3(26+14)/(2x2) i need help please

The absolute value equation for Kayleigh's location in relation to the high school is |21 - d| = 6. By solving this equation, we find that Kayleigh could live either 15 miles or 27 miles away from the school.

Part A: The absolute value equation would represent the distance between Kayleigh's house and the high school, which can either be closer or further away from the school compared to Donna's house. Given that Donna lives 21 miles from school, and Kayleigh lives 6 miles away from Donna, we can set an equation to indicate all possible locations of Kayleigh's home. The equation would be |21 - d| = 6, where 'd' represents the distance from Kayleigh's house to the high school.

Part B: To understand how far Kayleigh could live from her school, solve the equation above for 'd', you get two solutions: 21 - 6 = 15 miles and 21 + 6 = 27 miles. Therefore, Kayleigh could either live 15 miles or 27 miles away from the school.

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Answer:

write ✍️

Step-by-step explanation:

justification for the following line translation . line 1 3p-

Answer: 18 students

Step-by-step explanation: 43+27+48=118 118-100=18

The bird covered 34 metres of vertical space overall. Warm-blooded, feathery animals with the ability to fly are known as birds.

The bird soared 15 m above the water, dove 2 m vertically, and descended to a depth of the water that was 15 + 2 = 17 m below its starting height. The bird must fly 17 m vertically upwards in order to cover the same distance in the other direction in order to return to its starting height.

Consequently, the bird travelled a total vertical distance of 34 metres (17 metres when diving and 17 metres while returning to its starting height).

Hence, the bird covered 34 metres of vertical space overall.

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[tex]yx=5[/tex]

Explanation:

If [tex]y[/tex] varies inversely as [tex]x[/tex] then

         [tex]y\times x=k[/tex] for some constant [tex]k[/tex]

Since [tex]y = \frac{1}{8}[/tex] when [tex]x=40[/tex]

          [tex]\huge \text(\dfrac{1}{8}\huge \text)\times(40)=k[/tex]

          [tex]\longrightarrow k=5[/tex]

The value of x cannot be determined and the answer is undefined.

To find the value of x in this scenario, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, triangle ABC is a right triangle with AB as the hypotenuse. So, we can use the Pythagorean theorem to solve for AC, which is equal to the square root of (AB^2 - BC^2).

Using the given values, we have:
AC = sqrt(16^2 - 8^2)
AC = sqrt(256 - 64)
AC = sqrt(192)
AC ≈ 13.86

Now, we can use triangle ACD to solve for x. This triangle is also a right triangle with CD as the hypotenuse. So, we can use the Pythagorean theorem again to solve for AD, which is equal to the square root of (CD^2 - AC^2).

Using the given values, we have:
AD = sqrt(9^2 - 13.86^2)
AD = sqrt(81 - 192.23)
AD = sqrt(-111.23)

We can't take the square root of a negative number, so there is no real solution for x in this scenario.

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Answer:

145°

Step-by-step explanation:

The exterior angle 3 is equal to the sum of angle 1 and 2.

Using substitution we can say that:

5x + 4x + 10 = 10x -5

9x + 10 = 10x -5

10x-9x=10+5

x= 15

We have just found that the value of x is equal to 15.

By substituting this back to the angle 3 expression, 10x-5, we get

10(15) - 5

150 - 5

145°

There are 30 elements in region C.

What is the inclusion-exclusion principle?

The inclusion-exclusion principle is a counting technique used in combinatorics to find the size of a set that is a union of several other sets. It states that the size of the union of two or more sets is equal to the sum of their individual sizes minus the size of their intersection(s).

