Find the slope of the line passing through the points (-4, -9) and (5, -9).

The total cost of the meal if the meal costs $50 is $60

From the question, we have the following parameters that can be used in our computation:

Tip percent = 20%

From the question, we can use any value as the cost of the meal

So, we can use the following

Meal cost = $50

So, we have

Tip amount = Tip percent * Meal cost

This gives

Tip amount = 20% * 50

Tip amount = 10

Next, we have

Total meal cost = Tip amount + Meal cost

This gives

Total meal cost = 10 + 50

Evaluate

Total meal cost = 60

Hence, the total cost of the meal is $60

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The center and radius of the circle is (5, -3), and 5 respectively.

The standard form equation of a circle with center (h, k) and radius r is:

(x - h)² + (y - k)² = r²

Given the equation of the circle in the question:

( x - 5 )² + ( y + 3 )² = 25

To identify the center and the radius.

Comparing this equation ( x - 5 )² + ( y + 3 )² = 25 to the standard form of a circle equation, which is (x - h)² + (y - k)² = r², we can easily identify the center and radius of the circle.

( x - 5 )² + ( y + 3 )²

(x - h)² + (y - k)² = r²

The center of the circle is given by the values (h, k), which correspond to the opposite signs of the terms inside the parentheses.

In this case, the center of the circle is (5, -3), as we have (x - 5)² and (y + 3)².

The radius of the circle is the square root of the value on the right side of the equation.

In this case, the radius is √25, which simplifies to 5.

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Answer: 10:22

Step-by-step explanation:

9:55 + :05 = 10:00

10:00+ :22 = 10:22

The current balance for Jeff's savings account is $317.84.

To find the current balance for Jeff's savings account, we need to calculate the net effect of the deposits, withdrawals, and interest on his initial balance.

Initial balance: $396.80

Deposits: 3 × $15 = $45

Withdrawals: $125

Interest: $1.04

Adding the deposits and interest to the initial balance:

$396.80 + $45 + $1.04 = $442.84

Then, subtracting the withdrawal:

$442.84 - $125 = $317.84

Therefore, the current balance for Jeff's savings account is $317.84.

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The row of tiles would be 24 1/2 inches long.

B:The  row of 10 tiles would be 35 inches long.

A 1. To be able to know the length of the row of tiles, one need to multiply the length of one tile by the number of tiles in the said  row.

So, all tile measures 3 1/2 inches on each side, Hence  the length of one tile is 3 1/2 inches.

Based on the fact that there are 7 tiles in a row, we have to multiply the length of one tile by 7:

Length of the row = 3 1/2 inches/tile x 7 tiles

                              = 24 1/2 inches

So, the row of tiles is 24 1/2 inches long.

B 2. Also, to find the length of a row of 10 tiles, we need to multiply the length of one tile by 10.

Note that:

Length of one tile = 3 1/2 inches

Number of tiles in the row = 10

Hence:

Length of the row = 3 1/2 inches/tile x 10 tiles

                               = 35 inches

So, the row of 10 tiles is 35 inches long.

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Using the conversion factor, we can convert the liters into the number of gallons of 1. 057 gallons.

A conversion factor is a mathematical ratio that is used to convert one unit of measurement to another. It is a multiplier or divisor that relates two different units of measurement for the same quantity.

To convert 4 liters into gallons using the conversion factor 1 gallon = 3.785 liters, you divide the given value (in liters) by the conversion factor.

This means that the number of gallons in 4 liters would be:

= 4 liters ÷ 3. 785 liters/gallon

= 1. 057 gallons

Therefore, 4 liters is approximately equal to 1. 057 gallons.

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John William is entitled to purchase 15,000 shares before the offering is made to the public.

a. To calculate the percentage of the total vote John William will control, we need to find the proportion of shares he owns out of the total shares represented at the meeting.

John William owns 6,000 shares out of the 20,000 shares offered so far. Therefore, the proportion of shares he owns is 6,000/20,000, which simplifies to 3/10 or 0.3.

To convert this proportion to a percentage, we multiply by 100:

Percentage of the total vote = 0.3 * 100 = 30%

Thus, John William will control 30% of the total vote at the.         shareholder meeting.    

b. Preemptive rights allow existing shareholders to purchase new shares before they are offered to the public. To determine the number of shares John William is entitled to purchase, we need to calculate the proportion of his current ownership.

