Find the areas of the sectors formed by

Answer:

(SEE ATTACHMENT)

Step-by-step explanation:

The velocity of the plane expressed as a vector is:

(-155 m/s) i^ + (89.5 m/s) j^

We are given the parameters as:

Speed = 400 mph

Bearing = 150°

Converting to m/s gives:

400 mph = 179 m/s

Vertical component of velocity = 179 * sin 150 = 89.5 m/s

Horizontal component of velocity = 179 * cos 150 = -155 m/s

Thus, in vector form, we have:

(-155 m/s) i^ + (89.5 m/s) j^

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. Any fraction that is equal to 1/3 will, therefore, create a ratio with 2/6 .

An element of a whole is a fraction. 2. It represents a number that characterizes the components of a whole .The numerator and the denominator are the two components of a fraction. The numerator, which is the number above the line, indicates the number of equally-sized portions taken from the entire. The number below the line, known as the denominator, indicates how many equally sized portions the whole is divided into.

Equivalent fractions are fractions that express the same value but have various numerators and denominators. For instance, because they both represent half of a whole number 1, 2/4 and 3/6 are comparable fractions. We can multiply the numerator and denominator of a fraction by the same number 1, which will help us locate equivalent fractions. The fraction's value remains unchanged, but its form is altered.

Finding a fraction that is equal to 2/6 is necessary to build a proportion with this number.

By dividing the numerator and denominator by their greatest common factor, which is 2 , we can simplify 2/6 to achieve this.

This results in 1/3 . Any fraction that is equal to 1/3 will, therefore, create a ratio with 2/6 .

For instance, by multiplying both the numerator and denominator of 1/3 by 2, we obtain 2/6, which is the same as 1/3 , 4 as a result.

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linear inequality that describes the various combinations of the number of students s and adults:5s + 7a ≤ 199

A linear inequality is an expression that uses the inequality symbols (, >,, or ) to compare two numbers. It involves a polynomial of degree 1 or less and a linear expression. ax + b c, where a, b, and c are real numbers and a 0, can be used to represent a linear inequality in one variable. ax + by c, where a, b, and c are all real numbers and a and b are not equal to zero, can be used to represent a linear inequality in two variables.

Student entrance is $5, while adult admission costs $7. Let's say there are and there are grownups going on the field trip, respectively.

The following equation can be used to represent the total cost of admission for s students and an adults:

Total cost is 5s plus 7a.

Since the class can only spend up to $199 on the excursion, we can write:

5s + 7a ≤ 199

The various permutations of the number of students and adults who may accompany you on the field trip are described in this inequality.

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After nine years, Mrs. Perry's investment will have generated an interest of $1140.75.

To calculate the interest earned on Mrs. Perry's investment, we can use the simple interest formula:

I = P * r * t

where I is the interest earned, P is the principal (the initial amount invested), r is the interest rate as a decimal, and t is the time in years.

In this case, P = $3,900, r = 0.0325 (3.25% expressed as a decimal), and t = 9 years. Substituting these values into the formula, we get:

I = $3,900 * 0.0325 * 9

I = $1140.75

Therefore, Mrs. Perry will earn $1140.75 in interest on her investment at the end of 9 years.

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The length and width of the room are approximately 48.22 feet and 15.22 feet, respectively. We can write this as (48.22, 15.22).

Length and width are two dimensions that are commonly used to describe the size of a two-dimensional object, such as a rectangle, a piece of paper, or a room.Length typically refers to the longer dimension of an object, while width refers to the shorter dimension. In the case of a rectangle, the length is the longer side of the rectangle, and the width is the shorter side.

For example, if a rectangle has a length of 8 inches and a width of 5 inches, we can say that the dimensions of the rectangle are 8 inches (length) by 5 inches (width).

In the given question,

Let's denote the width of the room as x. Then, according to the problem, the length of the room is 33 feet more than the width, which means the length is x + 33.

