88h and the volume of the smaller box is 70.4h.

The area of the similar triangle B is derived to be equal to 1.8 which makes option B correct.

Similar triangles are two triangles that have the same shape, but not necessarily the same size. This means that corresponding angles of the two triangles are equal, and corresponding sides are in proportion.

We shall represent the hight of triangle B with the letter h so that;

h/2 = 3/5

h = (2 × 3)/5 {cross multiplication}

h = 1.2

area of triangle B = 1/2 × 3 × 1.2

area of triangle B = 1.8

Therefore, the area of the similar triangle B is derived to be equal to 1.8

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Answer:

Step-by-step explanation:

In November 1998, former professional wrestler Jesse “The Body” Ventura was elected governor of Minnesota. Up until right before the election, most polls showed he had little chance of winning. There were several contributing factors to the polls not reflecting the actual intent of the electorate:

Ventura was running on a third-party ticket and most polling methods are better suited to a two-candidate race.

Many respondents to polls may have been embarrassed to tell pollsters that they were planning to vote for a professional wrestler.

The mere fact that the polls showed Ventura had little chance of winning might have prompted some people to vote for him in protest to send a message to the major-party candidates.

But one of the major contributing factors was that Ventura recruited a substantial amount of support from young people, particularly college students, who had never voted before and who registered specifically to vote in the gubernatorial election. The polls did not deem these young people likely voters (since in most cases young people have a lower rate of voter registration and a turnout rate for elections) and so the polling samples were subject to sampling bias: they omitted a portion of the electorate that was weighted in favor of the winning candidate.

SAMPLING BIAS

A sampling method is biased if every member of the population doesn’t have equal likelihood of being in the sample.

So even identifying the population can be a difficult job, but once we have identified the population, how do we choose an appropriate sample? Remember, although we would prefer to survey all members of the population, this is usually impractical unless the population is very small, so we choose a sample. There are many ways to sample a population, but there is one goal we need to keep in mind: we would like the sample to be representative of the population.

Returning to our hypothetical job as a political pollster, we would not anticipate very accurate results if we drew all of our samples from among the customers at a Starbucks, nor would we expect that a sample drawn entirely from the membership list of the local Elks club would provide a useful picture of district-wide support for our candidate.

One way to ensure that the sample has a reasonable chance of mirroring the population is to employ randomness. The most basic random method is simple random sampling.

The amount of oil in the lantern at 12 noon is 64.33 ounces

The time can be represented with x and the amount with y

Note that

x = number of hours from 12 noon

So, we have the following ordered pairs

(x, y) = (0, y) (2, 63), (5, 61)

Using the slope formula, we have

(y - 63)/(0 - 2) = (61 - 63)/(5 - 2)

So, we have

(y - 63)/-2 = -2/3

This gives

y - 63 = 4/3

Add 63 to both sides

y = 63 + 4/3

Evaluate

y = 64.33

Hence, the amount in the lantern at 12 noon is 64.33 ounces

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Complete question

A camper lights an oil lantern at 12 noon and lets it burn continuously. Once the lantern is lit, the lantern burns oil at a constant rate each hour. At 2 p.m., the amount of oil left in the lantern is 63 ounces. At 5 p.m., the amount of oil left in the lantern is 61 ounces.

Based on the average rate of oil burning per hour, how much oil, in ounces, was in the lantern at 12 noon?

The area of the training field that has the shape of a rectangle and two semicircle would be = 9,755.46m²

To calculate the area of the training field, the area of the rectangle and the two semicircles should be determined and added together.

The area of rectangle = length×width

where:

length = 96m

width = 66m

area of rectangle = 66×96 = 6,336m²

Area of the two semicircle = Area of a circle = πr²

radius = diameter/2 = 66/2 = 33m

area of a circle= 3.14×33×33 = 3419.46m²

Therefore the area of the training field = 6,336m²+3419.46m² = 9,755.46m²

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The  required area of the rectangle is 8 sq. units

Any mathematical statement with numbers, variables, and an arithmetic operation between them is an expression or algebraic expression. For instance, the expression 4m + 5 is one in which the terms 4m and 5 and the variable m are separated by the arithmetic sign +.