John William owns 6,000 shares out of the total shares offered so far, which is 20,000. Therefore, his ownership proportion is 6,000/20,000, or 3/10.

If Peixotto Media plans to issue 50,000 new shares, John William's entitlement would be:

Entitlement = (3/10) * 50,000 = 15,000 shares.

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The circle equation x² + 8x + y² -10y - 32 = 0​ can be graphed using (x + 4)² + (y - 5)² = 73

From the question, we have the following parameters that can be used in our computation:

x² + 8x + y² -10y - 32 = 0​

Add 32 to both sides of the equation

This gives

x² + 8x + y² -10y = 32

Group the terms in two's

So, we have

(x² + 8x) + (y² -10y) = 32

When we complete the square on each group, we have

(x + 4)² + (y - 5)² = 16 + 25 + 32

Evaluate the like terms

(x + 4)² + (y - 5)² = 73

Hence, the circle equation can be graphed using (x + 4)² + (y - 5)² = 73

See attachment for the graph

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If Each card corresponds to a sample point with a 1/52 probability then the  sample points in the event an ace is selected is S = {ace of clubs, ace of diamonds, ace of hearts, ace of spades}

In a standard deck of 52 playing cards, there are four aces.

Each ace corresponds to a different suit: clubs, diamonds, hearts, and spades.

So, when we talk about the event of selecting an ace, we are referring to the act of choosing one of these four specific cards from the deck.

The sample points in the event "an ace is selected" are the individual cards that match this criteria.

Therefore, the sample points in this event are:

S = {ace of clubs, ace of diamonds, ace of hearts, ace of spades}

These four cards are the possible outcomes when we randomly select a single card from the deck, focusing only on the event of selecting an ace.

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The radius of the cylinder, considering it's lateral surface area, is given as follows:

r = 31.5 cm.

The lateral surface area of a cylinder of radius r and height h is given by the equation presented as follows:

Sl = 2πrh.

The parameters for this problem are given as follows:

Sl = 693π, h = 11.

Hence the radius of the cylinder, considering it's lateral surface area, is given as follows:

r = Sl/2πh

r = 693/22 (simplifying π in the numerator and denominator)

r = 31.5 cm.

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Option C (15 17 19) has a similar mean absolute deviation to the data set in the given dot plot.

The informational collection in the given speck plot has a mean of roughly 55 and a moderately little scope of values, with most of the information bunched around the mean. An informational index with a comparable mean outright deviation would have a comparative degree of variety in the information values.

Out of the given choices, the informational index that has a variety like the given informational collection is choice C: 15 17 19. This informational collection has a mean of roughly 17 and a little scope of values, with most of the information bunched around the mean.

The mean outright deviation of this informational collection is around 1.63, which is like the mean outright deviation of the given informational collection. Choice A (50 51 52 53 54 55 56 57 58 59 60 61) has a bigger scope of values and a bigger mean outright deviation contrasted with the given informational collection.

Choice B (4612) and Choice D (8 9 1011 and 55 57 59 61 63 65) have altogether unique mean qualities and scopes of values, and consequently their mean outright deviations are not quite the same as the given informational collection.

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Using DeMoivre’s theorem, the answer in regular form would be (5 + 5√3i)⁷ = -5000000 + 8660254.03i

The De Moivre's Theorem is used to simplify the computation of powers and roots of complex numbers and is used in together with polar form.

Convert the complex number to polar form. The polar form of a complex number is z = r(cos θ + isin θ),

r = |z|  magnitude of z

it becomes

r = √((5)² + (5√3)²) = 10

θ = arg(z) is the argument of z.

