Determine whether or not the given vectors in R" form a basis for R". V1 =(0,0,2), V2 =(5,2,9), V3 =(7,2,11)

Carrie can walk at a speed of 3 mph.

To find out how fast Carrie can walk, we can use the formula distance = speed × time. Let's call the speed at which Carrie walks x, and the time it takes her to walk 6 miles t.

Since she can ride 9 mph faster than she can walk, her biking speed will be x + 9. We can set up the following equations:

6 = x × t (for walking)
24 = (x + 9) × t (for biking)

Since the time it takes her to walk 6 miles and bike 24 miles is the same, we can set the two equations equal to each other:

x × t = (x + 9) × t

Simplifying the equation gives us:

x = x + 9

Subtracting x from both sides gives us:

0 = 9

This is not a valid solution, so we need to go back to our original equations and solve for t:

t = 6/x (for walking)
t = 24/(x + 9) (for biking)

Setting these two equations equal to each other gives us:

6/x = 24/(x + 9)

Cross-multiplying gives us:

6(x + 9) = 24x

Distributing the 6 gives us:

6x + 54 = 24x

Subtracting 6x from both sides gives us:

54 = 18x

Dividing by 18 gives us:

x = 3

So Carrie can walk 3 mph.

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The function values.

To evaluate the function g(x) = 8 - x / 8 + 7 at the indicated values, we need to substitute the values into the function and simplify.

g(2) = 8 - 2 / 8 + 7 = 6 / 15 = 2 / 5

g(-8) = 8 - (-8) / 8 + 7 = 16 / 15 = 16 / 15

g(1/2) = 8 - (1/2) / 8 + 7 = (15.5) / 15 = 31 / 30

g(a) = 8 - a / 8 + 7 = (15 - a) / 15

g(a - 8) = 8 - (a - 8) / 8 + 7 = (16 - a) / 15

g(x^2 - 8) = 8 - (x^2 - 8) / 8 + 7 = (16 - x^2) / 15

So, the function values are:
g(2) = 2 / 5
g(-8) = 16 / 15
g(1/2) = 31 / 30
g(a) = (15 - a) / 15
g(a - 8) = (16 - a) / 15
g(x^2 - 8) = (16 - x^2) / 15

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Answer:

Step-by-step explanation:

North Dakota is 34 times larger

                 Puerto Rico

North Dakota = 1.83x[tex]10^{5}[/tex]   [tex]Km^{2}[/tex]    

Puerto Rico =  5.33 x [tex]10^{3}[/tex]   [tex]Km^{2}[/tex]  

                       1.83 x [tex]10^{5}[/tex]

                       5.33 x [tex]10^3[/tex]        = 0.34 x [tex]10^2[/tex]

The solution for this system of equations is any ordered pair of the form (x, 3x-3), where x is any real number.

Two or more expressions with an Equal sign is called as Equation.

We can solve this system of equations using substitution. Solving the second equation for y, we get:

y = 3x - 3

Substituting this expression for y into the first equation, we get:

9x - 3(3x - 3) = 9

9x - 9x + 9 = 9

Therefore, the equation is true for any value of x.

Hence, the solution for this system of equations is any ordered pair of the form (x, 3x-3), where x is any real number.

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Answer:

[tex]\huge\boxed{\sf 116\ in.\²}[/tex]

Step-by-step explanation:

The composite figure is made up of two shapes.

= Length × Width

Where L = 13 in., W = 7 in.

= 13 × 7

= 91 in.²

[tex]\displaystyle =\frac{\pi r^2}{2} \\\\\underline{Where \ r:}\\\\= \frac{13-5}{2} \\\\= \frac{8}{2} \\\\= 4 \ in.\\\\So,\ the \ above\ equation \ becomes\\\\= \frac{(3.14)(4)^2}{2} \\\\= \frac{(3.14)(16)}{2} \\\\= (3.14)(8)\\\\= 25.13 \ in.^2[/tex]

= Area of rectangle + Area of semi-circle

= 91 + 25.13

= 116.13 in.²

[tex]\rule[225]{225}{2}[/tex]

The domain of a relation is the set of all the first elements of the ordered pairs in the relation, and the range of a relation is the set of all the second elements of the ordered pairs in the relation.

So, for the given relation {(5,0),(8,4),(5,2),(-2,4)}, the domain would be {5, 8, -2} and the range would be {0, 4, 2}.

