Are the two triangles similar?

Aquí hay algunos ejemplos de aplicaciones de ecuaciones que han tenido un gran impacto: Medicina, Ingeniería, Ingeniería, Economía.

Impacto:

1. Medicina: Las ecuaciones diferenciales son ampliamente utilizadas en la modelización y predicción de enfermedades y en el diseño de tratamientos médicos. La ecuación de Hodgkin-Huxley, por ejemplo, es un modelo matemático que describe cómo las señales eléctricas se propagan en las neuronas y ha sido fundamental para la comprensión y tratamiento de enfermedades neurológicas.

2. Ingeniería: Las ecuaciones diferenciales y la mecánica cuántica son ampliamente utilizadas en el diseño y construcción de puentes, edificios y otros proyectos de ingeniería. Las ecuaciones de Euler-Lagrange, por ejemplo, se utilizan en la modelización de sistemas mecánicos complejos, como los sistemas de suspensión de vehículos, lo que ha llevado a mejoras en la seguridad y el confort de los pasajeros.

3. Física: Las ecuaciones matemáticas han sido fundamentales para el desarrollo de la física moderna y la tecnología asociada. Las ecuaciones de Maxwell, por ejemplo, describen cómo los campos eléctricos y magnéticos interactúan y han sido fundamentales para el desarrollo de la electrónica moderna.

4. Economía: Las ecuaciones de oferta y demanda, las ecuaciones de costo-beneficio y otras ecuaciones económicas han sido fundamentales para la toma de decisiones empresariales y gubernamentales, lo que ha llevado a mejoras en la eficiencia y la productividad.

En resumen, las ecuaciones matemáticas han tenido un impacto significativo en muchos aspectos de la vida humana, desde la medicina y la ingeniería hasta la física y la economía, y su aplicación continuará siendo esencial para el progreso y desarrollo de la sociedad en el futuro.

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The key features of the parabola formed by the equation F(x) = -x² + 4x + 12 are: x-intercepts : x = 6 and x = -2, vertex: (2, -8) and opens downward.

What is parabola?

A parabola is a symmetrical plane curve that is shaped like an arch. It is a quadratic function and is defined by the equation y = ax² + bx + c, where a, b, and c are constants.

To identify the key features of the parabola formed by the equation F(x) = -x² + 4x + 12, we can start by finding the x-intercepts, which are the points where the graph of the parabola intersects the x-axis.

To find the x-intercepts, we set F(x) equal to zero and solve for x:

F(x) = -x² + 4x + 12 = 0

We can factor this quadratic equation:

F(x) = -(x² - 4x - 12) = 0

F(x) = -(x - 6)(x + 2) = 0

So the x-intercepts are x = 6 and x = -2.

Next, we can use the vertex form of the quadratic equation, which is:

y = a(x - h)² + k

where (h,k) is the vertex of the parabola and a is a constant that determines whether the parabola opens up or down.

To put F(x) into vertex form, we can complete the square:

F(x) = -x² + 4x + 12

F(x) = -(x² - 4x - 12)

F(x) = -(x² - 4x + 4 - 4 - 12)

F(x) = -(x - 2)² + (-8)

So the vertex of the parabola is (2, -8), and because the coefficient of x² is negative, the parabola opens downward.

Therefore, the key features of the parabola formed by the equation F(x) = -x² + 4x + 12 are:

x-intercepts: x = 6 and x = -2

vertex: (2, -8)

opens downward.

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The amount of space taken by the wooden blocks in the figure is 684 cubic inches.

A rectangular prism is a prism with rectangle-shaped bases (the top face and bottom face). There are three pairs of identical opposite faces on each of its six faces, making a rectangular prism's opposite faces all be the same. Its length, width, and height are its three dimensions. Rectangular tissue boxes, school notebooks, laptops, fish tanks, big buildings like freight containers, rooms, storage sheds, etc. are a few instances of rectangular prisms in daily life. The rectangular prism and its net, which represents the prism in two dimensions when its faces are spread apart on a plane, are shown in the accompanying figure.