To find n(C), we can use the inclusion-exclusion principle:

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)

We know that n(U) = 103, so we can use this to find n(B ∩ C) as follows:

n(B ∩ C) = n(B ∪ C) - n(B') = n(U) - n(B' ∩ C') - n(B' ∩ C) - n(B ∩ C') = 103 - 24 - 35 - n(B ∩ C')

Similarly, we can find n(A' ∩ C) and n(A ∩ B' ∩ C) as follows:

n(A' ∩ C) = n(U) - n(A ∪ C') = 103 - n(A) - n(C') + n(A ∩ C') = 103 - 32 - n(C') + 14 = 85 - n(C')

n(A ∩ B' ∩ C) = n(U) - n(A ∪ B' ∪ C') = 103 - n(A ∪ B') - n(C') + n(A ∩ B' ∩ C') = 103 - 69 - n(C') + 6 = 40 - n(C')

Now we can substitute these values into the inclusion-exclusion principle to get:

92 + 52 + n(C) - 11 - 14 - n(B ∩ C) - 32 - (85 - n(C)) - (40 - n(C)) + 6 = n(U)

Simplifying, we get:

n(C) - n(B ∩ C) - 21 = 0

n(C) = n(B ∩ C) + 21

To find n(B ∩ C), we can use the formula we derived earlier:

n(B ∩ C) = 103 - 24 - 35 - n(B ∩ C')

n(B ∩ C') = n(U) - n(B' ∪ C') = n(U) - n(B') - n(C') + n(B' ∩ C') = 103 - 52 - n(C') + 24 = 75 - n(C')

Substituting this into the equation for n(B ∩ C), we get:

n(B ∩ C) = 103 - 24 - 35 - (75 - n(C))

n(B ∩ C) = 39 - n(C)

Now we can substitute this into the equation for n(C) to get:

n(C) = (39 - n(C)) + 21

2n(C) = 60

n(C) = 30

Therefore, there are 30 elements in region C.

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Answer:

x = 14°

Step-by-step explanation:

The sum of the angles of a triangle is 180°

.

Let's make an equation according to this:

47° + 103° + (3x - 12)° = 180°

150° + 3x - 12° = 180°

3x = 180° + 12° - 150°

3x = 42° / : 3

x = 14°

With the given compound interest, the required quarterly payment is $38,745.74.

What is Compound interest?

Compound interest is the interest that is calculated on the initial principal as well as the accumulated interest of previous periods. In other words, it is the interest on interest. The interest is added to the principal at the end of each compounding period, and then the new total amount becomes the principal for the next period's interest calculation.

Now,

The amount financed is equal to the purchase price minus the down payment:

Amount financed = $5,000,000 - 0.15($5,000,000) = $4,250,000

To find the quarterly payment,

PV = Pmt * (1 - (1 + r)⁻ⁿ) / r

where PV is the present value (amount financed), Pmt is the quarterly payment, r is the quarterly interest rate, and n is the total number of quarterly payments (11 years x 4 quarters/year = 44 quarters).

The quarterly interest rate is equal to the annual interest rate divided by 4:

r = 0.091 / 4 = 0.02275

Substituting the values, we get:

$4,250,000 = Pmt * (1 - (1 + 0.02275)⁻⁴⁴) / 0.02275

Solving for Pmt, we get:

Pmt = $38,745.74

Therefore, the required quarterly payment is $38,745.74 (rounded to the nearest cent).

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Answer:

The circumcenter of the triangle with vertices A(2,6), B(8,6), and C(8,10) is (8,6).

Step-by-step explanation:

To find the circumcenter of the triangle with vertices A(2,6), B(8,6), and C(8,10), we can use the following steps:

Step 1: Find the midpoint of two sides

We first find the midpoint of two sides of the triangle. Let's take sides AB and BC:

Midpoint of AB: ((2 + 8)/2, (6 + 6)/2) = (5, 6)

Midpoint of BC: ((8 + 8)/2, (6 + 10)/2) = (8, 8)

Step 2: Find the slope of two sides

Next, we find the slope of the two sides AB and BC:

Slope of AB: (6 - 6)/(8 - 2) = 0

Slope of BC: (10 - 6)/(8 - 8) = undefined

Step 3: Find the perpendicular bisectors of two sides

Since the slope of AB is 0, its perpendicular bisector is a horizontal line passing through the midpoint of AB, which is y=6. Since the slope of BC is undefined, its perpendicular bisector is a vertical line passing through the midpoint of BC, which is x=8.