We know that the area of the rectangular room is 460 square feet. We can set up an equation using the formula for the area of a rectangle:

Area = length × width

Substituting the expressions we have for length and width, we get:

460 = (x + 33) × x

Expanding the right side, we get:

[tex]460 = x^2 + 33x[/tex]

Rearranging and setting the equation equal to zero, we get:

[tex]x^2 + 33x - 460 = 0[/tex]

We can solve this quadratic equation using the quadratic formula:

[tex]x = (-33 ± √(33^2 - 4*1(-460)) / (2×1)[/tex]

Simplifying under the square root, we get:

[tex]x = (-33 ± √(33^2 + 4×460×1)) / 2[/tex]

[tex]x = (-33 ± √19849) / 2[/tex]

[tex]x ≈ 15.22 or x ≈ -48.22[/tex]

Since the width of the room cannot be negative, we can ignore the negative solution. Therefore, the width of the room is approximately 15.22 feet.

Using the expression we found for the length of the room, we get:

Length = width + 33 = 15.22 + 33 ≈ 48.22 feet.

Therefore, the length and width of the room are approximately 48.22 feet and 15.22 feet, respectively. We can write this as (48.22, 15.22).

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Answer:

(0, -5)

Step-by-step explanation:

To solve the system of equations:

y = 3x - 5 (1)

2x - 5y = 25 (2)

We can use the substitution method or the elimination method to find the values of x and y that satisfy both equations.

Let's use the substitution method:

Start with equation (1): y = 3x - 5

Substitute this expression for y into equation (2) wherever y appears:

2x - 5(3x - 5) = 25

Distribute the -5 to both terms inside the parentheses:

2x - 15x + 25 = 25

Combine like terms:

-13x + 25 = 25

Subtract 25 from both sides to isolate x:

-13x = 0

Divide both sides by -13 to solve for x:

x = 0

Now, we can substitute the value of x we found into equation (1) to find the value of y:

y = 3(0) - 5

y = -5

So the solution to the system of equations is x = 0 and y = -5. The ordered pair representing the solution is (0, -5).

The probability of rolling a 6 given that the value is an even number is 1/3.

There are three even numbers on a die: 2, 4, and 6. The probability of getting an even number is therefore 3/6 or 1/2.

Since the problem specifies that the value is even, we only need to consider the outcomes 2, 4, and 6. The probability of rolling a 6 out of those three outcomes is 1/3.

Using conditional probability, the probability of rolling a 6 given that the value is even is:

P(6 | even) = P(6 and even) / P(even)

P(6 and even) = 1/6 (only 1 way to roll a 6)

P(even) = 1/2

P(6 | even) = (1/6) / (1/2) = 1/3

Therefore, the probability of rolling a 6 given that the value is an even number is 1/3.

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Step-by-step explanation:

[tex] \sqrt[3]{27} = 3 [/tex]

Because 3×3×3=27, or 3³=27

[tex] \sqrt[4]{81} = 3[/tex]

Because 3×3×3×3=81, or 3⁴=81

[tex] \sqrt[3]{ - 64} = - 4[/tex]

Because:

[tex] \sqrt[ 3]{ - 64} = - \sqrt[3]{ {4}^{3} } = - 4[/tex]

[tex] \sqrt[3]{1} = 1[/tex]

Because: 1×1×1=1, or 1³=1

Hope this helps!

The both expressions simplify to[tex]$-\frac{16}{143}$[/tex], and they are equivalent.

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 +.

First, let's evaluate the expression inside the first set of parentheses:

[tex]$\begin{align*}\frac{4}{13} \div \left(\frac{-11}{12}\right) &= \frac{4}{13} \times \left(\frac{12}{-11}\right) \&= -\frac{48}{143}\end{align*}[/tex]

Now, we can substitute this result back into the original expression:

[tex]$\begin{align*}\left(-\frac{48}{143}\right) \div \frac{1}{3} &= \left(-\frac{48}{143}\right) \times 3 \&= -\frac{144}{143}\end{align*}[/tex]

Next, let's evaluate the expression inside the second set of parentheses:

[tex]$\begin{align*}\frac{-11}{12} \div \frac{1}{3} &= \frac{-11}{12} \times 3 \&= \frac{-11 \times 3}{12} \&= \frac{-33}{12} \&= -\frac{11}{4}\end{align*}[/tex]

Now, we can substitute this result back into the original expression:

[tex]$\begin{align*}\frac{4}{13} \div \left(\frac{-11}{12} \div \frac{1}{3}\right) &= \frac{4}{13} \div \left(-\frac{11}{4}\right) \&= \frac{4}{13} \times \left(-\frac{4}{11}\right) \&= -\frac{16}{143}\end{align*}[/tex]

Therefore, both expressions simplify to[tex]$-\frac{16}{143}$[/tex], and they are equivalent.