The space taken by the figure can be determined using the volume of the figure.

The volume of rectangular prism is given as:

V = lwh

For the bottom prism we have:

V = (11)(12)(3)

V = 396 cubic inches.

For the top prism we have:

V = (12)(6)(4)

V = 288 cubic inches.

The total space occupied by the wooden block is:

V = 396 + 288

V = 684 cubic inches.

Hence, the amount of space taken by the wooden blocks in the figure is 684 cubic inches.

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The complete question is:

Answer:

Step-by-step explanation:

The answer would be 216cm2 or 33.48 in2

The monthly mortgage payment on a $500,000 6% 30-year mortgage is approximately $2,997.75.

Answer: 8/7x

Step-by-step explanation:

2x-3x/7x2

rewrite

2x-3/7x x 2

calculate

2x-6/7x

calculate

solution

8/7x

Answer:

Step-by-step explanation:

You want the distances from the ground to the top of the ladder, and from the wall to the base of the ladder when the 47-inch ladder makes an angle of 51° with the ground.

The trig functions Sine and Cosine relate the sides of a right triangle to the angle and the hypotenuse:

  Sin = Opposite/Hypotenuse

  Cos = Adjacent/Hypotenuse

Multiplying by the hypotenuse gives ...

  opposite (ground to top) = (47 in) × sin(51°) = 36.53 in

  adjacent (wall to base) = (47 in) × cos(51°) = 29.58 in

__

Additional comment

These trig relations are summarized in the mnemonic SOH CAH TOA.

Answer: are you mental

Step-by-step explanation:

Answer: First, we need to calculate how many inches the coin will sink in 1 second:

1/5 inch per second x 1 second = 1/5 inch

Next, we can multiply the sinking rate by the number of seconds to find the total distance the coin will sink:

1/5 inch x 7.5 seconds = 1.5 inches

Therefore, the coin will sink 1.5 inches in 7 1/2 seconds.

Step-by-step explanation:

Answer:

x = 3,

y = 3

Step-by-step explanation:

{5x + 6y = 33,

{-3x + 4y = 3;

Make x or y the subject from either the first or the second equation (it is advisable to choose one where the numbers can be written as a finite decimal fraction after division, I chose to make x the subject from the 1st equation):

5x = 33 - 6y / : 5

x = 6,6 - 1,2y

Replace x in the 2nd equation with its value from the 1st one (the underlined expression):

-3 × (6,6 - 1,2y) + 4y = 3

Multiply every term inside the bracket by the term on the outside:

-19,8 + 3,6y + 4y = 3

Collect like terms and add them together:

7,6y = 22,8 / : 7,6

y = 3

x = 6,6 - 1,2 × 3 = 3

Answer:

k = 150

Step-by-step explanation:

the measure of a secant- secant angle is half the difference of the intercepted arcs , that is

45 = [tex]\frac{1}{2}[/tex] (k - 60) ← multiply both sides by 2 to clear the fraction

90 = k - 60 ( add 60 to both sides )

150 = k

Answer:

2 2/3 hours.

Step-by-step explanation:

If her washs once a week, he will wash 4 times in 4 weeks.

2/3 * 4 = 8/3

2 2/3.

Answer: 24 grapes

Step-by-step explanation:

The nth term of the sequence 2, 5, 10, 17 is [tex]n^{2} + 1.[/tex]

The nth term of the sequence 2, 8, 18, 32 is [tex]2n^{2}.[/tex]

What is a sequence?

In mathematics, a sequence is an ordered list of numbers, usually defined by a rule or a pattern. every number in the sequence is called a term.. Finite sequences have a specific number of terms, while infinite sequences continue indefinitely.

To find the nth term of the sequence 2, 5, 10, 17, we need to observe the differences between the terms:

The difference between the first and second term is 5 - 2 = 3.

The difference between the second and third term is 10 - 5 = 5.

The difference between the third and fourth term is 17 - 10 = 7.