Step 4: Find the intersection of perpendicular bisectors

The circumcenter is the point where the two perpendicular bisectors intersect. The intersection point is (8,6).

Therefore, the circumcenter of the triangle with vertices A(2,6), B(8,6), and C(8,10) is (8,6).

For acute angles A and B it is known that sin A - 28/53 and tan B = 4/3. , , the value of cos(A+B) in simplest form is -15/53.

In geometry, an angle is the measure of the amount of turn between two lines that intersect at a point, called the vertex of the angle. Angles are typically measured in degrees, but can also be measured in radians.

Angles can be classified based on their measures. An angle that measures less than 90 degrees is called an acute angle, while an angle that measures exactly 90 degrees is called a right angle. An angle that measures greater than 90 degrees but less than 180 degrees is called an obtuse angle, while an angle that measures exactly 180 degrees is called a straight angle. Finally, an angle that measures greater than 180 degrees but less than 360 degrees is called a reflex angle.

We can start by using the identity:

cos(A + B) = cos A cos B - sin A sin B

To find cos A and sin A, we can use the identity:

sin² A + cos² A = 1

sin A = √(1 - cos² A)

We are given sin A = 28/53, so we can solve for cos A:

sin² A + cos² A = 1

(28/53)² + cos² A = 1

cos² A = 1 - (28/53)²

cos A = ± √[(53/53)² - (28/53)²]

Since A is acute, we take the positive root:

cos A = √(1 - (28/53)²)

To find sin B and cos B, we can use the identity:

tan B = sin B / cos B

We are given tan B = 4/3, so we can solve for sin B and cos B using the Pythagorean identity:

sin² B + cos² B = 1

sin B = (4/3) cos B

(4/3)² cos² B + cos² B = 1

cos² B = 1 / ((4/3)² + 1)

cos B = √(1 / ((4/3)² + 1))

sin B = (4/3) cos B

Now we can substitute all of these values into the formula for cos(A + B):

cos(A + B) = cos A cos B - sin A sin B

cos(A + B) = √(1 - (28/53)²) √(1 / ((4/3)² + 1)) - (28/53) (4/3) √(1 / ((4/3)² + 1))

cos(A + B) = (√((53² - 28²) / 53²)) (√(9 / (16 + 9))) - (4/3) (28/53) (√(9 / (16 + 9)))

cos(A + B) = (45/53) (√(9/25)) - (112/265) (√(9/25))

cos(A + B) = (9/53) - (24/53)

cos(A + B) = -15/53

Therefore, cos(A + B) = -15/53.

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Step-by-step explanation:

It is not clear from the given information which angles and sides are being referred to as "107⁰", "-x-8", "-42-7", and "3y-1". However, we can use some properties of rhombuses to solve for x, y, and z.

Opposite angles in a rhombus are equal. Therefore, if one angle is 107⁰, then the opposite angle is also 107⁰.

The diagonals of a rhombus are perpendicular bisectors of each other. This means that they intersect at a right angle and divide each other into two equal parts.

The diagonals of a rhombus bisect each other's angles. This means that the angles formed by each diagonal with the sides of the rhombus are equal.

Using these properties, we can set up some equations to solve for x, y, and z:

Let's assume that "-x-8" and "-42-7" are the lengths of the diagonals of the rhombus, and that "3y-1" is the length of one of the sides.