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Answer:

37

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

B is $3.273, because the 3 at the end is less than 5, it rounds down to $3.27, which is the value that you want.

Therfore, the answer is B.

Brainliest?

I would really appreciate it!

Hence, the inspector is incorrect in her decision to not pass inspection based only on the mean speed comparison. A more comprehensive analysis of the data is required to arrive at a fair decision. Standard Deviation = 7.34.

To compare the performances of each machine, we can use measures of center and variability. The measures of center describe the central tendency of the data, while the measures of variability describe how spread out the data are.

For the given data, we can use the following measures of center and variability:

Measures of Center:

Mean: It is the average value of the data and is calculated by adding up all the values and dividing by the number of observations.

Median: It is the middle value of the data when arranged in order. It is less sensitive to outliers than the mean.

Measures of Variability:

Range: It is the difference between the largest and smallest values in the data.

Standard Deviation: It measures the spread of the data from the mean. A larger standard deviation indicates greater variability.

To compare the performances of each machine, we can calculate the mean and standard deviation for each machine separately.

Machine X:

Mean = (18+13+20+15+17)/5 = 16.6

Standard Deviation = 2.87

Machine Y:

Mean = (19+1+15+18+17)/5 = 14

Standard Deviation = 7.34

Based on the above calculations, we can see that the mean speed of Machine X is indeed higher than that of Machine Y. However, the standard deviation of Machine Y is much larger than that of Machine X. This indicates that the data points for Machine Y are more spread out and have more variability than those of Machine X.

Therefore, we cannot simply conclude that Machine X is performing better than Machine Y based on the mean values alone. We need to consider the standard deviation as well to get a complete picture of their performances.

Hence, the inspector is incorrect in her decision to not pass inspection based only on the mean speed comparison. A more comprehensive analysis of the data is required to arrive at a fair decision.

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The algebraic values of x include:

Using the same reasoning as in the previous problem:

24² + 45² = x²

576 + 2025 = x²

2601 = x²

x = ±51

Since x represents a length, discard the negative solution. Therefore, the value of x is 51.

Using the Pythagorean Theorem:

30² + 40² = x²

900 + 1600 = x²

2500 = x²

x = ±50

Since x represents a length, discard the negative solution. Therefore, the value of x is 50.

Using the Pythagorean Theorem:

18² + 80² = x²

324 + 6400 = x²

6724 = x²

x = ±82

Since x represents a length, discard the negative solution. Therefore, the value of x is 82.

Using the Pythagorean Theorem:

18² + 4² = x²

324 + 16 = x²

340 = x²

x = √340 = 2√85

Therefore, the value of x is 2√85 in simplest radical form.

Using the Pythagorean Theorem:

8² + 5² = x²

64 + 25 = x²

89 = x²

x = √89

Therefore, the value of x is √89 in simplest radical form.

Using the Pythagorean Theorem:

20² + 18² = x²

400 + 324 = x²

724 = x²

x = √724 = 2√181

Therefore, the value of x is 2√181 in simplest radical form.

Using the Pythagorean Theorem:

25² + 3² = x²

625 + 9 = x²

634 = x²

x = √634

Therefore, the value of x is √634 in simplest radical form.

Using the Pythagorean Theorem:

3² + 6² = x²

9 + 36 = x²

45 = x²

x = √45 = 3√5

Therefore, the value of x is 3√5 in simplest radical form.

Using the Pythagorean Theorem:

22² + 4² = x²

484 + 16 = x²

500 = x²

x = √500 = 10√5

Therefore, the value of x is 10√5 in simplest radical form.

To determine if a triangle is a right triangle, we can use the Pythagorean Theorem, which states that for any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore:

a² + b² = c²

Substituting a = 8, b = 15, and c = 17:

8² + 15² = 17²

64 + 225 = 289

289 = 289

Since the equation is true, the triangle satisfies the Pythagorean Theorem and is therefore a right triangle.

Using the same method:

27² + 36² = 45²

729 + 1296 = 2025

2025 ≠ 2025

Since the equation is false, the triangle does not satisfy the Pythagorean Theorem and is therefore not a right triangle.