θ = atan2(b, a) = atan2(5√3, 5) = π/3

(5 + 5√3i) = 10 × (cos π/3 + i sin π/3)

De Moivre's theorem to raise the complex number to the 7th power

(5 + 5√3i)⁷

= 10⁷× (cos 7π/3 + i sin 7π/3)

= 10⁷ × (cos 2π/3 + i sin 2π/3)

Convert this back to rectangular form:

Real part = r cos θ = 10⁷× cos (2π/3) = -5000000

Imaginary part = r sin θ = 10⁷ × sin (2π/3) = 5000000√3 = 8660254.03i

∴  (5 + 5√3i)⁷ = -5000000 + 8660254.03i

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Answer:10^7 (1/2 - √3/2 i)

Step-by-step explanation:

To use DeMoivre's theorem, we first need to write the number in polar form. Let's find the magnitude and argument of the number:

Magnitude:

|5 + 5√(3i)| = √(5^2 + (5√3)^2) = √(25 + 75) = √100 = 10

Argument:

arg(5 + 5√(3i)) = tan^(-1)(√3) = π/3

So the number can be written in polar form as:

5 + 5√(3i) = 10(cos(π/3) + i sin(π/3))

Now we can use DeMoivre's theorem:

(5 + 5√(3i))^7 = 10^7 (cos(7π/3) + i sin(7π/3))

To simplify, we need to find the cosine and sine of 7π/3:

cos(7π/3) = cos(π/3) = 1/2

sin(7π/3) = -sin(π/3) = -√3/2

Explanation:

So the final answer in rectangular form is:

10^7 (1/2 - √3/2 i)

255 number of adults and 355 number of students bought ticket.

Here,  we have,

Let the number of adults bought tickets are x and the number of students that bought tickets is (x + 100).

Since it is given that 100 more students brought tickets than adults.

Now, each adult ticket costs $5 and each student's ticket costs $3.5 and the total collected value of tickets is $2517.5.

So, 5x + 3.5(x + 100) = 2517.5

⇒ 8.5x + 350 = 2517.5

⇒ 8.5x = 2167.5

⇒ x = 255

So, 255 number of adults and (255 + 100) = 355 number of students bought ticket. (Answer)

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Answer:

x+1

DOMAIN: x [tex]\neq[/tex] -5

Step-by-step explanation:

We can factor [tex]x^{2}[/tex]+6x+5 as (x+1)(x+5) since this expands to [tex]x^{2}[/tex]+5*x+1*x+5*1, which is [tex]x^{2}[/tex]+6x+5.

Now, we have:

[tex]\frac{(x+1)(x+5)}{(x+5)}[/tex]  

We can cancel out x+5, because this expression is the same as (x+1)*[tex]\frac{x+5}{x+5}[/tex], which is (x+1).

HOWEVER, the denominator cannot be equal to zero because that will make the expression undefined (you can't divide by zero). So, since x+5[tex]\neq[/tex]0, x[tex]\neq[/tex]-5.

The equation is verified, and we have shown that E_0 = cos⁴θ is equivalent to E_0 = E_0 (1/2 + cos2θ/2)².

We have,

To verify the equation

E_θ = E_0 cos^4θ = E_0 (1/2 + cos2θ/2)², we can expand the right-hand side and simplify it.

Starting with the right-hand side of the equation:

E_0 (1/2 + cos2θ/2)²

Expanding the square using the binomial expansion:

E_0 [(1/2)² + 2(1/2)(cos2θ/2) + (cos2θ/2)²]

Simplifying:

E_0 [1/4 + cos2θ + (cos2θ)²/4]

Combining the terms:

E_0 [1/4 + 4cos²θ/4 + (cos²θ)²/4]

Simplifying further:

E_0 [1/4 + 4cos²θ/4 + cos⁴θ/4]

Combining the terms again:

E_0 [1/4 + 4cos²θ/4 + cos⁴θ/4] = E_0 [1/4 + 4cos²θ/4 + cos⁴θ/4]

Simplifying the expression yields:

E_0 cos⁴θ = E_0 cos⁴θ

Therefore,

The equation is verified, and we have shown that E_0 = cos⁴θ is equivalent to E_0 = E_0 (1/2 + cos2θ/2)².

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value of the sides of the triangle are given as follows;

shorter leg = 5

longer leg = 12

hypotenuse = 13

Pythagoras Theorem states that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse.

