The yearly Avrage
4. An old furnace cost $850 per year to run. A new one costs $2,500 to buy and will save 34% annually in
energy costs to run it. In how many years will it pay for itself?

Answer:

B

Step-by-step explanation:

One half life there will be 50 %

   another half life and there will be 25 %  

        so    TWO half lives * 28.1 yrs/ half life = 56.2 yrs

(a) The 99% confidence interval for the population standard deviation at Bank A is 1.33 to 2.16.

(b) The 99% confidence interval for the population standard deviation at Bank B is 1.41 to 2.29.

To find the confidence interval for the population standard deviation, we can use the chi-square distribution. The formula for the chi-square distribution is:

(X²) / (n-1) = s²

where X² is the chi-square value, n is the sample size, and s is the sample standard deviation.

For Bank A:

n = 20

s = 1.252

Using the chi-square distribution table with a degree of freedom of n-1 = 19 and a 99% confidence level, we find the critical values to be 8.907 and 32.852. Substituting these values into the formula above, we get:

(19 x 1.252²) / 32.852 < o² < (19 x 1.252²) / 8.907

1.78 < o² < 4.67

1.33 < o < 2.16

Therefore, the 99% confidence interval for the population standard deviation at Bank A is 1.33 to 2.16.

For Bank B:

n = 20

s = 1.397

Using the chi-square distribution table with a degree of freedom of n-1 = 19 and a 99% confidence level, we find the critical values to be 8.907 and 32.852. Substituting these values into the formula above, we get:

(19 x 1.397²) / 32.852 < o² < (19 x 1.397²) / 8.907

2.00 < o² < 5.24

1.41 < o < 2.29

Therefore, the 99% confidence interval for the population standard deviation at Bank B is 1.41 to 2.29.

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Answer:

8 hours and 20 minutes

Step-by-step explanation:

Let's assume the distance between Lagos and Port Harcourt is approximately 500 kilometers. With an average speed of 60 km per hour, we can use the formula:

Time = Distance / Speed

Substituting the values, we get:

Time = 500 km / 60 km/h ≈ 8.33 hours

Therefore, it would take approximately 8 hours and 20 minutes for the car to travel from Lagos to Port Harcourt, assuming the distance is around 500 kilometers and the average speed is 60 km per hour. Please note that this is just an example calculation, and the actual time may vary depending on the distance and road conditions.

Answer: 98

Step-by-step explanation:

sailboats to dinghies = 5 : 7

if 5 is 70 then what is 7 to ?

well 70/5 is 14 so we had to multiply 5 by 14 to get 70.
That means we also have to multiply 7 by 14 to get 98

A line through points X and Y will be perpendicular bisector of AB.

By examining the figure, it is evident that the compass created arcs as follows:

An arc was drawn with A as the center and a radius greater than half the length of AB.

and, another arc was drawn with B as the center using the same radius.

The points where these arcs intersect are labeled as X and Y.

This construction clearly represents the creation of a perpendicular bisector for the line segment AB.

Therefore, if we draw a line segment passing through points XY, it will serve as the perpendicular bisector of AB.

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The table of at least 4 values of the function is

x 0   -6   6  3

y  36  0  0 27

The domain is All set of real values and the range: y ≤ 36

The x and y intercepts of the rainbow are x = (-6, 0) and (6, 0) y = (0, 36)

Given that we have

y = -x² + 36

From the graph, we have the following values

x      y

0    36

-6    0

6    0

Set x  = 3

So, we have

y = -3² + 36

Evaluate

y = 27

So, we have

x      y

0    36

-6    0

6    0

3    27

From the graph, we have

These values mean that;

For the x and y intercepts of the rainbow, we have

x = (-6, 0) and (6, 0)

y = (0, 36)

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The least number by which each of the given numbers should be multiplied so as to get a perfect square are 5 and 7.

Here, we have,

we have

a) 845 =5×13×13

The smallest whole number is 5, by which 845 should be multiplied so as to get a perfect square.