Using the same method:

9² + 11² = 4²

81 + 121 = 16

202 ≠ 16

Since the equation is false, the triangle does not satisfy the Pythagorean Theorem and is therefore not a right triangle.

To determine the classification of this triangle, compare the lengths of its sides. If all sides are less than the sum of the other two, then the triangle is acute; if one side is greater than or equal to the sum of the other two, then the triangle is obtuse; and if one side is equal to the sum of the other two, then the triangle is degenerate (a straight line).

Substituting a = 3, b = 4, and c = 6:

a + b = 3 + 4 = 7 < c

b + c = 4 + 6 = 10 > a

a + c = 3 + 6 = 9 > b

Since none of the sides are equal to or greater than the sum of the other two, the triangle is acute.

Substituting a = 9, b = 11, and c = 16:

a + b = 9 + 11 = 20 < c

b + c = 11 + 16 = 27 > a

a + c = 9 + 16 = 25 > b

Since none of the sides are equal to or greater than the sum of the other two, the triangle is acute.

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Image transcribed:

Algebra Find the value of x.

1. 12, 9, x

To start, use the Pythagorean Theorem. Then substitute 9 for a, 12 for b, and x for c.

2. 45 24 x

3. 30, 40, x

4. 80, 18, x

Algebra Find the value of x. Express your answer in simplest radical form.

8. 18, 4, x

9. 8, 5, x

10. 20, 18, x

11. 25, 3, x

12. 3, 6, x

13. 22, 4, x

Is each triangle a right triangle? Explain.

16. 15, 17, 8

17. 27, 45, 36

18. 9, 11, 4

The lengths of the sides of a triangle are given. Classify each triangle as acute, right, or obtuse.

19. 3,4,6

To start, compare to a+b. Substitute the greatest length for c.

20. 9, 11, 16

The given parallelogram is congruent by ASA property so option (A) and option (F) are correct.

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length , and opposite angles are equal in measures. Also, the interior angles on the same side of the transversal are supplementary. The sum of all the interior angles equals to 360 degrees.

The word congruent means having exactly the same size and shape.

∠KMT=∠PTM ,∠KTM=∠PMT it is given so inorder to prove ΔKMT≅ΔPTM congruent included side must be equal that is TM=MT

So, by ASA property triangles are congruent.

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1. The sample mean of Diego's television data is 1.73 hours.

2. Yes, if Diego repeats the process with more random samples of 30 students and calculates the mean of each sample, his set of sample means should have a normal distribution.

To find the sample mean of Diego's television data, we add up all the hours and divide by the total number of students:

The sample mean's (summation x) is:

= 1 + 3 + 2 + 0 + 4 + 1 + 2 + 1 + 5 + 3 + 0 + 4 + 2 + 3 + 2 + 3 + 1 + 0 + 4 + 5 + 2 + 1 + 3 + 3 + 2 + 4 + 1 + 5 + 1 + 0

= 52

So the sample mean (Summation x/n) is:

= 52 / 30

= 1.73333333333

= 1.73 hours

He should have normal distribution. This is known as the Central Limit Theorem, which states that as the sample size increases, the distribution of sample means approaches a normal distribution regardless of the shape of the original population distribution, as long as the samples are selected randomly and independently.

In this case, as Diego is selecting random samples of 30 students, the Central Limit Theorem would apply, and the distribution of sample means should be normal.

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The decimal used to figure the price of their clothing is 2.1. This means that if the cost price of a piece of clothing is, say, $50, the store would be selling it for:

$50 x 2.1 = $105.

What is meant by a decimal?

A decimal system is a number system that uses a base of 10 and includes a decimal point to represent fractions or parts of a whole.

What is meant by cost price?

Cost price is the original price or cost of a product or service. It includes all costs incurred to produce or acquire the item, such as materials, labor, and overhead.

According to the given information

Markup is usually defined as the percentage increase from the cost price to the selling price. So, if the store has a 210% markup on the price of their clothing, this means that the selling price is 210% greater than the cost price. Let's assume the cost price of an item of clothing is $x. Then, the selling price would be: selling price = cost price + markup

selling price = x + 210%x

selling price = x + 2.1x

selling price = 3.1xTherefore, the selling price of the clothing is 3.1 times the cost price. To write this markup as a decimal, we need to divide it by 100:markup as a decimal = 210/100

markup as a decimal = 2.1

This means that if the cost price of a piece of clothing is, say, $50, the store would be selling it for:

$50 x 2.1 = $105.