If the short leg = x

the longer length = x+7

the hypotenuse = x+7+1 = x+8

Using Pythagorean theorem

(x+8)² = x² + (x+7)²

x² +16x +64 = x²+ x² + 14x +49

collecting like terms

x²-2x²+16x -14x+64 -49= 0

-x² + 22x +15 = 0

x² -2x -15 = 0

x² -5x +3x -15 = 0

(x² -5x) (3x -15) = 0

x(x-5) +3(x-5)

(x+3)(x-5) = 0

x= -3 or +5

Since length is discreet we choose the positive answer

therefore x = 5

shorter leg = 5

longer leg = 5+7 = 12

hypotenuse = 5+7+1 = 13

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Answer:

It is true that in algebraic expressions there could have all four arithmetic operations

a. Jason will receive the strongest signal from BTS-4 because he is located in the Voronoi cell of BTS-4.

b. The coordinates of BTS-4 are (−8,4).

a. Jason will receive the strongest signal from BTS-4 because he is located in the Voronoi cell of BTS-4. A Voronoi cell is a region of space that is closer to a given point than any other point. In this case, the given point is BTS-4. The Voronoi diagram is a partitioning of the plane into Voronoi cells, one for each point.

b. The edge connecting vertices P and Q has the equation y=4, which means that it is a horizontal line that intersects the y-axis at 4. The coordinates of BTS-2 are (−8,6), which means that it is located 8 units to the left of the origin and 6 units above the origin. Therefore, BTS-4 must be located 8 units to the left of the origin and 4 units above the origin, which gives it the coordinates (−8,4).

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Answer:

1) B = 88°, a = 13.1, b = 26.9

2) C = 73°, a = 20.9, b = 12.6

Step-by-step explanation:

To solve for the remaining sides and angles of the triangle, given two sides and an adjacent angle, use the Law of Sines formula:

[tex]\boxed{\begin{minipage}{7.6 cm}\underline{Law of Sines} \\\\$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$\\\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are the angles. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides opposite the angles.\\\end{minipage}}[/tex]

Given values:

As the interior angles of a triangle sum to 180°:

[tex]\implies A+B+C=180^{\circ}[/tex]

[tex]\implies B=180^{\circ}-A-C[/tex]

[tex]\implies B=180^{\circ}-29^{\circ}-63^{\circ}[/tex]

[tex]\implies B=88^{\circ}[/tex]

Substitute the values of A, B, C and c into the Law of Sines formula and solve for sides a and b:

[tex]\implies \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]

[tex]\implies \dfrac{a}{\sin 29^{\circ}}=\dfrac{b}{\sin 88^{\circ}}=\dfrac{24}{\sin 63^{\circ}}[/tex]

Solve for a:

[tex]\implies \dfrac{a}{\sin 29^{\circ}}=\dfrac{24}{\sin 63^{\circ}}[/tex]

[tex]\implies a=\dfrac{24\sin 29^{\circ}}{\sin 63^{\circ}}[/tex]

[tex]\implies a=13.0876493...[/tex]

[tex]\implies a=13.1[/tex]

Solve for b:

[tex]\implies \dfrac{b}{\sin 88^{\circ}}=\dfrac{24}{\sin 63^{\circ}}[/tex]

[tex]\implies b=\dfrac{24\sin 88^{\circ}}{\sin 63^{\circ}}[/tex]

[tex]\implies b=26.9194211...[/tex]

[tex]\implies b=26.9[/tex]

[tex]\hrulefill[/tex]

Given values:

As the interior angles of a triangle sum to 180°:

[tex]\implies A+B+C=180^{\circ}[/tex]

[tex]\implies C=180^{\circ}-A-B[/tex]

[tex]\implies C=180^{\circ}-72^{\circ}-35^{\circ}[/tex]

[tex]\implies C=73^{\circ}[/tex]

Substitute the values of A, B, C and c into the Law of Sines formula and solve for sides a and b:

[tex]\implies \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]

[tex]\implies \dfrac{a}{\sin 72^{\circ}}=\dfrac{b}{\sin 35^{\circ}}=\dfrac{21}{\sin 73^{\circ}}[/tex]

Solve for a:

[tex]\implies \dfrac{a}{\sin 72^{\circ}}=\dfrac{21}{\sin 73^{\circ}}[/tex]

[tex]\implies a=\dfrac{21\sin 72^{\circ}}{\sin 73^{\circ}}[/tex]

[tex]\implies a=20.8847511...[/tex]

[tex]\implies a=20.9[/tex]

Solve for b:

[tex]\implies \dfrac{b}{\sin 35^{\circ}}=\dfrac{21}{\sin 73^{\circ}}[/tex]

[tex]\implies b=\dfrac{21\sin 35^{\circ}}{\sin 73^{\circ}}[/tex]

[tex]\implies b=12.5954671...[/tex]

[tex]\implies b=12.6[/tex]

Answer:

Giselle worked as a carpenter for 12 hours and as a blacksmith for 18 hours last week

Step-by-step explanation:

Let's assume that Giselle worked as a carpenter for x hours and as a blacksmith for y hours last week.