Now , we get,

845 ×5=5×13×13×5

4225=5×13×13×5

√4225=65

b) 1008=2×2××2×3×3×3

The smallest whole number is 7, by which 1008 should be multiplied so as to get a perfect square.

Now , we get,

1008×7=2×2×2×2×3×3×7×7

7056=2×2×3×7

=84

Answer : 5 and 7

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The standard absolute value graph has been shifted 7 units to the right and 3 units down to get this graph.  

That means we have answer B: f(x) = | x - 7 | - 3

Remember that the left-right shifts are opposite from what they would appear to be in the equation.

  | x - 7 | shifts it right 7 units.

  | x + 7 | shifts it left 7 units.

Answer:

18 ft.

Step-by-step explanation:

The Bakers make 85 glazed donuts.

To solve this problem, we need to determine how many glazed donuts and how many donut holes can be made from 80 pounds of dough. Then, we can compare the quantities of each to determine the difference.

First, we need to convert 80 pounds to ounces, since the amounts of dough for each type of donut are given in ounces. There are 16 ounces in a pound, so 80 pounds is equal to 80 x 16 = 1,280 ounces.

Next, we can set up two equations based on the amounts of dough needed for each type of donut:

Glazed donuts: 10 ounces of dough per donut

Donut holes: 2 ounces of dough per hole

Let's use "g" to represent the number of glazed donuts and "h" to represent the number of donut holes. We know that the total amount of dough used must be 1,280 ounces, so we can write:

10g + 2h = 1,280

We also know that the bakers divide the dough evenly between glazed donuts and donut holes, so the total number of donuts must be:

g + h = ?

We don't know the total number of donuts yet, but we do know that it must be an even number, since the dough is divided evenly between glazed donuts and donut holes.

To solve for g and h, we can use substitution or elimination. Let's use substitution. We can solve the first equation for one of the variables, such as h:

h = (1,280 - 10g) / 2

Then, we can substitute this expression for h in the second equation:

g + (1,280 - 10g) / 2 = ?

Simplifying this equation, we get:

g + 640 - 5g = ?

Combining like terms, we get:

5g + 640 = ?

Subtracting 640 from both sides, we get:

5g = -640

Dividing both sides by 5, we get:

g = -128

This doesn't make sense as a solution, since we can't have negative numbers of donuts. This means there must be an error in our calculations or assumptions.

Let's go back to the second equation:

g + h = ?

We know that the total number of donuts must be an even number, so we can write:g + h = 2n

where n is some positive integer. We can substitute this expression for g + h in the first equation:10g + 2h = 1,280

10g + 2(2n - g) = 1,280

Simplifying this equation, we get 14g - 4n = 1,28

Dividing both sides by 2, we get:7g - 2n = 640

Now we have two equations:7g - 2n = 640

g + h = 2n

We can use substitution or elimination to solve for g and h. Let's use elimination. We can multiply the first equation by 2 and add it to the second equation:14g - 4n = 1,280

2g + 2h = 4n

Multiplying the first equation by 2, we get:28g - 8n = 2,560

Adding this to the second equation, we get:30g = 2,560

Dividing both sides by 30, we get:g = 85.33

This means that the bakers make 85 glazed donuts. To find the number of donut holes, we can substitute this value

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Answer:

no, Irrational

Step-by-step explanation:

because the solution is a non-terminating decimal. 8.246

The three true statements about the function and its graph are The graph of the function is a parabola, The graph does not open down (it opens upwards). and None of the given points, (20, -8) or (0, 0), lie on the graph.

Based on the quadratic function f(x) = x^2 – 5x + 12:

The value of f(-10) = 82 is not true.

To find the value of f(-10), substitute x = -10 into the function: f(-10) = (-10)^2 - 5(-10) + 12 = 100 + 50 + 12 = 162.

The graph of the function is a parabola.

This statement is true. The quadratic function has a degree of 2, which means its graph will be a parabola.

The graph of the function opens down.