To find the domain, we simply take the first element of each ordered pair and put them in a set. Similarly, to find the range, we take the second element of each ordered pair and put them in a set.

Therefore, the domain of the relation is {5, 8, -2} and the range of the relation is {0, 4,2}.

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The equation that represents a linear function rule for the following set of ordered pairs is  A. y = -3x + 2

We can express the Equation of a linear function, as y = mx + b.

m= slope

b= y-intercept

We can determine the slope (m):

= (change in y)/(change in x )

then we can substitute the values as :

= (-7 -(-4)) / (3 - 2)

= -3/1

= -3

Then thevalue of b can be found through the substituting of  m = -3  into the  equation knowing that  (1, -1) = (x, y)

Then we have;

-1 = [-3(1) + b]

-1 = [-3 + b]

[-1 + 3] = b

b = 2

In this case ,  we can put m = -3 , b = 2

y = mx + b

y = [-3(x) + 2]

Therefore, we have

y = -3x + 2

Hence option A is correct.

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complete question:

Which equation represents a linear function rule for the following set of ordered pairs?

x y

1 -1

2 -4

3 -7

A. y = -3x + 2

B. y = -3x - 2

C. y = -2x + 3

D. y = -2x - 3

3 gallons are the starting value determined by the slope-intercept relation.

One of the most popular ways to represent a line's equation is in the slope intercept form of a straight line. When the slope of the straight line and the y-intercept are known, the slope intercept formula can be used to determine the equation of a line ( the y-coordinate of the point where the line intersects the y-axis). The equation of a line is the equation that each point on the line fulfils.

To determine how much gas is entering his car each second, we must determine the rate of change:

Rise / Run is the rate of change.

Rate of change: (11 - 3 - 11) / (48 - 0)

Change rate: 8/48 = 1/6 gallons

Making use of the slope-intercept relationship:

Y = b(x) + c(slope); c(initial gas amount)

Choose any (x, y) point pair on the table:

(0, 3)

3 = 1/6x + c 3 = 1/6(0) + c\s3 = 0 + c\sc = 3

As a result, the starting point is 3 gallons.

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The frequency of a repeated event is its number of instances for every unit of time. In below given way frequency table can be constructed.

The frequency of a repeated event is its number of instances for every unit of time. It differs from angular frequency and is sometimes made reference to as temporal resolution for clarification. The unit of frequency is hertz (Hz), or one occurrence per second.

The time elapsed between events is measured by the period, which is the opposite of the frequency. For instance, the period, T—the time between beats—is equal to half a second if a heart beats 120 times per minute. Frequency tables to represent the morning and afternoon dogs as two sets of data are:

range        frequency

0- 9             2

10- 19           3

20-29         1

30-39         2

40-49         1

50-59         1

Therefore, in above given way frequency table can be constructed.

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The answer of the number of offers you received are as follow,

1. Mean is equal to 1.96.

2. Standard deviation is equal to 1.17 (rounded to two decimal places) .

1.Mean (or expected value) is represented by E(X)

Formula of the expected value is,

E(X) = ∑xP(x)

Substitute the value from the table we get,

⇒E(X) = (0)(0.19) + (1)(0.12) + (2)(0.38) + (3)(0.16) + (4)(0.15)

⇒E(X)  = 0 + 0.12 + 0.76 + 0.48 + 0.6

⇒E(X)  = 1.96

2. Standard deviation of the number of offers received ,

⇒ Standard deviation 'σ' = √[ Σ(x - E(X))² P(x) ]

Substitute the values from the table we get,

⇒ Standard deviation 'σ'

= √[ (0 - 1.96)² (0.19) + (1 - 1.96)² (0.12) + (2 - 1.96)² (0.38) + (3 - 1.96)²(0.16) + (4 - 1.96)²(0.15) ]

= √[ 1.3728 ]

≈ 1.171665

≈1.17(rounded to two decimal places)

Therefore, the mean and the standard deviation of the number of offers received is approximately 1.96 and 1.17 (rounded to two decimal places) respectively.

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The above question is incomplete, the complete question is:

Suppose you applied for positions at four different companies and let X be the number of offers you received

X      :            0               1               2               3                4

P(x)  :             0.19         0.12           0.38         0.16            0.15

1.  Find the mean

2.  standard deviation

of the number of offers you received. Round your answer to two decimal places

Answer:

-11

Step-by-step explanation:

*The sum of all three angles of a triangle is 180*

So,

80+60

140 + (x + 51) = 180

-140 -140

x + 51 = 40

- 51 - 51

x = - 11

HOPE this helped

The solution is, the value of x is, x = 12.

