If sin t= 7/9, and t is in quadrant I, find the exact value of sin(2t) , cos(2t) , and tan(2t) algebraically without solving for t .

The reason why AC must be congruent to BD is the hypotenuse - leg theorem and CPCTC

According to the hypotenuse-leg (HL) theorem, a right triangle is congruent if each of its hypotenuse and one leg are congruent with their corresponding hypotenuse and one leg in another right triangle.

The problem have the hypotenuse and a leg of the two triangles stated to be equal. This completes the requirement for the theorem and hence the two triangles are said to be equal

Then applying the CPCTC theorem which says that if two triangles are congruent, every corresponding part of one triangle is congruent to the other side hence, the sides AC = BD

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Answer: x=4, AB=19, ED=57

Step-by-step explanation:

Trapezoid midsegment formula: [tex]\frac{b1+b2}{2}[/tex]=Midsegment

Therefore, (4x+3+18x-15)/2=38

Multiply both sides by 2 in order to get rid of the fraction. (Multiplication property of equality)

4x+3+18x-15=76

Combine like terms.

22x-12=76

Add 12 to both sides. (Addition property of equality)

22x = 88

Solve for x

x = 4

Now, substitute x in for segment AB and ED

AB = 4(4)+3

AB= 16+3

AB= 19

ED=18(4) -15

ED=72-15

ED=57

Check your work:

(ED+AB)/2 = 38

(57+19)/2 = 38

76/2 = 38

38 = 38

With the data points (-2,-3) and (-1,-1), the linear equation is obtained as Option B: y = 2x + 1.

What is a linear equation?

A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.

The data points are given for the equation.

Consider the first two points (-2,-3) and (-1,-1).

The slope intercept form of the linear equation is -

y = mx + b

Here m is the slope.

For calculation of slope the formula is -

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

Substitute the values in the equation -

m = [(-1) - (-3)] / [(-1) - (-2)]

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

m = 2/1 = 2

The equation becomes y = 2x + b.

Substitute the point (-2,-3) to get the value of b.

y = 2x + b

-3 = 2(-2) + b

-3 = -4 + b

b = -3 + 4

b = 1

Therefore, the linear equation is y = 2x + 1.

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Answer:

32/13

Step-by-step explanation:

To evaluate the expression Z₂ * Z3 / Z₁, we first need to find the complex numbers Z₂ * Z3 and Z₁.

Z₂ * Z3 = -2 + 3i * 4 + 2i = (-8 + 14i) + (-6 + 6i) = -8 + 8i

Z₁ = 3 - 2i

So, Z₂ * Z3 / Z₁ = (-8 + 8i) / (3 - 2i)

We can simplify this expression by multiplying the numerator and denominator by the complex conjugate of the denominator, which is (3 + 2i).

Z₂ * Z3 / Z₁ = (-8 + 8i) / (3 - 2i) * (3 + 2i) / (3 + 2i) = (-8 + 8i) * (3 + 2i) / (3 - 2i) * (3 + 2i)

Z₂ * Z3 / Z₁ = ((-8 * 3 + 8 * 2i) + (8 * 3 + 8 * 2i)) / (3^2 + (-2i)^2) = (16 + 16i) / 13

Therefore, Z₂ * Z3 / Z₁ = (16 + 16i) / 13.

Answer:

The triangle has an area of 192cm^2

Step-by-step explanation:

Given;

the formula to find the area of the triangle,

[tex]A=\frac{h_{b}b }{2}[/tex]

We can insert the values,

[tex]A=\frac{24*16}{2} \\\\\\A=\frac{384}{2} \\\\A=192cm^2[/tex]

The area of the triangle is 192cm^2.

Answer:

From the given information, we know that 2 red peppers produce a fire that is 25 meters long, and 10 green peppers produce a fire that is 15 meters long.

To find the length of the fire produced by the dragon eating 3 red peppers and 5 green peppers, we can use a proportion:

(3/2) * 25 + (5/10) * 15 = x

where x is the length of the fire produced by the dragon eating 3 red peppers and 5 green peppers.

Simplifying the equation, we get:

37.5 = x

Therefore, the length of the fire produced by the dragon eating 3 red peppers and 5 green peppers is 37.5 meters.

Step-by-step explanation:

Answer:

Step-by-step explanation:

To find the base of the exponential function F(x) = b^x, we need to use the information that its graph contains the point (1/3, 2). We can use this information to solve for b.