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Answer:

[tex]144\pi \: {cm}^{2} [/tex]

Step-by-step explanation:

Given:

d (diameter) = 12 cm

l = 18 cm

Find: A (surface area) - ?

First, we have to find the radius:

[tex]r = \frac{1}{2} \times d = \frac{1}{2} \times 12 = 6 \: cm[/tex]

Now, we can find the surface area:

[tex]a(surface) = \pi {r}^{2} + \pi \times r \times l [/tex]

[tex]a(surface) = \pi \times {6}^{2} + \pi \times 6 \times 18 = 36\pi + 108\pi = 144\pi \: {cm}^{2} [/tex]

Answer:

[tex]144\pi {cm}^{2} [/tex]

Step-by-step explanation:

The formula for the surface area of a cone is

[tex]s = \pi {r}^{2} + \pi \: rs[/tex]

Plugging given values r=6cm and s=18cm we get 144cm².

The volume of the glass must be greater than 9 cubic inches to accommodate the 9 ice cubes.

What is the volume of a cube?

The volume of a cube is given by the formula:

V = s³

where "s" is the length of any side of the cube.

In other words, to find the volume of a cube, we just need to raise the length of any side to the third power.

No, David is not right. If he fits 9 ice cubes, each with a volume of 1 cubic inch, into the glass, then the total volume of the ice cubes is 9 x 1 = 9 cubic inches.

If the glass had a volume of 9 cubic inches, there would be no room for the ice cubes since they would completely fill the glass.

Therefore, the volume of the glass must be greater than 9 cubic inches to accommodate the 9 ice cubes.

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The maximum amount that Siciliano's Produce Market can pay per square foot on an annual rate is $8 per square foot.

To find the maximum amount that Siciliano's Produce Market can pay per square foot on an annual rate, we need to convert the monthly budget into an annual budget.

Given:

Monthly budget: $5,000

Square footage needed: 7,500 square feet

To find the annual budget, we multiply the monthly budget by 12 (since there are 12 months in a year):

= Monthly budget * 12

= $5,000 * 12

= $60,000

Now, we can divide the annual budget by the total square footage needed to find the maximum amount that Siciliano's can pay per square foot on an annual rate:

= Annual budget / Total square footage

= $60,000 / 7,500 square feet

=  $8 per square foot.

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Answer: To solve this, you'll need to find a common denominator and simplify the equation before solving for x.

The answer is: x = 23/5 or 4.6

Answer:

venomous

Step-by-step explanation:

Most snakes are venomous, as venom is injected from their bite.

A poisonous organism is one whose toxins are transferred when they are eaten, which is not the case for most snakes.

Answer:

they are both

Step-by-step explanation:

Poisonous snake and venomous snakes are really the same except, snake Poison is a toxin that gets into the body by inhaling, swallowing, or absorption through the skin. Snake venom is when the toxin is injected into you, like a cobra that uses its fangs to inject venom.

A 3-digit number can be formed using the digits 2,3 and 4 without repetition and with repetition 6 and 27.

What is permutation?

A permutation of a set is, broadly speaking, a rearranging of its elements if the set is already ordered, or arranging its members into a sequence or linear order. The act or procedure of altering the linear order of an ordered set is referred to as a "permutation."

Here, we have

Given: 2,3 and 4

We have to find the 3-digit numbers that can be formed using the digits 2,3 and 4 without repetition and with repetition.

Without repetition: 2, 3, 4

Number of possible numbers

= 3×2×1

= 6

Hence, a 3-digit number can be formed using the digits 2,3, and 4 without repetition is 6.

With repetition

Number of possible numbers =

= 3×3×3

= 27

Hence, a 3-digit number can be formed using the digits 2,3, and 4 with a repetition is 27.

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There is no arithmetic progression that satisfies the given conditions.

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant value to the previous term. This constant value is called the common difference of the arithmetic progression.

In the given question,

Let the common difference of the arithmetic progression be d, and let the second, sixth, twenty-second, and last terms be denoted by a + d, a + 5d, a + 21d, and a + nd, respectively (where n is the number of terms in the arithmetic progression).