Notice that the differences between the terms are consecutive odd numbers starting from 3. This pattern matches the sequence of squares of consecutive integers. Specifically, the nth term of the sequence 2, 5, 10, 17 is:

[tex]n^{2} + 1.[/tex]

So the nth term of the sequence 2, 5, 10, 17 is [tex]n^{2} + 1.[/tex]

b. To find the nth term of the sequence 2, 8, 18, 32, we again need to observe the differences between the Sequences:

the terms of the sequence can be written as 2 times of the give sequence.

eg. 2(1) = 2

     2(4) = 8

     2(9) = 18

     2(16) = 32 and so on

hence, the nth term of the sequence will be 2 times of n² i.e. [tex]2n^{2}.[/tex]

So the nth term of the sequence 2, 8, 18, 32 is [tex]2n^{2}.[/tex]

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Every element x in S has a left inverse y that is also a right inverse, and therefore S is a group.

In abstract algebra, a semigroup is a mathematical structure consisting of a set equipped with an associative binary operation. That is, given any three elements a, b, and c in the set, the operation is such that (a * b) * c = a * (b * c), where * denotes the binary operation. This is known as the associative law of the operation.

Semigroups are a fundamental concept in algebraic structures, and they arise in many areas of mathematics and computer science, including group theory, ring theory, semigroup theory, and automata theory. Examples of semigroups include the set of non-negative integers under addition, the set of matrices of a fixed size under matrix multiplication, and the set of all finite strings over a given alphabet under concatenation.

One important property of semigroups is that they need not have an identity element, which is an element that acts as a neutral element under the binary operation.

Let S be a semigroup with a left identity e and a left inverse for every element x in S. We need to show that S is a group.

Let x be an arbitrary element of S. We want to show that x has a two-sided inverse.

Since e is a left identity, we have ex = x for all x in S.

Now, let y be the left inverse of x. This means that yx = e.

We can then use the associativity property of the semigroup to rearrange these equations:

(yx)x⁻¹ = ex⁻¹

y(xx⁻¹) = x⁻¹

ye = x⁻¹

y = x⁻¹

Thus, every element x in S has a left inverse y that is also a right inverse, and therefore S is a group.

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The identification of the solid shapes are;

1. Triangular prism

2. cylinder

3. sphere

4. cone

5. cuboid

6. pyramid

7. pyramid

8. sphere

9. cylinder

10. pyramid

11. cube

12. cube

13. cone

14. cylinder

15. sphere

16. cylinder

Solid shapes are three-dimensional (3D) geometric shapes that occupy some space and have length, breadth, and height. Solid shapes are classified into various categories. Some of the shapes have curved surfaces; some of them are in the shape of pyramids or prisms.

Examples of solid shapes include: prisms, pyramids, cone , cylinder, sphere cuboid , cube e.t.c

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The function for the value of the account after t years is: [tex]V(t) = 5900 * (1 + 0.056/365)^{(365*t)[/tex]and the percentage of growth per year is 5.74%.

A percentage is a way of expressing a quantity as a fraction of 100. It is often used to compare two quantities or to express a part of a whole.

In mathematics, a function is a rule that assigns to each element in one set (called the domain) exactly one element in another set (called the range).

The formula for calculating the value of the account after t years is:

[tex]V(t) = 5900 * (1 + 0.056/365)^{(365*t)[/tex]

where 0.056 is the annual interest rate (APR), divided by 100 to convert it to a decimal, and 365 is the number of days in a year.

To find the annual percentage yield (APY), we can use the formula:

[tex]APY = (1 + APR/n)^{n - 1[/tex]

where n is the number of times the interest is compounded per year. In this case, the interest is compounded daily, so n = 365.

[tex]APY = (1 + 0.056/365)^{365 - 1} = 0.0574\ or\ 5.74%[/tex]

Therefore, the function for the value of the account after t years is:

[tex]V(t) = 5900 * (1 + 0.056/365)^{(365*t)[/tex]

And the percentage of growth per year is 5.74%.

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Answer:We want to find the average rate of change of the graph on the interval 1 ≤ x ≤ 3.