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane.

here, we have,

from the given figure, we get,

the triangles are similar,

so, we know that, the sides are proportional to each other.

i.e. 8:4 = x: 6

or, x = 8 * 6 / 4

or, x  = 2 * 6

or, x = 12

Hence, The solution is, the value of x is, x = 12.

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The maturity value of the loan is $13487.58 and the interest paid on the loan is $5487.58.

The maturity value of a loan is the amount that will be due at the end of the loan period. To calculate the maturity value of the loan, we will use the formula:
Maturity Value = Principal x (1 + Interest Rate/Compounding Periods)^(Compounding Periods x Number of Years)
(a) Maturity value of the loan
Maturity Value = 8000 x (1 + 0.11/12)^(12 x 5)
Maturity Value = 8000 x (1 + 0.0091666667)^60
Maturity Value = 8000 x 1.685947578
Maturity Value = $13487.58
The maturity value of the loan is $13487.58.
(b) Interest are you paying on the loan
To calculate the interest paid on the loan, we will subtract the principal from the maturity value.
Interest = Maturity Value - Principal
Interest = 13487.58 - 8000
Interest = $5487.58
The interest paid on the loan is $5487.58.

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The new ordered pair of A’ are (-1, -2.25). Hence the correct option is b.

A cοοrdinate system in geοmetry is a technique that uses οne οr mοre integers οr cοοrdinates tο establish the accurate pοsitiοning οf geοmetrical οbjects οn a manifοld, such as Euclidean space. When lοcating a pοint οr οbject οn a twο-dimensiοnal plane, cοοrdinates—pairs οf numbers—are utilized.

The x and y cοοrdinates serve as a representatiοn οf a pοint's lοcatiοn οn a 2D plane. a grοup οf numbers used tο denοte certain lοcatiοns. The figure usually cοnsists οf twο integers. The frοnt-tο-back and tοp-tο-bοttοm distances are represented by the first and secοnd numbers, respectively. When there are 12 units belοw and 5 abοve, as in (12.5), the ratiο is 1.

Tο cοnduct a dilatiοn with a scale factοr οf 1/2 arοund the οrigin, multiply each pοint's cοοrdinates by 1/2. If the initial cοοrdinates οf pοint A are (x, y), then the dilated pοint A' cοοrdinates are:

A' = (1/2 * x, 1/2 * y)

We negate the x-cοοrdinate while keeping the y-cοοrdinate cοnstant tο represent the dilated pοint A' οver the y-axis. As a result, the final cοοrdinates οf A" are:

A'' = (-1/2 * x, 1/2 * y)

As a result, Anew "'s οrdered pair is (1/2 * x, 1/2 * y)

= ((1/2 × -2), (1/2 × -5))

= (-1, -2.25)

Thus, the new ordered pair of A’ are (-1, -2.25). Hence the correct option is b.

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The amount of chemical in the tank after 15 minutes is 154.14 grams.

To determine the amount of chemical in the tank after 15 minutes, we need to use the formula for the concentration of a solution:

C = m/V

Where C is the concentration of the solution, m is the mass of the chemical, and V is the volume of the solution.

Initially, the tank contains 12 litres of water and 24 grams of chemical A, so the initial concentration of the solution is:

C0 = 24/12 = 2 grams per litre

The solution flows into the tank at a rate of 4 grams per litre and 4 litres per minute, so the amount of chemical flowing into the tank per minute is:

4 grams per litre × 4 litres per minute = 16 grams per minute

The well-stirred mixture flows out of the tank at a rate of 2 litres per minute, so the amount of chemical flowing out of the tank per minute is:

C × 2 litres per minute = 2C grams per minute

The net change in the amount of chemical in the tank per minute is:

16 grams per minute - 2C grams per minute = 16 - 2C grams per minute

After 15 minutes, the net change in the amount of chemical in the tank is:

(16 - 2C) grams per minute × 15 minutes = 240 - 30C grams

The final amount of chemical in the tank is:

m = 24 + 240 - 30C = 264 - 30C grams

The final volume of the solution in the tank is:

V = 12 + 4 litres per minute × 15 minutes - 2 litres per minute × 15 minutes = 42 litres

The final concentration of the solution in the tank is:

C = m/V = (264 - 30C)/42

Solving for C, we get:

42C = 264 - 30C

72C = 264

C = 264/72 = 3.67 grams per litre

The final amount of chemical in the tank is:

m = C × V = 3.67 grams per litre × 42 litres = 154.14 grams

Therefore, the amount of chemical in the tank after 15 minutes is 154.14 grams.