Since the four terms form a geometric progression, we have:

[tex](a + d)(a + nd) = (a + 5d)(a + 21d)[/tex]

Expanding both sides and simplifying, we get:

a^2 + (n+1)d a + nd^2 = 26ad + 105d^2

Rearranging and simplifying, we get:

[tex]a^2 + (n+1)d a + nd^2 = 26ad + 105d^2\\a(n+1) = 79d[/tex]

Now we can use the fact that the last term is a + nd to find n in terms of a and d. The last term can be written as a + (n-1)d, but since the progression is increasing, we know that a + nd > a + (n-1)d, so n > 1. Therefore, we have:

a + nd = a + (n-1)d + nd > a + (n-1)d

Simplifying, we get:

[tex]a + nd = a + (n-1)d + nd > a + (n-1)d\\\\d > 0[/tex]

Now we can use the fact that the last term is a + nd to find n in terms of a and d. We have:

a + nd = a + (n-1)d

Simplifying, we get:

[tex]a + nd = a + (n-1)d\\nd = (n-1)d[/tex]

Dividing both sides by d (since d > 0), we get:

n = n-1

Solving for n, we get:

1 = -1

This is a contradiction, which means that there is no solution to this problem. Therefore, there is no arithmetic progression that satisfies the given conditions.

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According to the profit prediction model:

Using the finite differences method, we can find the differences between consecutive profits and then the differences between those differences until we reach a constant value, which indicates that the function is a polynomial of degree one more than the number of differences it took to reach the constant value.

The table of differences is:

Year, t | 1 | 2 | 3 | 4 | 5 | 6

Profit, p | 7 | 11 | 23 | 49 | 95 | 167

First diff. | 4 | 12 | 26 | 46 | 72

Second diff. | 8 | 14 | 20 | 26

Since the second differences are constant, we know that the polynomial model is of degree 3. We can use these differences to set up a system of equations and solve for the coefficients of the polynomial.

Let p(t) = at³ + bt² + ct + d be the polynomial model. Then:

p(1) = a + b + c + d = 7

p(2) = 8a + 4b + 2c + d = 11

p(3) = 27a + 9b + 3c + d = 23

p(4) = 64a + 16b + 4c + d = 49

Solving this system of equations:

a = 1

b = -3

c = 5

d = 4

So the equation of best fit is:

p = t³ - 3t² + 5t + 4

To predict the profit after 10 years, we substitute t = 10 into the equation:

p = 10³ - 3(10)² + 5(10) + 4 = 554

Therefore, the predicted profit after 10 years is $554.

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Image transcribed:

answer the following:

You have decided to start a dog-walking business. The table below shows the profits p (in dollars) of the business during your first 6 years. Use a graphing calculator and finite differences to find a polynomial model for the problem. Then use the model to predict your profit after 10 years.

Year, t | 1 | 2 | 3 | 4 | 5 | 6

Profit, p | 7 | 11 | 23 | 49 | 95 | 167

Equation of best fit:

p= ____ t³ + ____ t² + ____ t + _____

Profit after 10 years will be ______

Answer: Therefore, the account will be worth $7536 approximately 16.5 years after 2000, which would be in the year 2016.

Step-by-step explanation: lG: user0316070

[tex]\qquad \textit{division of two complex numbers} \\\\ \cfrac{r_1[\cos(\alpha)+i\sin(\alpha)]}{r_2[\cos(\beta)+i\sin(\beta)]}\implies \cfrac{r_1}{r_2}[\cos(\alpha - \beta)+i\sin(\alpha - \beta)] \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ z1=20[\cos(120^o)+i\sin(120^o)]~\hfill z2=4[\cos(30^o)+i\sin(30^o)] \\\\\\ \cfrac{z1}{z2}\implies \cfrac{20}{4}[\cos(120^o - 30^o)+i\sin(120^o - 30^o)]\implies 5[\cos(90^o)+i\sin(90^o)] \\\\\\ 5[0+i(1)]\implies 0+5i\implies \boxed{5i}[/tex]

The length of BD is 15 units. The measure of angle ADB is 59° and the measure of angle DEC  is 59°.

What is meant by length?

Length is a physical dimension that measures the extent of an object or distance between two points. It is a scalar quantity, typically measured in units such as meters or feet, and is an essential concept in geometry and other branches of mathematics and science.