This statement is not true. The coefficient of the x^2 term in the quadratic function is positive (1), so the parabola opens upwards.

The graph contains the point (20, -8).

This statement is not true. To verify if the point (20, -8) is on the graph, substitute x = 20 into the function: f(20) = (20)^2 - 5(20) + 12 = 400 - 100 + 12 = 312. Therefore, the point (20, -8) does not lie on the graph.

The graph contains the point (0, 0).

This statement is not true. To verify if the point (0, 0) is on the graph, substitute x = 0 into the function: f(0) = (0)^2 - 5(0) + 12 = 0 - 0 + 12 = 12. Therefore, the point (0, 0) does not lie on the graph.

Therefore, the three true statements about the function and its graph are:

The graph of the function is a parabola.

The graph does not open down (it opens upwards).

None of the given points, (20, -8) or (0, 0), lie on the graph.

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The answers to all parts is shown below.

a. According to the graph, the concentration after 8 hours is approximately 22 mg/L.

b. The concentration increases from 0 to approximately 30 mg/L over the first 4 hours, and then decreases from approximately 30 mg/L to 16 mg/L between 4 and 20 hours.

c. The drug is at its maximum concentration after approximately 4 hours, with a maximum concentration of approximately 30 mg/L.

d. According to the graph, it takes approximately 12 hours for the concentration to decrease from the maximum concentration of approximately 30 mg/L to 16 mg/L.

e. After 1 week, the graph predicts that the concentration will be very close to 0 mg/L. After 1 month, the concentration is predicted to be essentially 0 mg/L.

f. During the first 4 hours, the concentration increases from 0 to approximately 30 mg/L.

Between 4 and 8 hours, the concentration decreases slightly from approximately 30 mg/L to 28 mg/L.

Between 8 and 12 hours, the concentration decreases more rapidly from 28 mg/L to approximately 22 mg/L.

Between 12 and 20 hours, the concentration decreases gradually from 22 mg/L to 16 mg/L.

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The volume of the second right triangular prism is 7488.96 cubic meters.

For the second right triangular prism, we have:

Base: A right triangle with base 10 m and height 12 m

Height: 8 m

To find the volume, we can use the same formula as before:

Volume = (1/2) x base x height x altitude

Using the Pythagorean theorem, we can find that the altitude is:

altitude = √(10² + 12²) = √(244) ≈ 15.62

Now we can substitute the given values into the volume formula:

Volume = (1/2) x 10 x 12 x 8 x 15.62

Volume ≈ 7488.96 m^3

Therefore, the volume of the second right triangular prism is 7488.96 cubic meters.

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Based on the above. (see image attached), The value of GF in the rhombus is option A: 26.4

A four-sided shape with equal side lengths is known as a rhombus. Equally sided four-sided figure is also referred to as an equilateral quadrilateral, because equilateral denotes that its sides have the same length.

Note that the properties of a rhombus are :

So, note that:

DH = HF = 16

GH = HE = 21

Therefore, by Using Pythagoras theorem we can look for GF by:

GF = √21² + 16²

= √441 + 256

= √697

= 26.4007575649

= 26.4

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The value of missing side is,

⇒ x = 2.54

We have to given that;

A triangle is shown.

And, To find the missing side of triangle.

Now, By using trigonometry formula, we get;

⇒ tan 23° = Opposite / Base

⇒ tan 23° = x / 6

⇒ 0.424 = x / 6

⇒ x = 0.424 × 6

⇒ x = 2.54

Thus, The value of missing side is,

⇒ x = 2.54

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The domain of the composite function (f - g)(-1) is derived to be 12 which makes c the correct option.

A function is composite when the co- domain of the first mapping is the domain of the second mapping

We shall evaluate the domains of the function (f - g)(-1) as follows:

Observing from the table of values, at the point x = -1 we have;

f(x) = 5 and g(x) = -7

So;

(f - g)(-1) = 5 - (-7)

(f - g)(-1) = 5 + 7

(f - g)(-1) = 12

Therefore, the value for domain of the composite function (f - g)(-1) is derived to be 12.

