Given the function f(x) = x3 - 2x2 - 19x + 20, the zeros of the function are

Answer/Step-by-step explanation:

5. ✔️Exterior angle = angle outside the triangle = W

✔️Remote interior angle of the triangle to the exterior angle W = opposite angles to angle W which are X and Y

✔️m<X + m<Y = m<W (exterior angle theorem of a triangle)

✔️m<W + m<Z = 180° (linear pair/angles on a straight line)

6. m<6 = 115°

m<5 = 120°

✔️m<2 = 180° - m<5° (linear pair/angles on a straight line)

m<2 = 180° - 120°

m<2 = 60°

✔️m<3 = 180° - m<6° (linear pair/angles on a straight line)

m<3 = 180° - 115°

m<3 = 65°

✔️m<1 = 180° - (m<2 + m<3) (sum of triangle theorem)

m<1 = 180° - (60° + 65°)

m<1 = 55°

✔️m<4 = 180° - m<1 (linear pair/angles on a straight line)

m<4 = 180° - 55°

m<4 = 125°

Answer:

The depreciation tax shield for this project in year 4 is $178.24.

Explanation:

To calculate the depreciation tax shield for this project in year 4, we need to first determine the depreciation expense for year 4 using the MACRS three-year class life category and the double-declining balance (DDB) method.

From Table 12.8 on page 415 of the textbook, we can see that the depreciation rate for year 1 is 33.33%, for year 2 it is 44.45%, for year 3 it is 14.81%, and for year 4 it is 7.41%.

Using the DDB method, we can calculate the depreciation expense for each year as follows:

Year 1: Depreciation expense = $14,000 x 33.33% = $4,667

Year 2: Depreciation expense = ($14,000 - $4,667) x 44.45% = $3,554

Year 3: Depreciation expense = ($14,000 - $4,667 - $3,554) x 14.81% = $830

Year 4: Depreciation expense = ($14,000 - $4,667 - $3,554 - $830) x 7.41% = $557

The total depreciation expense over the four years is the sum of the individual year's depreciation expenses, which is:

$4,667 + $3,554 + $830 + $557 = $9,608

Now, we can calculate the depreciation tax shield in year 4. The depreciation tax shield is the amount of the depreciation expense that reduces the firm's taxable income, multiplied by the tax rate. In year 4, the depreciation tax shield is:

Depreciation tax shield = Depreciation expense in year 4 x Tax rate

Depreciation tax shield = $557 x 32% = $178.24

Therefore, the depreciation tax shield for this project in year 4 is $178.24.

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Answer:

The sample size will be "\(102.5494 \sim 102\)". A further explanation is given below.

Step-by-step explanation:

Let the sample size will be:

= n

The given values are:

Standard deviation,

= 1.4

ME,

= 0.3

As we know,

⇒ \(n=(Z \frac{a}{2}\times \frac{SD}{E})^2\)

On putting the estimated values, we get

⇒    \(=(2.17\times \frac{1.4}{0 .3})^2\)

⇒    \(=(\frac{3.038}{0.3})^2\)

⇒    \(=102.5494 \sim 102\)

Answer:

a)  963g

b) 17

Step-by-step explanation:

i) Now we must first find the common difference of the AP

d = 7.8 -6 = 1.8

Now the sum of the AP gives the total amount of chemical used for 30 experiments

Sn = n/2 [2a + (n-1)d]

n= 30, a = 6, d= 1.8

Sn = 30/2[2(6) + (30-1) (1.8)]

Sn = 15[12 + 52.2]

Sn = 963g

ii)For the G.P the common ration is (r)= 7.8/6 = 1.3

Since r>1, sum of GP = a (r^n -1)/r-1

If sum =  1800g

1800= 6(1.3^n -1)/1.3 - 1

1800 = 6(1.3^n -1)/0.3

1800 * 0.3  = 6(1.3^n -1)

540 = 6(1.3^n -1)

540/6 = (1.3^n -1)

90 = 1.3^n -1

90 + 1 = 1.3^n

1.3^n = 91

log 1.3^n = log 91

nlog1.3 = log 91

n = log91/log 1.3

n = 1.959/0.114

n = 17

Answer:

a)  963g

b) 17

Step-by-step explanation:

i) Now we must first find the common difference of the AP

d = 7.8 -6 = 1.8

Now the sum of the AP gives the total amount of chemical used for 30 experiments

Sn = n/2 [2a + (n-1)d]

n= 30, a = 6, d= 1.8

Sn = 30/2[2(6) + (30-1) (1.8)]