Since the diagonals bisect each other's angles, we know that the angles formed by each diagonal with the side of the rhombus are equal. Let's call each of these angles "z":

-z + 107⁰ + z = 180⁰ (sum of angles in a triangle)

107⁰ = 180⁰ - 2z

2z = 73⁰

z = 36.5⁰

Now let's use the fact that the diagonals are perpendicular bisectors of each other:

(-x-8)/2 = (-42-7)/2

-x-8 = -49

-x = -41

x = 41

Finally, let's use the fact that the sides of a rhombus are equal:

-x-8 = 3y-1

41-8 = 3y-1

33 = 3y

y = 11

Therefore, the values of x, y, and z in the rhombus are:

x = 41

y = 11

z = 36.5⁰

The width of the garden is 4 feet.

What is Rectangle ?

A rectangle is a quadrilateral with four right angles, opposite sides that are parallel, and equal in length. It is a type of parallelogram where the adjacent sides are equal in length, but the opposite sides are not necessarily equal.

Let's start by using the formula for the perimeter of a rectangle:

Perimeter = 2(length + width)

We know that the perimeter is 34 feet, so we can plug in that value and simplify:

34 = 2(l + w)

17 = l + w

We also know that the length is 5 more than 2 times the width, so we can write an equation for that:

l = 2w + 5

We can substitute this expression for l into the first equation:

17 = (2w + 5) + w

Simplifying, we get:

17 = 3w + 5

12 = 3w

w = 4

Therefore, the width of the garden is 4 feet.

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The area of the trapezoid [tex]36^{2}[/tex] centimeters

To use the formula A = ¹h(b₁ + b₂) to find the area of a trapezoid, you need to know the height (h) and the lengths of the two parallel bases (b₁ and b₂).
For example,

The formula for finding the area of a trapezoid is A = ¹h(b₁ + b₂),

Where A represents the area of the trapezoid, h represents the height, and b₁ and b₂ represent the lengths of the two parallel bases.

By plugging in the given values for the height and bases and performing the necessary arithmetic operations, you can find the area of the trapezoid.

It's important to remember to use the correct units and follow the order of operations to obtain the correct result.

If the height of the trapezoid is 9 cm, the lengths of the two bases are 3 cm and 5 cm, you can plug these values into the formula to find the area :
A = ¹h(b₁ + b₂)

= ¹(9)(3 + 5)

= ¹(9)(8)

= 36 square centimeters
So, the area of this trapezoid is 36 square centimeters.
If you have different values for the height and bases, you can use the same formula to find the area.

Just make sure to use the correct units and follow the order of operations (parentheses first, then multiplication and division, then addition and subtraction).

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Tom would bank Ksh 277,272.73 and John would bank Ksh 693,939.39 at the end of the business day with 25% markup.

The markup is the discrepancy between a product's selling price and its cost price. Adding a particular percentage to the cost price to arrive at the selling price of a product is a frequent practise in business. Depending on the sector, the level of competition, and other elements, the markup % may change.

Since it influences a product's or service's profitability, markup is significant in business. If the markup is too low, the company could not have a sufficient profit to pay its bills and continue operating. The product could not be competitive with other similar items on the market if the markup is too high.

Let us suppose the cost price and sales price = x.

Thus,

For Tom:

Cost of 5 A + Cost of 2B = 5x + 2x = 7x

Sale of 10 B = 10(2x) = 20x

Here,

Profit = Sale - Cost = 20x - 7x = 13x

For the given value of profit we have:

Tom's profit = Ksh 230,000

13x = 230,000

x = 17,692.31

Simillarly for Jhon:

or John:

Sale of 3 A = 3(5x) = 15x

Cost of 13 B = 13x

Profit = Sale - Cost = 13x - 15x = -2x

John's profit = Ksh 110,000

-2x = 110,000

x = -55,000

Now, the markup is 25%,:

New cost price for John = 17,692.31

New sale price for John = 17,692.31 + 0.25(17,692.31) = 22,115.39

Actual sale price for John = 22,115.39 - 0.15(22,115.39) = 18,798.08

The new profit is thus:

New cost price for John = 17,692.31

New sale price for John = 17,692.31 + 0.25(17,692.31) = 22,115.39

Actual sale price for John = 22,115.39 - 0.15(22,115.39) = 18,798.08

Hence, Tom would bank Ksh 277,272.73 and John would bank Ksh 693,939.39 at the end of the business day.