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The annual payment for the loan is $48,867.89.

To find how much is the annual payment for the loan, we must first find the total cost of the land and the loan amount.

The total cost of the land is $4,100 × 215 = $881,500.

The down payment is 25% of the total cost, so the down payment is $881,500 × 0.25 = $220,375.

The loan amount is the total cost minus the down payment, so the loan amount is $881,500 - $220,375 = $661,125.

To find the annual payment, we can use the formula A = P(1 + r)^n / [(1 + r)^n - 1], where A is the annual payment, P is the loan amount, r is the annual interest rate (APR), and n is the number of years.

Plugging in the values, we get:

A = $661,125(1 + 0.0675)^20 / [(1 + 0.0675)^20 - 1]

A = $661,125(1.0675)^20 / [(1.0675)^20 - 1]

A = $661,125(3.0878) / (3.0878 - 1)

A = $2,040,756.55 / 2.0878

A = $977,357.72

So the annual payment is $977,357.72 / 20 = $48,867.89.

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Answer: The second and last one

Step-by-step explanation:

The value of \( k \) that makes \( \vec{a} \) and \( \vec{b} \) orthogonal is \( k=1 \).

Two vectors are orthogonal if their dot product is zero. The dot product of two vectors \( \vec{a}=\langle a_1,a_2\rangle \) and \( \vec{b}=\langle b_1,b_2\rangle \) is given by \( \vec{a}\cdot\vec{b}=a_1b_1+a_2b_2 \).

In this case, we have \( \vec{a}=\langle 1,-3\rangle \) and \( \vec{b}=\langle 3, k\rangle \). So the dot product is:

\( \vec{a}\cdot\vec{b}=(1)(3)+(-3)(k)=3-3k \)

We want this dot product to be zero, so we can set it equal to zero and solve for \( k \):

\( 3-3k=0 \)

\( 3k=3 \)

\( k=1 \)

Therefore, the value of \( k \) that makes \( \vec{a} \) and \( \vec{b} \) orthogonal is \( k=1 \).

Answer: \( \boxed{k=1} \).

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The equation (3x⁴z² - 8x⁵)(-6xz⁶) is rewritten without parenthesis as 48x⁶z⁶ - 18x⁵z⁸

An equation is an expression that shows the relationship between two or more numbers and variables. Equations can either be linear, quadratic, cubic and so on depending on the degree.

Given the equation:

(3x^(4)z^(2)-8x^(5))(-6xz^(6))

Simplifying gives:

= (3x⁴z² - 8x⁵)(-6xz⁶)

Opening the parenthesis gives:

= -18x⁵z⁸ + 48x⁶z⁶

= 48x⁶z⁶ - 18x⁵z⁸

The equation is equivalent to 48x⁶z⁶ - 18x⁵z⁸

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Answer:

9

Step-by-step explanation:

7/8 divided by 1/8 is 7

7+2 = 9

Brainliest?

Answer:

One possible way to complete the array is:

    8  |  2 0 8

       |

    ------

       |

 2 |  2 6

 5 |  5 2

 1 |  1 7

 6 |  2 0

The answer is 26, so 208 ÷ 8 = 26.

To complete the array, we start by dividing the hundreds digit (2) by 8, which gives 0 with a remainder of 2. We write the remainder (2) in the ones place of the first row. Then we bring down the tens digit (0) and add it to the remainder to get 20. We divide 20 by 8, which gives 2 with a remainder of 4. We write the remainder (4) in the tens place of the second row, and bring down the ones digit (8) to the ones place of the second row. Finally, we add the digits in the second row to get 26, which is the quotient of the division

Step-by-step explanation:

From the diagram of a Regular hexagon ABCDEFG in inscribed into circle O, we can state that the radius of circle O is 8/√3 cm

1. (a) The radius of circle O is equal to the length of OB, which is also equal to the length of OA.

Since OA = OB = OC, and OC is the perpendicular bisector of AB, triangle OAB is equilateral.