Sn = 15[12 + 52.2]

Sn = 963g

ii)For the G.P the common ration is (r)= 7.8/6 = 1.3

Since r>1, sum of GP = a (r^n -1)/r-1

If sum =  1800g

1800= 6(1.3^n -1)/1.3 - 1

1800 = 6(1.3^n -1)/0.3

1800 * 0.3  = 6(1.3^n -1)

540 = 6(1.3^n -1)

540/6 = (1.3^n -1)

90 = 1.3^n -1

90 + 1 = 1.3^n

1.3^n = 91

log 1.3^n = log 91

nlog1.3 = log 91

n = log91/log 1.3

n = 1.959/0.114

n = 17

Answer: 75⁰°

Answer: 75⁰°Step-by-step explanation:

Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:

Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180

Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify

Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180

Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides

Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides7x = 175

Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides7x = 175divide each side by 7

Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides7x = 175divide each side by 7x = 25

Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides7x = 175divide each side by 7x = 25plug 25 in for x to find the angle measure

Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides7x = 175divide each side by 7x = 25plug 25 in for x to find the angle measure3(25) = 75

Answer:

Step-by-step explanation:

Given

Corresponding angles

Find the value of ∠D

Find the value of x

Answer:

median is 2

mean is 6.333

Step-by-step explanation:

arranging values in ascending or descending order you pick the middle for odd numbers which is 2. mean is sum of the total data divided by the number of items in the set which is 2+8+4+10+6+8 divide by 6 which is 6.333

Answer:

\(48x^{13}\ ft^2\)

Step-by-step explanation:

Step 1:  Determine the area of the rectangle

To find the area of the rectangle we just need to do Length * Width which will give us the area in square feet (it really depends on the units that they are giving the L and W in).

\((8x^6) * (6x^7)\)

\(((8*6)(x^6*x^7))\)

\((48*x^{6+7})\)

\(48x^{13}\)

Answer:  \(48x^{13}\ ft^2\)

7-digit:9999999

9-digit: 1000000

19, 15.5

20, 25

21, 1/16

22, 4

23, 11

24, 8

25, 19.5

26, 30

27, 4

28, 56

29, 6.5

30, 4

31, 21

32, 112

33, 3,200

here are they

Area of a triangle = 1/2 x base x height:

Area = 1/2 x 8 x 3x = 12x

Perimeter = sum of the 3 sides:

4x-3 + x +9 + 8 = 5x +14

Answer: C. 12x; 5x+14

Answer:

The point of intersection of both diagonals is (0,1)

Step-by-step explanation:

for the parallelogram, we have the diagonals GJ and HK

Firstly, we need to get the equation of the lines that join these two points

For GJ

the general equation is;

y = mx + b

m is slope and b is y-intercept

We have the points

(-2,3) and (2,-1)

The slope m is as follows;

m = (y2-y1)/(x2-x1)

m = (-1-3)/(2-(-2) = -4/4 = -1

So we have;

y = -x + b

To get b, we substitute any of the two points

Let us use (-2,3)

x = -2 and y = 3

so we have;

3 = -1(-2) + b

3 = 2 + b

b = 3-2 = 1

so we have the equator GJ as y = -x + 1

We repeat same for HK

(4,4) and (-4,-2)

m = (-2-4)/(-4-4) = -6/-8 = 3/4

so we have;

y = 3x/4 + b

use (4,4) to get b

4 = 3/4(4) + b

4 = 3 + b

b = 1

so we have the equate;

y = 3x/4 + 1

multiply through by 4

4y = 3x + 4

To get the point of intersection, we have to solve the equations of the diagonals simultaneously

the equations are;

4y = 3x + 4

y = -x + 1

Multiply equation ii by 4 and i by 1

4y = 3x + 4

4y = -4x + 4

Subtract ii from i

0 = 7x

x = 0

To get y, use any of the equations;

y = -x + 1

y = 0 + 1

y = 1

So the point of intersection of both diagonals are;

(0,1)

Answer:

c = 3

Step-by-step explanation:

\( 32^{2c} = 8^{c + 7} \)

\( (2^5)^{2c} = (2^3)^{c + 7} \)

\( 2^{10c} = 2^{3c + 21} \)