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Answer:

∠ADB

Step-by-step explanation:

Acute angles measure less than 90°.

So, here ∠ADB is acute angle,

the base of the triangle below is b. bc.

Answer:If cos(\theta) = \sqrt(3)/2, then we can use the Pythagorean identity sin^2(\theta) + cos^2(\theta) = 1 to find sin(\theta):

sin^2(\theta) + (\sqrt(3)/2)^2 = 1

sin^2(\theta) + 3/4 = 1

sin^2(\theta) = 1 - 3/4

sin^2(\theta) = 1/4

sin(\theta) = +/- 1/2

Since \theta is an acute angle, sin(\theta) must be positive. Therefore, sin(\theta) = 1/2.

We can then use the identity tan(\theta) = sin(\theta)/cos(\theta) to find tan(\theta):

tan(\theta) = sin(\theta)/cos(\theta)

tan(\theta) = (1/2)/(\sqrt(3)/2)

tan(\theta) = 1/\sqrt(3)

Finally, we can use the reciprocal and quotient identities to find the values of sec(\theta) and csc(\theta):

sec(\theta) = 1/cos(\theta)

sec(\theta) = 2/\sqrt(3)

csc(\theta) = 1/sin(\theta)

csc(\theta) = 2

Step-by-step explanation:

Answer:

[tex]\sin\theta=\dfrac{1}{2}[/tex]

[tex]\tan\theta=\dfrac{\sqrt{3}}{3}[/tex]

[tex]\csc\theta=2[/tex]

[tex]\sec\theta=\dfrac{2\sqrt{3}}{3}[/tex]

[tex]\cot\theta=\sqrt{3}[/tex]

Step-by-step explanation:

The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse of a right triangle:

[tex]\cos\theta= \sf \dfrac{adjacent}{hypotenuse}=\dfrac{\sqrt{3}}{2}[/tex]

Therefore, the length of the side adjacent angle θ is √3 and the length of the hypotenuse is 2.

[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]

We can use Pythagoras Theorem to calculate the length of the side opposite angle θ:

[tex](\sqrt{3})^2+O^2=2^2[/tex]

[tex]3+O^2=4[/tex]

[tex]O^2=1[/tex]

[tex]O=1[/tex]

Therefore, the length of the side opposite angle θ is 1.

Now we have the lengths of the three sides of the right triangle, we can find the other trigonometric function of angle θ.

[tex]\boxed{\begin{minipage}{8cm}\underline{Trigonometric functions}\\\\$\sf \sin\theta=\dfrac{O}{H}\quad\cos\theta=\dfrac{A}{H}\quad\tan\theta=\dfrac{O}{A}$\\\\\\$\sf\csc\theta=\dfrac{H}{O}\quad\sec\theta=\dfrac{H}{A}\quad\cot\theta=\dfrac{A}{O}$\\\\\\where:\\\phantom{ww}$\bullet$ $\theta$ is the angle.\\\phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle.\\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle.\\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse.\\\end{minipage}}[/tex]

Given values:

Substitute these values into the six trigonometric functions:

[tex]\sin\theta=\dfrac{O}{H}=\dfrac{1}{2}[/tex]

[tex]\cos\theta=\dfrac{A}{H}=\dfrac{\sqrt{3}}{2}[/tex]

[tex]\tan\theta=\dfrac{O}{A}=\dfrac{1}{\sqrt{3}}=\dfrac{\sqrt{3}}{3}[/tex]

[tex]\csc\theta=\dfrac{H}{O}=\dfrac{2}{1}=2[/tex]

[tex]\sec\theta=\dfrac{H}{A}=\dfrac{2}{\sqrt{3}}=\dfrac{2\sqrt{3}}{3}[/tex]

[tex]\cot\theta=\dfrac{A}{O}=\dfrac{\sqrt{3}}{1}=\sqrt{3}[/tex]