Thus, using the formula for the length of the side of a regular hexagon, we get:

AB = 8cm

OA = OB = OC = AB/√3

OA = OB = OC = 8/√3 cm

Radius of circle O = OA = OB = OC = 8/√3 cm

We only need to focus on triangle AOB to calculate the radius of the circle.

Since OA, OB, and AB are all known (OA and OB being equal to each other, and AB being given as 8 cm), we can use the Pythagorean theorem to solve for the length of AO (which is the same as the length of BO), and hence the radius of the circle.

b. The area of circle O can be calculated using the formula:

Area = πr²

Area = π(8/√3)²

Area = 64π/3 square cm.

The measure of arc AB is equal to twice the measure of angle BOC, since arc AB subtends angle BOC:

m(arc AB) = 2m∠BOC

m(arc AB) = 2(60°)

m(arc AB) = 120°f.

The measure of arc ACE is equal to the sum of the measures of angles BOC and COE, since arc ACE subtends both of these angles:

m(arc ACE) = m∠BOC + m∠COE

m(arc ACE) = 60° + 60°

m(arc ACE) = 120°g.  

The length of arc AB is equal to the circumference of circle O times the fraction of the circle subtended by arc AB:

Length of arc AB = (m(arc AB)/360°) x 2πr

Length of arc AB = (120°/360°) x 2π(8/√3) cm

Length of arc AB = (1/3) x (16π/√3) cm

Length of arc AB = (16π)/(3√3) cm.

The area of sector AOB is equal to the fraction of the circle subtended by arc AB times the area of circle O:

Area of sector AOB = (m(arc AB)/360°) x πr²

Area of sector AOB = (120°/360°) x π(8/√3)^2 cm²

Area of sector AOB = (1/3) x (64π/3) cm²

Area of sector AOB = 64π/9 square cm

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Answer:

1162261467

Step-by-step explanation:

Note that this is a geometric sequence.

1,3,9,27
The pattern is to multiply the number by 3 to get to the next term.
a = 1

r = 3

n=20

arⁿ⁻¹ = nth term

You then substitute the values of a,r and n into the nth term:

arⁿ⁻¹=3¹⁹

3¹⁹=1162261467 which is the 20th term of the sequence

Hoped this helped

Answer:

To find the 20th term of the sequence 1, 3, 9, 27, we need to first identify the pattern of the sequence. It appears that each term of the sequence is obtained by multiplying the previous term by 3. This means that the sequence is a geometric sequence with first term (a) = 1 and common ratio (r) = 3.

Using the formula for the nth term of a geometric sequence:

an = a * r^(n-1)

where an is the nth term of the sequence, a is the first term, r is the common ratio, and n is the term number.

Substituting the values we have:

a = 1, r = 3, and n = 20

an = 1 * 3^(20-1) = 1 * 3^19 = 1162261467

Therefore, the 20th term of the sequence 1, 3, 9, 27 is 1162261467.

Step-by-step explanation:

brainliest pls

Answer: $21.60

Step-by-step explanation: This is a similar way I solve this problem..

You take 20%, notice is it 20% of 100%, our 100% in this case is $27.00, our 20% is missing.. so.. divide 27.00 by 5, because 5 20%'s is 100%, after you divide by 5 you will end up with $5.40, so we know that 20% of $27.00 is $5.40..

Take $27.00 and take $5.40 away, and you will be left with 21.60

20%x5=100%

27.00/5=5.40

27.00-5.40=21.60

Coretta's pair of jeans is $21.60 when it is on sale for 20% off.

The income tax amount of 6% will be equal to $252 on a salary of $4200.

A number or ratio that can be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its entirety and then multiply it by 100. The percentage therefore refers to a component per hundred. Per 100 is what the word percent implies.

A percentage is a figure or ratio expressed as a fraction of 100 in mathematics. Although the abbreviations "pct.", "pct.", and occasionally "pc" are also used, the percent symbol, "%," is frequently used to indicate it. A percentage is a figure without dimensions and without a standard measurement.

We must divide the value by the overall value to find the percentage, and then multiply the resulting number by 100.

A tax placed on people or organisations in relation to their income or profits is known as an income tax. Tax rates multiplied by taxable revenue are typically used to calculate income taxes. Tax rates can change depending on the taxpayer's characteristics and sort of income.

The word "income tax" refers to a category of tax that governments levy on income generated by businesses and people under their jurisdiction.