Since the graph passes through the point (1/3, 2), we can set x = 1/3 and y = 2 in the equation F(x) = b^x:

b^(1/3) = 2

Taking the cube root of both sides, we get:

b = (2)^3 = 8

So the base of the exponential function F(x) = b^x is b = 8, which means the correct answer is C) 8.

Answer:

5,840,000

Step-by-step explanation:

Let P be the population after x years. The population after 24 years is 2P. So, we have:

P = 365,000 * 2^(x/24)

To find the population after 96 years, we plug in x = 96:

P = 365,000 * 2^(96/24) = 365,000 * 2^4 = 365,000 * 16 = 5,840,000

So, the population of the city will be 5,840,000 after 96 years.

Answer:

Step-by-step explanation:

The domain of f  * g is the set of all real numbers x such that x ≤ 6.

In interval notation, we can write:

Domain of f * g = (-∞, 6].

What is the domain of the function?

The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x). The range of a function is the set of values that the function assumes. This set is the values that the function shoots out after we plug an x value in.

The domain of the function f(x) = √(6-x) is the set of all real numbers x such that 6-x ≥ 0, which is x ≤ 6.

The domain of the function g(x) = x² - x is the set of all real numbers x, since there are no restrictions on the input.

To find the domain of the function f * g, we need to consider the values of x for which the expression f(x) * g(x) is defined.

We have:

f(x)g(x) = √(6-x)(x² - x)

The expression is defined only if both factors are defined.

For the first factor, √(6-x) to be defined, we need 6-x ≥ 0. This means that x ≤ 6.

For the second factor, x² - x to be defined, there are no restrictions on the input.

Therefore, the domain of f * g is the set of all real numbers x such that x ≤ 6.

In interval notation, we can write:

Domain of f * g = (-∞, 6].

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The solution for x is not possible because the information is not enough, the correct option is D.

Its a parallelogram but all sides of same length.

Rhombus is a parallelogram whose all sides are of equal lengths.

Its diagonals are perpendicular to each other and they cut each other in half( thus, they're perpendicular bisector of each other).

Its vertex angles are bisected by its diagonals.

The triangles on either side of the diagonals are isosceles and congruent.

We are given that;

m∠AEB of rhombus ABCD is 7x+6

Now,

To find the value of x we need atleast one more information

Since the measure of diagonals are not given and the length of sides are also not provided.

So, that the calculation could not be made.

Therefore, the information of rhombus is insufficient.

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The side length of the square with the same area as the circle is given as follows:

9.75 cm.

The area of a circle of radius r is given by π multiplied by the radius squared, as follows:

A = πr²

The radius for this problem is of 5.5 cm, hence the area of the circle is given as follows:

A = π x 5.5²

A = 95.03 cm².

For a square of side length s, the area is given by the side length squared, that is:

A = s².

Hence the desired side length is obtained as follows:

s² = 95.03

s = sqrt(95.03)

s = 9.75 cm.

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Answer:

To solve it, you need to use probability. Let me show you how.

We can use a tree diagram to represent the possible outcomes of the last ball. Here is the diagram:

```

Haris Rauf

/        \

yorker   full toss

0.8       0.2

/  \      /  \

boundary no boundary boundary no boundary

0.2  0.8  0.9  0.1

```

The probability of a boundary is the sum of the probabilities of the branches that end with a boundary. To find the probability of each branch, we multiply the probabilities along the branch. For example, the probability of a yorker and a boundary is 0.8 × 0.2 = 0.16.

So, the probability of a boundary is:

0.8 × 0.2 + 0.2 × 0.9 = 0.16 + 0.18 = 0.34

Therefore, the probability that a boundary will be scored of the last ball is 0.34 or 34%.

Step-by-step explanation:

The amount of natural oil needed to make 869 liters of Petrolyn oil is given as follows:

1520.75 liters.

The necessary amount of natural oil is obtained applying the proportions in the context of this problem.

Considering the ratio given in the problem, a rule of three is established, as follows:

7 liters of natural oil - 4 liters of Petrolyn oil.

n liters of natural oil - 869 liters of Petrolyn oil.

Applying cross multiplication, the desired amount is obtained as follows:

4n = 7 x 869

n = 7 x 869/4

n = 1520.75 liters.