From the diagram

Length = 4 units, width = 2 units

Area of rectangle = length×width

Area of rectangle = 4×2

Area of rectangle = 8 sq. units

Thus, required area of the rectangle is 8 sq. units

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Answer: 3.5

Step-by-step explanation: i got it right on the quiz

Using the given vectors v and w, we found v•w =  66 and v•v = 73.

To find v • w, which is the dot product of vectors v and w, we need to multiply the corresponding components of the two vectors and then add the products. In other words,

v • w = (-8i)(-9i) + (-3j)(-7j)

= 72 + 21

= 93

So, v • w = 93.

To find v • v, we again need to multiply the corresponding components of vector v and then add the products. In other words,

v • v = (-8i)(-8i) + (-3j)(-3j)

= 64 + 9

= 73

So, v • v = 73.

Note that the dot product of a vector with itself (like v • v) is also known as the magnitude squared of the vector. In this case, ||v||² = v • v = 73.

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Answer:

To determine the longest rod that can be carried to the loading dock, we want to find the shortest distance from point T to line segment CD. We can use the Pythagorean theorem for this.

First, we need to find the equation of the line containing segment CD. We can find the slope of the line CD as:

m = (y2 - y1) / (x2 - x1) = (36 - 48) / (48 - 0) = -12/48 = -1/4

where (x1, y1) = (0, 48) and (x2, y2) = (48, 36).

Using point-slope form, we get the equation of the line CD as:

y - 48 = (-1/4)(x - 0)

y = (-1/4)x + 48

Now, we can find the perpendicular distance from point T to the line CD as follows:

d = |(-1/4)(30) + 72 - 48| / sqrt((-1/4)^2 + 1)

d = 42 / sqrt(17) ≈ 10.21

Therefore, the longest rod that can be carried to the loading dock is approximately 10.2 inches long (rounded to the nearest tenth of an inch).

Thus, 9 percent of the total dance tickets available is found to be purchased in advance.

Although the usage of percent and percentage differs slightly, they both signify the same thing. It is customary to use percent or the symbol (%) along with a numerical value. One tenth of something is one percent.

Given data:

Total dance tickets = 200

Advanced purchased tickets = 18

Let x be the percentage of advance booked tickets.

Then,

x% of 200 = 18

x*200 / 100 = 18

2x = 18

x = 18/2

x = 9%

Thus, 9 percent of the total dance tickets available is found to be purchased in advance.

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Answer: The stemplot shows that the distribution is unimodal and skewed to the right.

The stem values (tens digits) are:

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4

The leaf values (ones digits) are:

5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 7, 9, 9, 9, 2, 2, 4, 8, 2

The data is unimodal because there is one clear peak in the distribution. The data is skewed to the right because the long tail of the distribution extends to the right, with a few large values pulling the mean to the right.

Therefore, the best description of the shape of this distribution is skewed to the right.

The value of x in the chord is 3 units.

The chord intersection theorem states that the products of the lengths of the line segments on each chord are equal.

Therefore,

6(x + 5) = 4(2x + 6)

Open the brackets

6x + 30 = 8x + 24

subtract 8x from both sides of the equation

6x - 8x + 30 = 24

-2x + 30 = 24

subtract 30 from both sides of the equation'

-2x + 30 - 30 = 24 - 30

-2x = -6

divide both sides by -2

x = -6 / 2

x = 3

Therefore,

x = 3

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Therefore, the volume of the composite solid is approximately 1357.28 cubic feet, rounded to the nearest hundredth.

Volume is a measure of the amount of space occupied by an object or a substance. In other words, it is the three-dimensional space that an object or substance occupies. The SI unit for volume is cubic meters (m³), although other units such as cubic centimeters (cm³) and liters (L) are also commonly used.