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The situation can be modeled by the following system of linear inequalities:

x ≥ 6 (The number of male volunteers should be at least 6)

x + y ≤ 15 (The total number of volunteers should be no more than 15)

y ≥ 0 (The number of female volunteers should be non-negative)

What is inequality model?

A formula exists for inequality. Less than, larger than, or not equal are used in place of the equal sign when there are inequalities. Rules for resolving disparities are unique. Here are a few examples of inequalities that are mentioned. When disparities are connected, the centre inequality can be jumped over.

The situation can be modeled by the following system of linear inequalities:

x ≥ 6 (The number of male volunteers should be at least 6)

x + y ≤ 15 (The total number of volunteers should be no more than 15)

y ≥ 0 (The number of female volunteers should be non-negative)

Explanation:

The first inequality states that the number of male volunteers, represented by x, should be greater than or equal to 6. This ensures that the science teacher has at least 6 male volunteers.

The second inequality states that the total number of volunteers, represented by x + y, should be no more than 15. This ensures that the science teacher does not exceed the maximum number of volunteers required for the competition.

The third inequality states that the number of female volunteers, represented by y, should be non-negative. This ensures that there are no negative values for the number of female volunteers.

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We can cοnclude that there is a significant difference between the treatments.

This methοd is abοut determining "likely" οr "unlikely" by determining the prοbability assuming the null hypοthesis were true οf οbserving a mοre extreme test statistic in the directiοn οf the alternative hypοthesis than the οne οbserved.

Using the p-value apprοach, we can calculate the p-value fοr the F-statistic.

we see that F = 7.28. The degrees οf freedοm fοr treatments and errοr are df1 = 2 and df2 = 38, respectively. we can find that the p-value is apprοximately 0.0026.

Using the critical-value apprοach, we can find the critical value οf F fοr a significance level οf 0.05 and degrees οf freedοm df1 = 2 and df2 = 38. Frοm a table οr calculatοr, we find that the critical value is apprοximately 3.18.

Since the p-value (0.0026) is less than the significance level (0.05), we reject the null hypοthesis. Alternatively, since the F-statistic (7.28) is greater than the critical value (3.18), we reject the null hypοthesis.

Therefοre, we can cοnclude that there is a significant difference between the treatments.

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Let's add the fractions first and then we can estimate the sum.

[tex]= \dfrac{2}{3} +\dfrac{3}{4}[/tex]

change to common denominators

[tex]= \dfrac{8}{12} +\dfrac{9}{12}[/tex]

add numerators only

[tex]\dfrac{17}{12}[/tex] or [tex]1\dfrac{5}{12}[/tex] or [tex]1.42[/tex]

ANSWER: 13 is the best estimate of the answer. Normally we round up to 1.4 with a number like 1.42, but since that is not a choice, we can estimate 1 as the answer.

Answer:

hope it helps.

In parabola, the vertex of b(x) is 8 units to the right of the vertex of .

What is parabola in math?

A parabola is a U-shaped plane curve in which every point is situated at an equal distance from both the focus, a fixed point, and the directrix, a fixed line. All of the parabola-related ideas are discussed here since it is a crucial component of the conic section subject. When f(x)=ax² +bx +c, where a, b, and c are real integers and a0, the function is said to be quadratic.

f(x) = x²

b(x) = f(x - 8)

f(x) = x²

f(x- 8) = (x- 8)²

f(X-8) = x² - 16x + 64

b(X) =  f(x- 8) = x² - 16x + 64

Comparing  b(X)  = x² - 16x + 64 with  y = ax² + bx + c

 a = 1, b = -16 , c = 64

-b/2a =  8  ...........1

Hence, the x-coordinate of vertex is 8.

putting x = 8 in b(x)

b(8) = 8² - 16 * 8 + 64 = 0

     So, the vertex of b(x)  is (8,0)

Comparing f(x) = x²  with y = ax² +bx+c

  a = 1 , b= 0 , c = 0

  -b/2a = 0 .........2

 Hence, the x-coordinate of vertex is 0.

putting  x = 0 in f(X)

f(0) = 0² = 0

So, the vertex of  f(x) = (0,0)

Thus, the vertex of b(x) is 8 units to the right of the vertex of .