\( 10c = 3c + 21 \)

\( 7c = 21 \)

\( c = 3 \)

Answer:

(10, 5π/3)

Step-by-step explanation:

To convert rectangular coordinates (5, -5√3) to polar form, we can use the following formulas:

r = √(x^2 + y^2)

θ = arctan(y/x)

Substituting the given values, we get:

r = √(5^2 + (-5√3)^2) = √(25 + 75) = √100 = 10

θ = arctan((-5√3)/5) = arctan(-√3) = -π/3

Note that the value of θ is in the fourth quadrant, which corresponds to a negative angle. However, we need to express the angle θ in the range 0 ≤ θ < 2π. To do this, we can add 2π to the angle if it is negative:

θ = -π/3 + 2π = (5π/3)

Therefore, the rectangular coordinates (5, -5√3) in polar form are (10, 5π/3).

Answer:

The dimensions of the lid are 8mm by 19mm.

w = 8

l = 19

Step-by-step explanation:

\(l=w+11\\l\times w=152\)

where l is length and w is width. This can be solved as a system of equations.

\(l\times w=152\\(w+11)\times w=152\\w(w+11)=152\\w^2+11w=152\\w^2+11w-152=0\)

At this point, it gets a little tough. I might be unnecessarily overcomplicating things, but this is the only way I see to solve the problem.

=================== Skip down below if you don't care about factoring

You need to factor the newly created trinomial.

\(ax^2+bx+c\)

With a trinomial in this form, you need to find 2 numbers that add together to make b and multiply together to make ac.

Here, we need 2 numbers that add to 11 and multiply to -152. First, factor 152:

1, 152

2, 76

4, 38

8, 19

Then the reverse of all of those is true too, of course:

19, 8

38, 4

etc

In our case, we're looking for -152, so one of our factors will be negative. We're also looking for factors that add up to 11. Looking at these factors, you can see that 19 - 8 = 11, so our factors are 19 and -8.

Finally, you can use those to factor our trinomial. Split up the middle number (11w) into two:

\(w^2+11w-152=0\\w^2-8w+19w-152=0\)

And now, you can factor by grouping:

\(w^2-8w+19w-152=0\\w(w-8)+19(w-8)=0\\(w+19)(w-8)=0\)

===================

Now that the number is factored, you can finally find w:

\((w+19)(w-8)=0\)

Here, you can see that the equation will be true when w = -19 or w = 8. Those are our solutions, but we can't have a negative distance, so it's just

\(w=8\)

Going all the way back to the top, now you can use the width to find the length.

\(l=w+11\\l=8+11\\l=19\)

That one was much easier.

The dimensions of the lid are 8mm by 19mm.

Finally, check that with both of the original equations to make sure it's correct.

\(l=w+11\\19=8+11\\19=19\\\\l\times w=152\\19\times8=152\\152=152\)

Answer:

Step-by-step explanation:

To compare these two values, we first need to convert the mixed number 7 3/8 to an improper fraction. To do so, we multiply the whole number (7) by the denominator of the fraction (8), then add the numerator (3), and put the result over the denominator:

7 3/8 = (7 x 8 + 3) / 8 = 59/8

Now we can rewrite the inequality as:

(59/8) × (4/5) < 59/8

To simplify the left-hand side of the inequality, we multiply the numerators and denominators:

(59/8) × (4/5) = (59 × 4) / (8 × 5) = 236/40 = 59/10

So the inequality becomes:

59/10 < 59/8

To compare these fractions, we need to find a common denominator. The least common multiple of 8 and 10 is 40, so we can convert both fractions to have a denominator of 40:

59/10 = (59 x 4) / (10 x 4) = 236/40

59/8 = (59 x 5) / (8 x 5) = 295/40

Now we can see that 236/40 < 295/40, which means that:

59/10 < 59/8

Therefore, the inequality 7 3/8 × 4/5 < 7 3/8 is true.

the two planes will be at the same altitude of approximately 5675.48 feet after 49.45 seconds.

A plane is a flat two-dimensional surface that extends infinitely in all directions. In geometry, a plane is defined as a surface that is completely flat and has no thickness. It is often represented as a coordinate system with two perpendicular axes, usually labeled as the x-axis and y-axis.