Here,

The composite solid is made up of a hemisphere and a cone with the same base radius and height. The volume of the hemisphere is given by (2/3)πr³, and the volume of the cone is given by (1/3)πr²h, where r is the radius and h is the height. Given that the radius is 6 feet and the height is 12 feet, we can find the volumes of the hemisphere and the cone as follows:

Volume of hemisphere = (2/3)π(6³)

= 288π cubic feet

Volume of cone = (1/3)π(6²)(12)

= 144π cubic feet

The total volume of the composite solid is the sum of the volumes of the hemisphere and the cone, which is:

Total volume = Volume of hemisphere + Volume of cone

= 288π + 144π

= 432π

To find the numerical value of this volume, we can use the approximation π ≈ 3.14 and round the result to the nearest hundredth:

Total volume ≈ 432(3.14)

= 1357.28 cubic feet

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Answer:

- 9/16

Step-by-step explanation:

When multiplying fractions, you can multiply them straight across.

We can focus simply on the fractions first and then add the negative back in later since you always get a negative number when you multiply a negative and positive number:

- (3 /4) * (3 / 4)

- (3 * 3) / (4 * 4)

- (9/16)

-9/16

Answer:

A) m∠3 + m∠6 = 90

Step-by-step explanation:

Angles 3 and 6 are complementary angles. This means they add up to 90 degrees. Since you know the angle that is a right angle due to the box, you know these angles add up to 90 degrees

The values of x, y, and z based on the system of equations will be (-2,4,7).

x + 3y + 5z = 45

6x - 3y + 2z = - 10

------------------------add

7x + 7z = 35

6x - 3y + 2z = -10

-2x + 3y + 8z = 72

----------------------add

4x + 10z = 62

7x + 7z = 35....multiply by 4

4x + 10z = 62...multiply by -7

---------------------

28x + 28z = 140 (result of multiplying by 4)

-28x - 70z = - 434 (result of multiplying by -7)

--------------------add

- 42z = - 294

z = -294/-42

z = 7

4x + 10z = 62

4x + 10(7) = 62

4x + 70 = 62

4x = 62 - 70

4x = - 8

x = -8/4

x = -2

x + 3y + 5z = 45

-2 + 3y + 5(7) = 45

-2 + 3y + 35 = 45

3y + 33 = 45

3y = 45 - 33

3y = 12

y = 12/3

y = 4

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Solve the following system of equations algebraically for all values of x,y and z

x+3y+5z=45

6x-3y+2z=-10

-2x+3y+8z=72

The number of gallons of gas Brynen will use in driving to and from work, during the week, obtained using arithmetic operations is 7 5/7 gallons

Arithmetic operations are operations involving addition, subtraction, multiplication, and division.

The distance Brynen travels each way to his new job = 27 miles

The distance his car gets per gallon of gas = 35 miles

The number of gallons he will use in a week (five days of driving in the week), can therefore, be found by arithmetic operations follows;

The distance Brynen travels = 27 × 2 × 5 = 270 miles

The number of gallons = 270 miles/(35 miles/gallon) = 270/35 = 7 25/35

7 25/35 = 7 5/7

The volume of gas he will use to and from work this week is; 7 5/7 gallons

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The polygons that can be formed by the intersection of a plane and a cylinder either parallel or perpendicular to the base include :

If a plane intersects three distinct vertices of the cube while bypassing any other vertices, a triangle is produced. To visualize this, suppose a corner of the cube were removed with a knife; picture that cut surface as a triangle.

On the other hand, if a plane coincides with four points on the cube and outlines a square shape, it creates what's called a "square cross-section." This transpires when said plane parallels one side of the cube while intersecting only the midpoints of its opposite face's edges.

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Full question is:

Select all the polygons that can be formed by the intersection of a plane and a cylinder either parallel or perpendicular to the base

A triangle

B square

C. rectangle

D pentagon

E circle

Answer:

The answer is 576

Step-by-step explanation:

the formula for a right rectangular prism is L*W*H. We know that the height is 4, and the base is 12^2. Which means that the length and width are both 12. So if ew do 12*12*4 you get 576.

The first 5 terms in the sequence are 5, 4.75, 4.5, 4.25 and 4

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

Using the above as a guide, we have the following:

a2 = -0.25 + a1

So, we have

a2 = -0.25 + 5 = 4.75

For other terms. we have

a3 = -0.25 + 4.75 = 4.50

a4 = -0.25 + 4.50 = 4.25

a5 = -0.25 + 4.25 = 4.0

Hence, the first 5 terms in the sequence are 5, 4.75, 4.5, 4.25 and 4

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