We know that she earns $20 per hour as a carpenter and $25 per hour as a blacksmith, and that she worked a total of 30 hours, so we can write two equations based on this information:

x + y = 30 (total hours worked)

20x + 25y = 690 (total earnings)

To solve for x and y, we can use substitution or elimination. Let's use substitution.

From the first equation, we can solve for x in terms of y:

x = 30 - y

Substitute this expression for x in the second equation:

20(30 - y) + 25y = 690

Simplify and solve for y:

600 - 20y + 25y = 690

5y = 90

y = 18

So Giselle worked as a blacksmith for 18 hours last week.

To find how long she worked as a carpenter, we can substitute y = 18 into the first equation and solve for x:

x + 18 = 30

x = 12

Therefore, Giselle worked as a carpenter for 12 hours last week.

The type of retirement account is an investment option for ANY young person is  401(k). option C

A 401(k) retirement account is described as an option for investing money, created and arrangement for youths with access to any lucrative job that pays a reasonable amount of money.

This retirement account creates room for  individuals to contribute a portion of their tax income towards retirement savings, and also  contributes to grow tax-deferred until withdrawal.

However, pensions are employer-funded retirement plans and may be unavailable to all young individuals

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In this figure, we have been given a circle. The area of the circle is 314 sq ft.

The number of square units required to fill a circle is known as its area. The region occupied inside the boundaries of a round item or 2d figure is defined as the area in general. The measurement is done in square units, with square metres (m2) being the usual unit.

There are predefined formulas for calculating area for squares, rectangles, circles, triangles, and so on.

Area of a circle = πr²

We have been given r as 10 ft

and the value of π as 3.14, so we will just put it in the formula

Area of circle = 3.14 × 10²

= 3.14 × 100

314 sq ft.

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The statements that complete the function transformations are left, up, stretch and stretch

From the question, we have the following parameters that can be used in our computation:

f(x) = (x + 94)²

Also, we have

f(x) = x²

This implies that f(x) = (x + 94)² is obtained by shifting f(x) = x² left by 94 units

Next, we have

f(x) = x² + 94

This implies that f(x) = x² + 94² is obtained by shifting f(x) = x² up by 94 units

Next, we have

f(x) = 94√x from f(x) = √x

This implies that f(x) = 94√x can be obtained from vertically stretching f(x) = √x by a factor of 94

Also, the graph of f(x) = √94x can be obtained from horizontally stretching f(x) = √x by a factor of 1/94

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Answer:

a) The truth table for F is as follows:

A B C D F

0 0 0 0 1

0 0 0 1 0

0 0 1 0 0

0 0 1 1 1

0 1 0 0 0

0 1 0 1 1

0 1 1 0 1

0 1 1 1 0

1 0 0 0 0

1 0 0 1 1

1 0 1 0 1

1 0 1 1 0

1 1 0 0 1

1 1 0 1 0

1 1 1 0 0

1 1 1 1 1

b) Using the Quine-McCluskey method, we can find the minimal sum-of-products form of F:

F = ¬A ∧ ¬B ∧ D ∨ ¬A ∧ C ∧ D ∨ A ∧ B ∧ ¬C ∧ D ∨ A ∧ ¬B ∧ C ∧ D

c) Using Boolean algebra, we can simplify the Boolean expression of F as follows:

F = (¬A ∧ ¬B ∧ D) ∨ (¬A ∧ C ∧ D) ∨ (A ∧ B ∧ ¬C ∧ D) ∨ (A ∧ ¬B ∧ C ∧ D)

= (¬A ∧ (¬B ∧ D ∨ C ∧ D)) ∨ (A ∧ (B ∧ ¬C ∧ D ∨ ¬B ∧ C ∧ D))

= (¬A ∧ (¬B ∨ C) ∧ D) ∨ (A ∧ ((B ∧ ¬C) ∨ (¬B ∧ C)) ∧ D)