Answer:

y ≤ - [tex]\frac{2}{7}[/tex] x

Step-by-step explanation:

the blue region indicates where the solutions lie

that is below the given line

the equation of the line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (0, 0) and (x₂, y₂ ) = (- 7, 2) ← 2 points on the line

m = [tex]\frac{2-0}{-7-0}[/tex] = [tex]\frac{2}{-7}[/tex] = - [tex]\frac{2}{7}[/tex]

the line crosses the y- axis at (0, 0 ) ⇒ c = 0

y = - [tex]\frac{2}{7}[/tex] x ← equation of line

the line is solid thus the region below is denoted by y ≤ , that is

y ≤ - [tex]\frac{2}{7}[/tex] x

This acquisition of used buses reflects Sheffield Transportation's commitment to providing reliable and cost-effective transportation options for their customers.

In recent years, Sheffield Transportation has acquired three used buses for their transportation fleet. The decision to purchase these buses was likely made to meet the growing demand for transportation services in the area, while also keeping costs down by opting for used vehicles instead of new ones.

It is important to note that when purchasing used buses, Sheffield Transportation would have had to ensure that the vehicles were in good condition and met safety standards before putting them into service. Overall, this acquisition of used buses reflects Sheffield Transportation's commitment to providing reliable and cost-effective transportation options for their customers.

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The expression that is equivalent to [tex]49^{\frac{3}{2}}[/tex] is given as follows:

D. 343.

The power of a power rule is used when a single base is elevated to multiple exponents.

Then the simplified expression is obtained keeping the base, while the exponents are multiplied.

The expression for this problem is given as follows:

[tex]49^{\frac{3}{2}}[/tex]

49 is the square of 7, hence:

49 = 7².

Then the expression can be simplified as follows:

[tex]7^2^{\frac{3}{2}}[/tex]

The multiplication of the exponents is of:

2 x 3/2 = 3.

(when we apply the power of power rule, we keep the base and multiply the exponents)

Hence:

7³ = 343.

The problem is given by the image presented at the end of the answer.

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Answer:

(-1, 1.5).

Step-by-step explanation:

To find the midpoint of line segment AB with endpoints A(2, 5) and B(-4, -2), you can use the midpoint formula. The midpoint formula states that the midpoint (M) of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by:

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Let's calculate the midpoint using this formula:

x-coordinate of midpoint:

x = (2 + (-4)) / 2 = -2 / 2 = -1

y-coordinate of midpoint:

y = (5 + (-2)) / 2 = 3 / 2 = 1.5

Therefore, the midpoint of line AB is M(-1, 1.5).

Answer:

[tex]\sin\theta=\dfrac{\sqrt{5}}{5}[/tex]

[tex]\cos\theta=\dfrac{2\sqrt{5}}{5}[/tex]

[tex]\tan\theta=\dfrac{1}{2}[/tex]

[tex]\csc\theta=\sqrt{5}[/tex]

[tex]\sec\theta=\dfrac{\sqrt{5}}{2}[/tex]

Step-by-step explanation:

The cotangent ratio is the reciprocal of the tangent ratio.

[tex]\cot \theta = \dfrac{1}{\tan \theta}[/tex]

Therefore, if cot θ = 2 then:

[tex]\dfrac{1}{\tan \theta}=2[/tex]

[tex]\tan \theta=\dfrac{1}{2}[/tex]

The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle:

[tex]\tan\theta = \sf \dfrac{opposite}{adjacent}[/tex]

Therefore, the length of the side opposite angle θ is 1 and the length of the side adjacent angle θ is 2.

[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]

We can use Pythagoras Theorem to calculate the length of the hypotenuse, H:

[tex]1^2+2^2=H^2[/tex]

[tex]1+4=H^2[/tex]

[tex]H^2=5[/tex]

[tex]H=\sqrt{5}[/tex]

Therefore, the length of the hypotenuse is √5.

Now we have the lengths of the three sides of the right triangle, we can find the other trigonometric function of angle θ.