In this question,

Salary amount= $4200

Income tax= 6%

The amount to be paid in income tax= 6% of $4200

                                                             = (6/100)*4200

                                                             = $252

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The addition of the two supplementary angles will be equal to 180 degrees,

The tilt is the partition caught between lines, surfaces, or vectors that intersect. Degrees are another mode to reveal the slant. For a full rotation, the arc is 360°.

A tilt is a figure in Geometric shapes created by two rays, called the flanks of the arc, that share a common termination, called the apex of the angle.

Two slants are said to be supplementary tilts if their totality is 180 degrees.

Let ∠1 and ∠2 be the supplementary angles. Then the equation is written as,

∠1 + ∠2 = 180°

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The shape with a series of parallel cross sections that are congruent circles is a cylinder.

The cross-section that results from cutting a cylinder parallel to its base is a circle that is congruent to all other parallel cross-sections. This is true for any plane that is perpendicular to the cylinder's base. The only shape that has parallel cross-sections that are congruent circles is a cylinder, for this reason.

Two parallel, congruent circular bases that lay on the same plane make up the three-dimensional shape of a cylinder. A curved rectangle connecting the bases makes up the cylinder's lateral surface. Congruent circles are produced when a cylinder is cut in half parallel to its base.

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(a) m∠BAC = 73°

(b) The value of y is 2.

A triangle that has two equal lengths of sides and two equal measures of angles is called an isosceles triangle.

(a) Given:

m∠BEA = 62°.

Assuming ABCD is a square.

The diagonal of the square divides the square into two isosceles right-angled triangles.

So, m∠BAD = 90° and m∠ABD = m∠ADB = m∠ABE = 45°.

So, the angle measure of ∠BAC,

m∠BAC = 180° - (45 + 62)

m∠BAC = 180° - 107°

m∠BAC = 73°

(b) Given:

BE = 6y + 2 and CE = 4y + 6.

Assuming ABCD is a square.

The diagonals of squares divide the diagonals into two equal parts.

So, BE = CE

6y + 2  = 4y + 6.

2y = 4

y = 2

Therefore, the value of y is 2.

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The revised mean is thus 20.8, which is consistent with choice (B).

How can you calculate the mean?

Simply dividing the total number of values in a data collection by the aggregate of all of the values yields it. The computation can be performed on unprocessed data or data that has been combined into a frequency chart.

We must first compute the average of the dataset such as the new number, then split by the overall number of data points in order to determine the innovative mean within a week of adding a count to the dataset.

Let's call the four original numbers in the dataset A, B, C, and D. We know that their mean is 19, so we can write:

(A + B + C + D)/4 = 19

Multiplying both sides by 4 gives us:

A + B + C + D = 76

Now, if we add the number 28 to the dataset, we get a new sum:

A + B + C + D + 28

To find the new mean, we need to divide this sum by the total number of values in the dataset, which is 5 now:

(A + B + C + D + 28)/5

Substituting in our earlier expression for A + B + C + D, we get:

(76 + 28)/5 = 104/5 = 20.8

Therefore, the new mean is 20.8, which corresponds to option (B).

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16. The expression of the product as a sum: 7 cos(6x)cos(7x) = 7/2 * (cos(13x) + cos(x))
17. The expression of the sum as a product: sin(2x) - sin(7x) = -2 cos(9x/2) sin(5x/2)

The product as a sum can be written using the formula for the product of two cosines:

cos a * cos b = 1/2 * (cos(a + b) + cos(a - b))

Using this formula, we can write the product 7 cos(6x)cos(7x) as a sum:

7 cos(6x)cos(7x) = 7/2 * (cos(6x + 7x) + cos(6x - 7x))
= 7/2 * (cos(13x) + cos(-x))
= 7/2 * (cos(13x) + cos(x))


The sum as a product can be written using the formula for the difference of two sines:

sin a - sin b = 2 cos((a + b)/2) sin((a - b)/2)

Using this formula, we can write the sum sin(2x) - sin(7x) as a product:

sin(2x) - sin(7x) = 2 cos((2x + 7x)/2) sin((2x - 7x)/2)
= 2 cos(9x/2) sin(-5x/2)
= 2 cos(9x/2) (-sin(5x/2))
= -2 cos(9x/2) sin(5x/2)

So the final answers are:
7 cos(6x)cos(7x) = 7/2 * (cos(13x) + cos(x))
sin(2x) - sin(7x) = -2 cos(9x/2) sin(5x/2)

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