= (¬A ∧ ¬B ∧ C ∧ D) ∨ (¬A ∧ B ∧ ¬C ∧ D) ∨ (A ∧ B ∧ C ∧ D) ∨ (A ∧ ¬B ∧ ¬C ∧ D)

Therefore, the simplified Boolean expression of F in terms of the three variables A, B, and C is:

F = (¬A ∧ ¬B ∧ C ∧ D) ∨ (¬A ∧ B ∧ ¬C ∧ D) ∨ (A ∧ B ∧ C ∧ D) ∨ (A ∧ ¬B ∧ ¬C ∧ D)

Step-by-step explanation:

The shape of the fence is a rhombus.

Option C is the correct answer.

We have,

To determine the shape of the fence with the given coordinates, we can analyze the properties of different quadrilaterals.

Square: All four sides are equal in length, and all angles are right angles.

Rectangle: Opposite sides are equal in length, and all angles are right angles.

Rhombus: All four sides are equal in length, but the angles are not necessarily right angles.

Trapezoid: At least one pair of opposite sides are parallel.

Let's analyze the given coordinates to determine the shape of the fence:

Side 1: Distance between (-16, 1) and (-6, 5) = √(((-6) - (-16))² + (5 - 1)²) = √(10² + 4²) = √(116)

Side 2: Distance between (-6, 5) and (4, 1) = √((4 - (-6))² + (1 - 5)²) = √(10² + (-4)²) = √(116)

Side 3: Distance between (4, 1) and (-6, -3) = √((-6 - 4)² + (-3 - 1)²) = √((-10)² + (-4)²) = √(116)

Side 4: Distance between (-6, -3) and (-16, 1) = √((-16 - (-6))² + (1 - (-3))²) = √((-10)² + 4²) = √(116)

We can see that all four sides of the fence have equal lengths, which means it is either a square, rectangle, or rhombus.

To further differentiate between these shapes, we need to consider the angles.

The angle between side 1 and side 2:

tan(angle) = (5 - 1)/(-6 - (-16)) = 4/10 = 0.4

angle ≈ 21.8°

The angle between side 2 and side 3:

tan(angle) = (1 - 5)/(4 - (-6)) = (-4)/10 = -0.4

angle ≈ -21.8°

The angle between side 3 and side 4:

tan(angle) = (-3 - 1)/(-6 - (-16)) = (-4)/10 = -0.4

angle ≈ -21.8°

The angle between side 4 and side 1:

tan(angle) = (1 - (-3))/(-16 - (-6)) = 4/10 = 0.4

angle ≈ 21.8°

All four angles are approximately 21.8° or -21.8°. This indicates that the fence is a rhombus since its sides are equal in length but its angles are not right angles.

Therefore,

The shape of the fence is a rhombus.

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The limit of f(x) as x approaches 10 is -25, and the value of k is -25.

For the left-hand limit,

lim(x->10-) f(x)

= lim(x->10-) (0.1x² + 20x + 15)

For the right-hand limit,

lim(x->10+) f(x)

= lim(x->10+) (0.25x³ + k)

Since we want the left-hand limit to be equal to the right-hand limit

lim(x->10-) f(x) = lim(x->10+) f(x)

lim(x->10-) (0.1x² + 20x + 15)= lim(x->10+) (0.25x³ + k)

0.1(10)² + 20(10) + 15= 0.25(10)³ + k

10 + 200 + 15 = 0.25(1000) + k

225 = 250 + k

k = 225 - 250

k = -25

Therefore, the limit of f(x) as x approaches 10 is -25, and the value of k is -25.

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The basic ratio of minutes of painting to minutes of instruction is 7 : 6

From the question, we have the following parameters that can be used in our computation:

Painting = 63 minutes

Instruction = 54 minutes

The ratio can be represented as

Ratio = Painting : Instruction

When the given values are substituted in the above equation, we have the following equation

Painting : Instruction = 63 : 54

Simplify

Painting : Instruction = 21 : 18

Simplify

Painting : Instruction = 7 : 6

This  ratio cannot be simplified

Hence, the solution is 7 : 6

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The constraints involve three variables (Y, I, and Z), it is not feasible to graphically represent the entire Feasible region in three dimensions.