The correct option is B: -4

We know that for a given function f(x) the average rate of change on the interval a ≤ x ≤ b is given by:

Here we have the interval  1 ≤ x ≤ 3, and by looking at the graph we can see that:

f(3) = 1

f(1) = 9

Then the average rate of change is:

So the correct option is B.

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Step-by-step explanation:

The requried, Reduction to the lowest terms of 4 1/5 ÷ 2 1/3 is  9/5.

To divide mixed numbers, we need to convert them to improper fractions, then multiply the first fraction by the reciprocal of the second fraction.

Converting the mixed numbers to improper fractions:

4 1/5 = 21/5

2 1/3 = 7/3

Multiplying by the reciprocal:

(21/5) ÷ (7/3) = (21/5) * (3/7) = 9/5

Therefore, 4 1/5 ÷ 2 1/3 = 9/5.

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A sample size of at least 54,444 guests to ensure a 95% confidence level and a margin of error of 3% and a sample size of at least 217777 guests to ensure a 95% confidence level and a margin of error of ±1.5%.

A. To calculate the sample size needed for a 95% confidence level and a margin of error of 3%, we can use the following formula:

n = (Z^2 * p * q) / E^2

Where:

n = sample size

Z = Z-score (which corresponds to the level of confidence) = 1.96 (for a 95% confidence level)

p = proportion of the population that has the characteristic we are interested in (we don't know this value, so we use 0.5 for a conservative estimate)

q = 1 - p

E = margin of error

Plugging in the values, we get:

n = (1.96^2 * 0.5 * 0.5) / 0.03^2 = 54,444.44

Therefore, we need a sample size of at least 54,444 guests to ensure a 95% confidence level and a margin of error of 3%.

B. If Ouranos Resorts wants to reduce the margin of error to ±1.5%, the required sample size will increase. Using the same formula as above, we get:

n = (1.96^2 * 0.5 * 0.5) / 0.015^2 = 217777

Therefore, we need a sample size of at least 217777 guests to ensure a 95% confidence level and a margin of error of ±1.5%. As we can see, the required sample size has increased by a factor of 4, which means that the survey will be more expensive and time-consuming to conduct.

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The solution set (rounded to four decimal places) is:  {x ≈ 5.8986}

Starting from the given equation:

[tex]e^(2x-7) - 5 = 27132[/tex]

Adding 5 to both sides:

[tex]e^(2x-7) = 27137[/tex]

Taking the natural logarithm of both sides:

[tex]ln(e^(2x-7)) = ln(27137)[/tex]

Using the property that ln(e^a) = a:

[tex]2x - 7 = ln(27137)[/tex]

Adding 7 to both sides:

[tex]2x = ln(27137) + 7[/tex]

Dividing by 2:

[tex]x = (ln(27137) + 7)/2[/tex]

Therefore, the solution set expressed in terms of natural logarithms is:

[tex]{x | x = (ln(27137) + 7)/2}[/tex]

Using a calculator to approximate the solution:

x ≈ 5.8986

Therefore, the solution set (rounded to four decimal places) is:

{x ≈ 5.8986}

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102.57 cm³ is the volume of the part of the cone that is filled with sand.

We can start by using the formula for the volume of a cone:

V = (1/3) × π × r² × h

where V is the volume, r is the radius of the circular base, h is the height, and π is a mathematical constant approximately equal to 3.14. Let's assume that the cone-shaped paper cup has a height of h and a radius of r. Since the cup is 2/3 full of sand, the volume of sand in the cup is 2/3 of the total volume of the cone. Therefore, we can express the volume of sand in terms of the total volume of the cone as:

V sand = (2/3) × V

Substituting the formula for the volume of a cone into the above equation, we get:

V sand = (2/3) × (1/3) × π × r² × h

Simplifying the equation, we get:

V sand = (2/9) × π × r² × h

Therefore, the volume of the part of the cone that is filled with sand is (2/9) × π × r² × h.