Assuming that the two planes maintain their rates of ascent, we can use the following equations to determine when the planes will be at the same altitude:

altitude of plane A = 4000 + 33.5t

altitude of plane B = 3146 + 50.75t

where t represents time in seconds.

To find the time at which the two planes will be at the same altitude, we can set the two equations equal to each other and solve for t:

4000 + 33.5t = 3146 + 50.75tSubtracting 3146 from both sides, we get:

854 + 33.5t = 50.75t

Subtracting 33.5t from both sides, we get:

854 = 17.25t

Dividing both sides by 17.25, we get:

t = 49.45 seconds

Therefore, it will take approximately 49.45 seconds for the two planes to be at the same altitude. To find the altitude at which they will meet, we can substitute this value of t into either equation. Using the equation for plane A, we get:

altitude of plane A = 4000 + 33.5(49.45) = 5645.08 feet

Using the equation for plane B, we get:

altitude of plane B = 3146 + 50.75(49.45) = 5675.48 feet

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Answer:

The length = 120 feet

Step-by-step explanation:

Givens

Let the length  = x

Let the width = 3/4  x

Perimeter = 420 feet

Formula

Perimeter = 2*L + 2*w

Solution

2*x + 2(3/4) x = 420

2x + 1.5x = 420

3.5 x = 420

x = 120

The statement about probability that is correct would be that the probability of landing on yellow is less than the probability of landing on blue. That is option B.

Probability is defined as the total number of possible outcome of an event.

The repeated colour which are numbered from 1 to 8 are as follows:

Therefore, the probability of landing on yellow is less than the probability of landing on blue because 1/4 is less than 3/8.

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Answer: B: The probability of landing on yellow is less than the probability of landing on blue.

Step-by-step explanation:

Answer:

37

Step-by-step explanation:

The expression (sinx + cosx)^2 + (sinx - cosx)^2 simplifies to

To simplify the expression (sinx + cosx)^2 + (sinx - cosx)^2 using trigonometric identities, we can expand and simplify the expression.

Expanding the squared terms

(sin^2x + 2sinxcosx + cos^2x) + (sin^2x - 2sinxcosx + cos^2x)

Using the trigonometric identity sin^2x + cos^2x = 1, we can simplify further:

(1 + 2sinxcosx + 1) + (1 - 2sinxcosx + 1)

Simplifying the expression, we have:

2 + 2sinxcosx + 2

Combining like terms, we get:

4 + 2sinxcosx

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The correct answer is $131250

Explanation:

Talia's earnings during one year are equal to her base yearly salary ($65000) added to her commission. Now, to know the commission it is necessary to find the 2.65% of her total sales ($2.5 million)

1. Divide the total sales or 2,500,000 by 100 considering this represents the total or 100%

2,500,000 ÷ 100 = 25000

2. Multiply this number by the specific percentage (2.65)

25000 x 2.65 = 66250

Now add the commission and the yearly salary

65000 + 66250 = 131250

This means the total earnings of Talia were $131250

2.65% = 0.0265

0.0265 x 2,500,000 = 66250

66250 + 65000 = 131,250

The equations that has the solution x = all real numbers is 5(3x)+ 7x = 3x + 15 - x and 5(3x) + 7x = 3x + 10 - x,

Algebraically speaking, an equation is a statement that shows the equality of two mathematical expressions. For instance, the two equations 3x + 5 and 14, which are separated by the 'equal' sign, make up the equation 3x + 5 = 14.

A formula that expresses the connection between two expressions on each side of a sign. Typically, it has a single variable and an equal sign.

Here given equations,

5(3x) + 6x = 3x + 15 + 2x

15x + 6x = 3x + 15 + 2x

15 x + 6x - 3x -2x = 15

16 x = 15

x = 15/16

5(3x).+ 7x = 3x + 15 - x

15 x + 7x = 3x + 15 - x

15 x + 7x - 3x + x = 15

22x - 3x + x = 15

20 x = 15

x = 15/20

x = 3/4

5(3x) + 7x = 3x + 10 - x

15x + 7x - 3x + x = 10

20x = 10

x = 10/20

x = 1/2

5(3x) + 7x = x + 15 - 3x

15x +7x - x + 3x = 15

24x = 15

x = 15/24

x = 5/8

Therefore the equations that has the solution x = all real numbers are :

5(3x).+ 7x = 3x + 15 - x and 5(3x) + 7x = 3x + 10 - x .