(Option A is false.)

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So, the answer in its simplest form is 1/200.a) Here's the completed frequency tree: Drink tea.

So, the number of people who drink at least 3 cups of tea each day is:[tex]10 * 0.2 = 2[/tex]Therefore, the fraction of the 400 people who drink at least 3 cups of tea each day is: 2 / 400This fraction can be simplified by dividing both the numerator and the denominator by their greatest common factor, which is 2:1 / 200So, the answer in its simplest form is 1/200.

A fraction is a numerical quantity representing a part of a whole or a ratio between two quantities. It is expressed as one number (the numerator) divided by another number (the denominator), separated by a horizontal line. The numerator represents the number of parts being considered, while the denominator represents the total number of equal parts that make up the whole. For example, in the fraction 3/4, the numerator is 3, which represents three parts out of a total of four equal parts. Fractions can be used to represent many different types of quantities, such as parts of a whole, ratios, percentages, and probabilities.

b) Out of the 400 people, 10 said yes to drinking tea, and 20% of those (or 0.20*10 = 2) said they drink at least 3 cups each day. Therefore, the fraction of people who drink at least 3 cups of tea each day is 2/400, which simplifies to 1/200.

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Hence, the banquet's overall payroll came to $1,495 as the number of captains by their pay rate, and the number of hours worked.

We must first determine the rate of pay for each group of employees before multiplying that number by the total number of employees for the dinner. The pay rate for captains is as follows because they are paid 60% above the rest of the staff: Compensation for captains is $17 per hour plus 60% of that amount, or $17 plus $10.20, for a total hourly wage of $27.20. The pay rate for the remaining employees is: $17 per hour is the pay rate for waitresses, bartenders, and hosts. Now, we can determine the banquet's overall payroll using the formula below.

given

We must first determine the pay rate for each group of employees before multiplying that number by the total number of employees for the dinner.

The pay rate for captains is as follows because they are paid 60% more than the rest of the staff:

Compensation for captains is $17 per hour plus 60% of that amount, or $17 plus $10.20, for a total hourly wage of $27.20.

The pay rate for the remaining employees is:

$17 per hour is the pay rate for servers, bartenders, and hosts.

Now, we can determine the banquet's overall payroll using the formula below:

Total payroll is calculated by multiplying the number of servers by their pay rate, the number of bartenders by their pay rate, the number of hosts by their pay rate, the number of captains by their pay rate, and the number of hours worked.

Total payroll is (14 x $17) plus (5 x $17) plus (4 x $17) plus (2 x $27.20) x 5 = $238 plus $85 plus $68 plus $272 x 5

Total compensation: $1,495

Hence, the banquet's overall payroll came to $1,495 as the number of captains by their pay rate, and the number of hours worked.

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Both the lines are parallel hence, they cannot intersect each other and value of H= (0,0).

A mathematical statement known as an equation consists of two algebraic expressions separated by equal signs (=) on either side.

It demonstrates the equality of the relationship between the printed statements on the left and right.

Left side equals right side in all formulas.

To find the values of unknowable variables, which stand in for unknowable quantities, you can solve equations.

A statement is not an equation if it lacks the equals sign.

When two expressions have the same value, a mathematical statement known as an equation will include the symbol "equal to" between them.

Hence, Both the lines are parallel hence, they cannot intersect each other and value of H= (0,0).

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The correct answer for this would be: Triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem.

A parallelogram is a straightforward quadrilateral with two sets of parallel sides in Euclidean geometry. A parallelogram's facing or opposing sides are of equal length, and its opposing angles are of similar size.