Since we do not have the values of r and h, we cannot find the exact volume of the sand. However, we can use the given options to make an educated guess.

Let's try substituting the values of r and h from the given options into the equation and see which option gives us a value close to 2/3 of the total volume of the cone.

Option F: V sand = (2/9) × π × (3.3)² × 4.5 ≈ 102.57 cm³

Option G: V sand = (2/9) × π × (4.4)² × 3.0 ≈ 153.86 cm³

Option H: V sand = (2/9) × π × (5.5)² × 3.0 ≈ 307.72 cm³

Option J: V sand = (2/9) × π × (6.6)² × 2.25 ≈ 461.58 cm³

Option F gives us a value close to 2/3 of the total volume of the cone, so the answer is (F) 102.57 cm³

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Based on the information, the ordinate to the origin is -3.

In mathematics, an "ordinate" refers to the vertical coordinate of a point on a graph, while the "origin" typically refers to the point (0, 0) where the x- and y-axes intersect. Therefore, the phrase "ordinate to the origin" means the vertical distance of a point from the origin, which is simply the value of its y-coordinate.

For example, if you have a point with coordinates (3, 5), the ordinate to the origin would be 5, because that is the distance from the point to the x-axis (which passes through the origin).

The first question is incomplete. Regarding the second question, the equation y = -6x - 3 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept (or the ordinate to the origin). Therefore, the slope is -6 and the y-intercept is -3. So the ordinate to the origin is -3.

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Answer:

Centre = (3, 5)

Radius = [tex]\sqrt{131}[/tex]

Step-by-step explanation:

Given equation of a circle:

[tex]-6x + x^2 = 97 + 10y - y^2[/tex]

To find the centre and radius of the given equation of a circle, rewrite it in standard form.

[tex]\boxed{\begin{minipage}{6.3cm}\underline{Equation of a Circle - Standard Form}\\\\$(x-h)^2+(y-k)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(h, k)$ is the centre. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

First, rearrange the equation so that the terms in x and y are on the left side and the constant is on the right side:

[tex]x^2 - 6x + y^2 - 10y = 97[/tex]

Complete the square for the x and y terms by adding the square of half the coefficient of the term in x and y to both sides:

[tex]\implies x^2 - 6x +\left(\dfrac{-6}{2}\right)^2+ y^2 - 10y +\left(\dfrac{-10}{2}\right)^2= 97+\left(\dfrac{-6}{2}\right)^2+\left(\dfrac{-10}{2}\right)^2[/tex]

Simplify:

[tex]\implies x^2 - 6x +\left(-3\right)^2+ y^2 - 10y +\left(-5\right)^2= 97+\left(-3\right)^2+\left(-5\right)^2[/tex]

[tex]\implies x^2 - 6x +9+ y^2 - 10y +25= 97+9+25[/tex]

[tex]\implies x^2 - 6x +9+ y^2 - 10y +25= 131[/tex]

Now we have created two perfect square trinomials on the left side of the equation:

[tex]\implies (x^2 - 6x +9)+ (y^2 - 10y +25)= 131[/tex]

Factor the perfect square trinomials:

[tex]\implies (x-3)^2+ (y-5)^2= 131[/tex]

If we compare this equation with the standard form, we see that the centre of the circle is (3, 5) and its radius is the square root of 131.

Therefore:

Required value of x is 20/3 inches.

What is area of the square?

Side × Side

Given, the area of the painting, which is 360 square inches, and we know that the side of the square is (3x + 2) inches. So we can set up an equation,

(3x + 2)² = 360

Expanding the left side,

9x²+ 12x + 4 = 360

Subtracting 360 from both sides,

9x² + 12x - 356 = 0

Now we can use the quadratic formula to solve for x,

[tex]x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} [/tex]

In this case, a = 9, b = 12, and c = -356. Plugging these values into the formula, we get:

[tex]x = \frac{ - 12 \pm \sqrt{ {12}^{2} - 4 \times 9} }{2 \times 9} \\ = \frac{ - 12 \pm \sqrt{144 - 36} }{18} [/tex]

By solving,we will get [tex]x = - \frac{7}{3} \: or \: \frac{20}{3} [/tex]

Since the side length of the square must be positive, we can ignore the negative solution and conclude that the value of x is 20/3 inches.