The linear programming model for  decision variables, objective function, and constraints.

Decision Variables:the number of gallons of frozen yogurt purchased each week as Y and the number of gallons of ice cream purchased each week as I.

Objective Function:

The objective is to maximize the total profit. The profit from frozen yogurt can be calculated as 3.60Y, and the profit from ice cream can be calculated as 4.15I. Thus, the objective function is:

Maximize Profit: Z = 3.60Y + 4.15I

Constraints:

1. Budget constraint: The total cost of frozen yogurt and ice cream should not exceed the weekly budget of $90.

  Cost Constraint: 0.75Y + 0.93I ≤ 90

2. Freezer space constraint: The total gallons of frozen yogurt and ice cream should not exceed the available freezer space of 115 gallons.

  Space Constraint: Y + I ≤ 115

3. Ice cream sales constraint: The weekly ice cream sales should be at least twice the frozen yogurt sales.

  Ice Cream Sales Constraint: I ≥ 2Y

4. Non-negativity constraint: The number of gallons for both products cannot be negative.

  Non-negativity Constraint: Y ≥ 0, I ≥ 0

The model using graphical analysis, we can plot the constraints on a graph and find the feasible region. Then, we can evaluate the objective function at each corner point of the feasible region to determine the optimal solution.

Since the constraints involve three variables (Y, I, and Z), it is not feasible to graphically represent the entire feasible region in three dimensions.

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To prove that (secx+tanx)² = (cscx+1)/(cscx-1), we will start with the left-hand side (LHS) of the equation and simplify it step by step until it matches the right-hand side (RHS) of the equation.

LHS: (secx+tanx)²

Using the trigonometric identities secx = 1/cosx and tanx = sinx/cosx, we can rewrite the LHS as:

LHS: (1/cosx + sinx/cosx)²

Now, let's find a common denominator and simplify:

LHS: [(1+sinx)/cosx]²

Expanding the squared term, we get:

LHS: (1+sinx)² / cos²x

Next, we will simplify the denominator:

LHS: (1+sinx)² / (1 - sin²x)

Using the Pythagorean identity sin²x + cos²x = 1, we can replace 1 - sin²x with cos²x:

LHS: (1+sinx)² / cos²x

Now, let's simplify the numerator by expanding it:

LHS: (1+2sinx+sin²x) / cos²x

Next, we will simplify the denominator by using the reciprocal identity cos²x = 1/sin²x:

LHS: (1+2sinx+sin²x) / (1/sin²x)

Now, let's simplify further by multiplying the numerator and denominator by sin²x:

LHS: sin²x(1+2sinx+sin²x) / 1

Expanding the numerator, we get:

LHS: (sin²x + 2sin³x + sin⁴x) / 1

Now, let's simplify the numerator by factoring out sin²x:

LHS: sin²x(1 + 2sinx + sin²x) / 1

Using the fact that sin²x = 1 - cos²x, we can rewrite the numerator:

LHS: sin²x(1 + 2sinx + (1-cos²x)) / 1

Simplifying further, we get:

LHS: sin²x(2sinx + 2 - cos²x) / 1

Using the fact that cos²x = 1 - sin²x, we can rewrite the numerator again:

LHS: sin²x(2sinx + 2 - (1-sin²x)) / 1

Simplifying the numerator, we have:

LHS: sin²x(2sinx + 1 + sin²x) / 1

Now, let's simplify the numerator by expanding it:

LHS: (2sin³x + sin²x + sin²x) / 1

LHS: 2sin³x + 2sin²x / 1

Finally, combining like terms, we get:

LHS: 2sin²x(sin x + 1) / 1

Now, let's simplify the RHS of the equation and see if it matches the LHS:

RHS: (cscx+1) / (cscx-1)

Using the reciprocal identity cscx = 1/sinx, we can rewrite the RHS:

RHS: (1/sinx + 1) / (1/sinx - 1)

Multiplying the numerator and denominator by sinx to simplify, we get:

RHS: (1 + sinx) / (1 - sinx)

Now, we can see that the LHS and RHS are equal:

LHS: 2sin²x(sin x + 1) / 1

RHS: (1 + sinx) / (1 - sinx)

Therefore, we have proven that (secx+tanx)² = (cscx+1)/(cscx-1).

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