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Answer:

Step-by-step explanation:

a) To determine the endogenous variables, we need to identify the variables that are determined within the model equation. In the given model, the endogenous variable is Y (output or national income).

b) Let's find Y, C, and T step-by-step:

Start with the equation Y = C + I0 + g0.

Substitute C from the equation C = a + b(y - T).

Y = (a + b(y - T)) + I0 + g0.

Substitute T from the equation T = d + tY.

Y = (a + b(y - (d + tY))) + I0 + g0.

Expand the equation:

Y = a + by - bd - btY + I0 + g0.

Rearrange the equation to isolate Y:

Y + btY = a + by - bd + I0 + g0.

Y(1 + bt) = a + by - bd + I0 + g0.

Y = (a + by - bd + I0 + g0) / (1 + bt).

Now, Y is expressed in terms of the exogenous variables a, b, d, I0, g0, and the endogenous variable Y itself, along with the parameter t.

To find C and T, we can substitute the obtained Y value back into the respective equations:

Substitute Y into the equation C = a + b(y - T):

C = a + b(y - T) = a + b(y - (d + tY)) = a + by - bd - btY.

C = a + by - bd - bt[(a + by - bd + I0 + g0) / (1 + bt)].

Now, C is expressed in terms of the exogenous variables a, b, d, I0, g0, and the endogenous variable Y, along with the parameter t.

Substitute Y into the equation T = d + tY:

T = d + tY = d + t[(a + by - bd + I0 + g0) / (1 + bt)].

Now, T is expressed in terms of the exogenous variables d, t, and the endogenous variable Y, along with the parameters a, b, I0, and g0.

It's important to note that in the given model, there is only one endogenous variable, Y (national income/output). C and T are determined based on the values of Y and the exogenous variables.

Answer:

1)  695.3 m²

2)  8 ft

3)  172.0 in²

Step-by-step explanation:

To find the area of a regular polygon, we can use the following formula:

\(\boxed{\begin{minipage}{5.5cm}\underline{Area of a regular polygon}\\\\$A=\dfrac{s^2n}{4 \tan\left(\dfrac{180^{\circ}}{n}\right)}$\\\\\\where:\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\ \phantom{ww}$\bullet$ $s$ is the side length.\\\end{minipage}}\)

Given the polygon is an octagon, n = 8.

Given each side measures 12 m, s = 12.

Substitute the values of n and s into the formula for area and solve for A:

\(\implies A=\dfrac{(12)^2 \cdot 8}{4 \tan\left(\dfrac{180^{\circ}}{8}\right)}\)

\(\implies A=\dfrac{144 \cdot 8}{4 \tan\left(22.5^{\circ}\right)}\)

\(\implies A=\dfrac{1152}{4 \tan\left(22.5^{\circ}\right)}\)

\(\implies A=\dfrac{288}{\tan\left(22.5^{\circ}\right)}\)

\(\implies A=695.29350...\)

Therefore, the area of a regular octagon with side length 12 m is 695.3 m² rounded to the nearest tenth.

\(\hrulefill\)

The sum of an interior angle of a regular polygon and its corresponding exterior angle is always 180°.

If the exterior angle of a polygon measures 40°, then its interior angle measures 140°.

To determine the number of sides of the regular polygon given its interior angle, we can use this formula, where n is the number of sides:

\(\boxed{\textsf{Interior angle of a regular polygon} = \dfrac{180^{\circ}(n-2)}{n}}\)

Therefore:

\(\implies 140^{\circ}=\dfrac{180^{\circ}(n-2)}{n}\)

\(\implies 140^{\circ}n=180^{\circ}n - 360^{\circ}\)

\(\implies 40^{\circ}n=360^{\circ}\)

\(\implies n=\dfrac{360^{\circ}}{40^{\circ}}\)

\(\implies n=9\)

Therefore, the regular polygon has 9 sides.

To determine the length of each side, divide the given perimeter by the number of sides:

\(\implies \sf Side\;length=\dfrac{Perimeter}{\textsf{$n$}}\)

\(\implies \sf Side \;length=\dfrac{72}{9}\)

\(\implies \sf Side \;length=8\;ft\)

Therefore, the length of each side of the regular polygon is 8 ft.