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The mean of the data for minutes spent every day = 47mins.

The mean in statistics is calculated by multiplying all the values in a set of data by the total number of values for a given set of observations. To put it another way, all that is necessary to determine the mean of a set of data is to sum up all the values and divide the total by the number of values.

The data as per the question is:

33, 59, 68, 32, 43.

Now mean will be = sum of all data/ number of data

= (33 + 59 + 68 + 32 + 43)/5

= 235/5

= 47mins.

Hence, the mean of the data is 47mins.

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The range would grow by 12 if the data included a second person who is 46 years old.

The difference between the data set's maximum and smallest values is known as the range.

In the given data set, the minimum value is 15 and the maximum value is 34. So, the range is:

Range = maximum value - minimum value

Range = 34 - 15

Range = 19

If we add another age of 46 to the data set, then the new maximum value would be 46 and the new range would be:

New range = new maximum value - minimum value

New range = 46 - 15

New range = 31

So, the range would increase by 12 if another age of 46 is added to the data.

Therefore, the correct answer is: The range would increase by 12.

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As a result, the distance between points A and B is roughly 630.3 feet as  the letters "d" for the distance to the lighthouse from point A and "x" for the distance .

Trigonometric ratios, commonly referred to as trigonometric functions, are mathematical relationships between the ratios of the side lengths of a right triangle and its angles. The first three fundamental trigonometric ratios are: The length of the side opposite the angle to the length of the hypotenuse is known as the sine (sin). The cosine (cos) function measures how long the adjacent side is in relation to the hypotenuse. The length of the side that is opposite the angle to the length of the side that is next to it is referred to as the tangent (tan). The reciprocals of sine, cosine, and tangent, respectively, are cosecant (csc), secant (sec), and cotangent (cot), which are additional trigonometric functions. Trigonometric ratios are employed in a number of disciplines, such as mathematics, physics, engineering, and navigation.

given

Trigonometry can be used to resolve this issue. Let's use the letters "d" for the distance to the lighthouse from point A and "x" for the distance to point B. Next, we have:

tan(15°) = (lighthouse height) / d

tan(6°) is equal to (lighthouse height) / (d + x).

In the first equation, "d" can be solved as follows:

D is equal to (lighthouse height) / tan(15°).

This is what we get when we enter it into the second equation:

tan(6°) is equal to (lighthouse height) / (lighthouse height / tan(15°) + x).

tan(6°) is equal to tan(15°) / (tan(15°) / (lighthouse height) + x/d)

The result is obtained by multiplying both sides by (tan(15°) / (height of lighthouse) + x/d):

Tan(6°) + Tan(15°) / (Lighthouse Height + x/d) = Tan(15°)

We can now determine how to solve for "x"

x is equal to d*(tan(6°)*(height of lighthouse)/tan(15°)-1)

When we enter the values from the issue, we obtain:

D=(lighthouse height)/tan(15°) = 3892.72 feet

630.3 feet are equal to x = 3892.72 * (tan(6°) * (height of lighthouse) / tan(15°) - 1)

As a result, the distance between points A and B is roughly 630.3 feet as  the letters "d" for the distance to the lighthouse from point A and "x" for the distance .

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The truth value is OFT T which is option D.

Truth value is a term used in logic to describe the truth or falsehood of a statement or proposition. Every statement or proposition is either true or false, which are the two possible truth values. In symbolic logic, the truth values are usually represented by the letters "T" for true and "F" for false

The truth value of p - q depends on the truth values of p and q, and the meaning of the "-" operator.

Assuming that "-" represents the logical operator of implication, then p - q means "if p is true, then q is true", or "p implies q".

In this case, both p and q are false, which means that p implies q is true. So the truth value of p - q is T (True).

Therefore, the correct answer is OFT T.

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