According to the ASA theorem, if the included side of one triangle and its two angles are comparable to the parts of another triangle, then the triangles are said to be congruent.

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Answer: No Solutions

Step-by-step explanation:

The square root of -125 if undefined

Therefore, the answer in no solutions

Answer:

Area_sector = 9.8 square inches

Step-by-step explanation:

A sector is a piece of a circle. Like a pie slice. We can use the angle given to find out what fraction of the circle we are looking for.

There are 360° in a circle. The sector is 125°. So the fraction of the circle we are looking for is (125/360) Of course this can be reduced, but we should just let the calculator handle it.

Now, the area of the whole circle is:

A = pi•r^2

A = pi • 3^2

The radius is given. r = 3

A = 9pi

They said to use 3.14 for pi

A = 28.26

This is the area of the whole circle, so we need to multiply by that fraction to find the area of the pie slice (sector)

Area_sector =

(125/360)28.26

= (0.3472222)28.26

= 9.8125

Round to the nearest tenth.

~= 9.8

Answer:

13%

Step-by-step explanation:

I have a test on  this  an i'm missing Cake batter. But I have sweet coconut and it says its 13%.

The total amount that he will pay for the car will be $17,093.40, assuming a loan amount of $15,000, a down payment of $4,300, a 5-year loan term, and a 5.25% APR.

Assuming that Dean has used the $4,300 as a down payment, his loan amount would be the total cost of the car minus the down payment, which is not given in the question.

However, assuming that the loan amount is $15,000 (which would be the case if the total cost of the car was $19,300), the monthly payment would be $284.89.

To calculate the total amount that Dean pays for the car over the course of the loan, we can multiply the monthly payment by the number of payments (60 payments over 5 years) and add the down payment:

Total amount = (Monthly payment x Number of payments) + Down payment

Total amount = ($284.89 x 60) + $4,300

Total amount = $17,093.40

Therefore, if Dean gets the loan and pays it off as scheduled, the total amount that he will pay for the car will be $17,093.40.

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The probability that the duration is at least 4 is 0.0053 (rounded to four decimal places).

Probability is represented by a number between 0 and 1, with 0 denoting an impossibility and 1 denoting a certainty.

(a) Let X be the duration of the contest. Then X can take values from 1 to 6, where 1 indicates that A or B wins the first game, and 6 indicates that the contest ends in a draw after 5 successive draws.

We can find the PMF of X by considering the possible outcomes for each value of X.

P(X = 1) = P(A wins) + P(B wins) = 0.2 + 0.5 = 0.7

P(X = 2) = P(A loses, B loses, A wins or B wins) = 0.3 * 0.3 * (0.2 + 0.5) = 0.054

P(X = 3) = P(A loses, B loses, A loses, B loses, A wins or B wins) = [tex]0.3^4[/tex] * (0.2 + 0.5) = 0.01323

P(X = 4) = P(A loses, B loses, A loses, B loses, A loses, B loses, A wins or B wins) = [tex]0.3^6[/tex] * (0.2 + 0.5) = 0.00243

P(X = 5) = P(A loses, B loses, A loses, B loses, A loses, B loses, A loses, B loses, A wins or B wins) = [tex]0.3^8[/tex] * (0.2 + 0.5) = 0.0004374

P(X = 6) = P(5 successive draws) = [tex]0.3^5[/tex] = 0.00243

Therefore, the PMF of X is:

X 1 2          3          4         5                 6

P(X) 0.7 0.054 0.01323 0.00243 0.0004374 0.00243

(b) To find the probability that the duration is at least 4, we need to add the probabilities of X = 4, X = 5, and X = 6.

P(X >= 4) = P(X = 4) + P(X = 5) + P(X = 6) = 0.00243 + 0.0004374 + 0.00243 = 0.0052974

Therefore, the probability that the duration is at least 4 is 0.0053 (rounded to four decimal places).

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