[tex]\boxed{\begin{minipage}{8cm}\underline{Trigonometric functions}\\\\$\sf \sin\theta=\dfrac{O}{H}\quad\cos\theta=\dfrac{A}{H}\quad\tan\theta=\dfrac{O}{A}$\\\\\\$\sf\csc\theta=\dfrac{H}{O}\quad\sec\theta=\dfrac{H}{A}\quad\cot\theta=\dfrac{A}{O}$\\\\\\where:\\\phantom{ww}$\bullet$ $\theta$ is the angle.\\\phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle.\\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle.\\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse.\\\end{minipage}}[/tex]

Given values:

Substitute these values into the six trigonometric functions:

[tex]\sin\theta=\dfrac{O}{H}=\dfrac{1}{\sqrt{5}}=\dfrac{\sqrt{5}}{5}[/tex]

[tex]\cos\theta=\dfrac{A}{H}=\dfrac{2}{\sqrt{5}}=\dfrac{2\sqrt{5}}{5}[/tex]

[tex]\tan\theta=\dfrac{O}{A}=\dfrac{1}{2}[/tex]

[tex]\csc\theta=\dfrac{H}{O}=\dfrac{\sqrt{5}}{1}=\sqrt{5}[/tex]

[tex]\sec\theta=\dfrac{H}{A}=\dfrac{\sqrt{5}}{2}[/tex]

[tex]\cot\theta=\dfrac{A}{O}=\dfrac{2}{1}=2[/tex]

Answer:

a. The theoretical probability of drawing a spade from a standard deck of cards is 13/52, which reduces to 1/4, or 0.25, or 25%.

b. The theoretical probability of drawing a face card (jack, queen, or king) from a standard deck of cards is 12/52, which reduces to 3/13, or approximately 0.231, or 23.1%.

c. The theoretical probability of drawing a non-face card from a standard deck of cards is 40/52, which reduces to 10/13, or approximately 0.769, or 76.9%.

d. The theoretical probability of drawing a black jack from a standard deck of cards is 2/52, which reduces to 1/26, or approximately 0.038, or 3.8%.

e. The theoretical probability of drawing a red or black card from a standard deck of cards is 52/52, which reduces to 1/1, or 100%.

f. The theoretical probability of drawing a red face card from a standard deck of cards is 6/52, which reduces to 3/26, or approximately 0.115, or 11.5%.

The arc angle in the circle is as follows:

mPTR = 63 degrees.

The tangent intersection theorem states that If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of its intercepted arc.

Therefore,

m∠PSR = 1  /2 (mPTR - mPR)

mPTR = ?

mPR = 149 degrees

m∠PSR = 106

Therefore,

106 = 1 / 2 (x - 149)

106 = 0.5x - 74.5

106 - 74.5 = 0.5x

31.5 = 0.5x

divide both sides by 0.5

x = 31.5 / 0.5

x = 63 degrees

Therefore,

mPTR = 63 degrees

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[tex]6\left[\cos\left( \frac{11\pi }{6} \right) +i\sin\left( \frac{11\pi }{6} \right) \right] \\\\\\ 6\left[\cfrac{\sqrt{3}}{2}~~,~-\cfrac{1}{2}i \right]\implies (3\sqrt{3}~~,~-3i) ~~ \approx ~~ \stackrel{ \textit{\LARGE S} }{(5.2~~,~-3i)}[/tex]

Answer:

The mode of this set of numbers is 12. If that is what you are looking for.

It will take them approximately 2.9 hours to finish the puzzle together. Therefore, options B, C are the correct answers.

Given that, Maria and Nemzet are about to try and complete a jigsaw puzzle.

Together attempting this puzzle alone, it would take Maria 7 hours to complete it and it would take Nemzet 5 hours

Here, 1/7 + 1/5 = 1/t

(5+7)/35 = 1/t

12/35 = 1/t

t=35/12

t=2.91

t=2.9 hours

Therefore, options B, C are the correct answers.

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