What is meant by measure?

Measure refers to a numerical value that represents the size, quantity, or extent of a mathematical object or set. It can be used to describe the length, area, volume, angle, or other properties of geometric figures, functions, or sets.

According to the given information

If EC = 15, find BD:

Now, consider triangle BDC. We know that angle BDC is 31 degrees, so angle BCD is 180 - 90 - 31 = 59 degrees.

Using the sine rule in triangle BCD, we have:

BD/sin(BDC) = CD/sin(BCD)

=> BD/sin(31) = x/sin(59)

using the sine rule in triangle AEC, we have:

AC/sin(theta) = CE/sin(180 - 2theta)

=> AC/sin(theta) = 15/sin(2theta)

But we also know that AC = BD ,So

BD/sin(theta) = 15/sin(2*theta)

Now, using the double angle formula for sine, we have:

sin(2theta) = 2sin(theta)*cos(theta)

Substituting this in the above equation, we get:

BD/sin(theta) = 15/(2*sin(theta)cos(theta))

=> BD = 15cos(theta)/2

But we also know that:

cos(theta) = AC/2EC = BD/2EC

Substituting this in the above equation, we get:

BD = 15BD/(22EC)

=> BD = 15*BD/60

=> BD = 15

Therefore, BD = 15 units.

Find angle ADB:

Consider triangle ABD. We know that AE is the bisector of diagonal AC, so angle BAE = angle DCE.

Also, since opposite angles of a rectangle are equal, we know that angle BAE = angle ADB.

Therefore, angle ADB = angle DCE.

Now we have:

angle DCE = 180 - angle BCE - angle BCD

=> angle DCE = 180 - 90 - 31

=> angle DCE = 59 degrees

Therefore, angle ADB = 59 degrees.

Find angle DEC:

Consider triangle DEC. We know that AE is the bisector of diagonal BD, so angle AEB = angle DEB.

Also, since opposite angles of a rectangle are equal, we know that angle AEB = angle AED.

Therefore, angle DEC = angle DEB - angle AED.

Now we have:

angle DEC = angle DEB - angle BEC

=> angle DEC = 180 - angle BCD - angle BEC

=> angle DEC = 180 - 31 - 90

=> angle DEC = 59 degrees

Therefore, angle DEC = 59 degrees.

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Based on their given probabilities, e would expect the color blue or green to be spun approximately 75 times in this experiment.

To find the expected number of times the color blue or green will be spun, we first need to add their probabilities:

0.25 + 0.125 = 0.375

This means that, on average, we would expect to see blue or green spun 0.375 times for every spin. To find the expected number of times this will happen over 200 spins, we can multiply the probability by the number of spins:

0.375 x 200 = 75

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keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the equation above

[tex]3x+y=5\implies y=\stackrel{\stackrel{m}{\downarrow }}{-3}x+5\qquad \impliedby \qquad \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}[/tex]

so we're really looking for the equation of a line that has a slope oif -3 and it passes through (2 , -4)

[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{-4})\hspace{10em} \stackrel{slope}{m} ~=~ - 3 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-4)}=\stackrel{m}{- 3}(x-\stackrel{x_1}{2}) \implies y +4 = - 3 ( x -2) \\\\\\ y+4=-3x+6\implies {\Large \begin{array}{llll} y=-3x+2 \end{array}}[/tex]

now, keeping in mind that perpendicular lines have negative reciprocal slopes

[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ -3 \implies \cfrac{-3}{1}} ~\hfill \stackrel{reciprocal}{\cfrac{1}{-3}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{1}{-3} \implies \cfrac{1}{ 3 }}}[/tex]

so for this one, we're looking for the equation of a line whose slope is 1/3 and it passes through (2 , -4)

[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{-4})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{1}{3} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-4)}=\stackrel{m}{ \cfrac{1}{3}}(x-\stackrel{x_1}{2}) \implies y +4 = \cfrac{1}{3} ( x -2) \\\\\\ y+4=\cfrac{1}{3}x-\cfrac{2}{3}\implies y=\cfrac{1}{3}x-\cfrac{2}{3}-4\implies {\Large \begin{array}{llll} y=\cfrac{1}{3}x-\cfrac{14}{3} \end{array}}[/tex]