\(\hrulefill\)

The area of a regular polygon can be calculated using the following formula:

\(\boxed{\begin{minipage}{5.5cm}\underline{Area of a regular polygon}\\\\$A=\dfrac{s^2n}{4 \tan\left(\dfrac{180^{\circ}}{n}\right)}$\\\\\\where:\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\ \phantom{ww}$\bullet$ $s$ is the side length.\\\end{minipage}}\)

A regular pentagon has 5 sides, so n = 5.

If its perimeter is 50 inches, then the length of one side is 10 inches, so s = 10.

Substitute the values of s and n into the formula and solve for A:

\(\implies A=\dfrac{(10)^2 \cdot 5}{4 \tan\left(\dfrac{180^{\circ}}{5}\right)}\)

\(\implies A=\dfrac{100 \cdot 5}{4 \tan\left(36^{\circ}\right)}\)

\(\implies A=\dfrac{500}{4 \tan\left(36^{\circ}\right)}\)

\(\implies A=\dfrac{125}{\tan\left(36^{\circ}\right)}\)

\(\implies A=172.047740...\)

Therefore, the area of a regular pentagon with perimeter 50 inches is 172.0 in² rounded to the nearest tenth.

Answer:

1.695.29 m^2

2.8 feet

3. 172.0477 in^2

Step-by-step explanation:

1. The area of a regular octagon can be found using the formula:

\(\boxed{\bold{Area = 2a^2(1 + \sqrt{2})}}\)

where a is the length of one side of the octagon.

In this case, a = 12 m, so the area is:

\(\bold{Area = 2(12 m)^2(1 + \sqrt{2}) = 288m^2(1 + \sqrt2)=695.29 m^2}\)

Therefore, the Area of a regular octagon is 695.29 m^2

2.

The formula for the exterior angle of a regular polygon is:

\(\boxed{\bold{Exterior \:angle = \frac{360^o}{n}}}\)

where n is the number of sides in the polygon.

In this case, the exterior angle is 40°, so we can set up the following equation:

\(\bold{40^o=\frac{ 360^0 }{n}}\)

\(n=\frac{360}{40}=9\)

Therefore, the polygon has n=9 sides.

Perimeter=72ft.

We have

\(\boxed{\bold{Perimeter = n*s}}\)

where n is the number of sides in the polygon and s is the length of one side.

Substituting Value.

72 feet = 9*s

\(\bold{s =\frac{ 72 \:feet }{ 9}}\)

s = 8 feet

Therefore, the length of each side of the polygon is 8 feet.

3.

Solution:

A regular pentagon has five sides of equal length. If the perimeter of the pentagon is 50 in, then each side has a length = \(\bold{\frac{perimeter}{n}=\frac{50}{5 }= 10 in.}\)

The area of a regular pentagon can be found using the following formula:

\(\boxed{\bold{Area = \frac{1}{4}\sqrt{5(5+2\sqrt{5})} *s^2}}\)

where s is the length of one side of the Pentagon.

In this case, s = 10 in, so the area is:

\(\bold{Area= \frac{1}{4}\sqrt{5(5+2\sqrt{5})} *10^2=172.0477 in^2}\)

Drawing: Attachment

The statements that are true regarding the expression with the exponents -1, -3, and 1 are;

An exponent is a quantity or power to which a value, expression or number is to be raised.

The specified expression can be presented as follows;

8⁻¹ × 8⁻³ × 8

Therefore; 8⁻¹ × 8⁻³ × 8 = 8⁽⁻¹ ⁺ ⁻³ ⁺ ¹⁾ = 8⁻³ = 1/8³

1/8³ = 1/512

Similarly, we get; 8⁷ × 8⁻¹⁰ = 8⁽⁷ ⁻ ¹⁰⁾ = 8⁻³ = 1/8³

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Using equations, the time for the main engine 14.4 seconds and 21.6 seconds for the auxiliary engine

Let's call the time each engine takes to consume its allotted fuel supply as "t₁" for the main engine and "t₂" for the auxiliary engine.

From the first piece of information, we know:

t₁ = (2/3)t₂

From the second piece of information, we know that the combined fuel consumption time for both engines is 36 seconds:

t₁ + t₂ = 36

Now we can substitute the first equation into the second:

t₁ + (2/3)t₂ = 36

Combining like terms:

t₂ = 21.6

Finally, substituting t₂ back into the first equation:

t₁ = (2/3)(21.6) = 14.4

So the main engine alone could be fired for 14.4 seconds and the auxiliary engine alone could be fired for 21.6 seconds.

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