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[tex]\cot\left(\frac{\pi}{2}-\theta\right)=\tan(\theta) \hspace{5em}\cos\left(\frac{\pi}{2}-\theta\right)=\sin(\theta) \\\\\\ \textit{Symmetry Identities} \\\\ \sin(-\theta )=-\sin(\theta) \hspace{5em}\cos(-\theta )=\cos(\theta )\hspace{4em} \tan(-\theta)=-\tan(\theta ) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cot\left(\frac{\pi}{2}-x\right)=-1.84\implies \tan(x)=-1.84 \\\\\\ \tan(-x)\implies -\tan(x)\implies -(-1.84)\implies \text{\LARGE 1.84} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\sin(\theta )=-0.57\implies \cos\left(\frac{\pi}{2}-\theta\right)=-0.57 \\\\\\ \boxed{\alpha = \frac{\pi}{2}-\theta} \hspace{5em}\cos(-\alpha)= \cos(\alpha) \\\\\\ \cos\left[ -\left(\frac{\pi}{2}-\theta\right) \right]\implies \cos\left(\theta -\frac{\pi}{2}\right)=\cos\left(\frac{\pi}{2}-\theta\right) \implies \cos\left(\theta -\frac{\pi}{2}\right)=\text{\LARGE -0.57}[/tex]

There are 2(3/16) cups of shredded cheese on each pizza.

A fraction is written in the form of a numerator and a denominator where the denominator is greater that the numerator.

Example: 1/2, 1/3 is a fraction.

We have,

Large pizza = 1(1/4) cups of sauce

Now,

1(3/4) of the sauce = shredded cheese

The amount of shredded cheese.

= 1(3/4) x 1(1/4)

= 7/4 x 5/4

= 35/16

= 2(3/16)

Thus,

The amount of shredded cheese is 2(3/16).

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The unknown in the similar triangles are as follows;

x = 15 units

x = 16 units

Two triangles are said to be similar when they have two corresponding angles congruent and the sides proportional to each other.

Let's solve for x in the triangles as follows:

using the proportion,

2.

9 / x = x / 25

cross multiply

25 × 9 = x²

x² = 225

x = √225

x = 15 units

3.

x / 12 = 12 / 9

cross multiply

9x = 144

x = 144 / 9

x = 16 units

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The graph of the quadratic function is given below.

Quadratic functions are polynomial functions consisting of variables and exponents and the degree of the variable is 2.

The general form of a quadratic function is f(x) = ax² + b x + c.

Given is a quadratic function.

y = 3x² + 1

We know that graph of a quadratic function is a parabola.

Since the leading coefficient is +3, positive. So the graph opens upward.

Vertex = (-b/2a, f(-b/2a))

-b/2a = -0/(2 × 3) = 0

f(-b/2a) = (3 × 0) + 1 = 1

(0, 1) is the vertex.

x = 1, then y = 3 + 1 = 4

x = -1, then y = 3 + 1 = 4

x = 2, then y = 12 + 1 = 13

x = -2, then y = 12 + 1 = 13

x = 3, then y = 27 + 1 = 28

x = -3, then y = 27 + 1 = 28

Some of the points are (-3, 28), (-2, 13), (-1, 4). (0, 1), (1, 4), (2, 13) and (3, 28).

Hence the graph through the given points is given below.

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We cannot make any definitive inferences about the entire school population, however, so the answer is E. None of the above.

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

The total number of students are 10+11+23+6=50

Based on the survey of 50 students about their preference for an additional sports team, we can observe that lacrosse is the most popular sport among the surveyed students, with 23 supporters.

It is likely that there would be enough interest to form an ice hockey team, as there were 11 supporters.

We cannot make any definitive inferences about the entire school population, however, so the answer is E. None of the above.

Hence, We cannot make any definitive inferences about the entire school population, so the answer is E. None of the above.

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The table shown below represents the results from a sample group of 50 students

surveyed about a preference for an additional sports team.

Sports Team    Number

Gymnastics       10

Ice hockey         11

Lacrosse             23

Swimming           6

If the school has 500 students, what can be inferred from the survey?

A. There would not be enough students to form a 20-person ice hockey

team.

B. About half of the students prefer lacrosse to the other sports.

C. Fewer than ten students in the school prefer swimming to the other

sports.

D. About 10% of the students prefer gymnastics.

E. None of these

Equivalence means to be same, whether it be value, temperature, size, etc.

First, let's convert [tex]\frac{3}{4}[/tex] into a different fraction.

We know this because each side of the fraction [tex]\frac{3}{4}[/tex] can be multiplied by 25 to get [tex]\frac{75}{100}[/tex].

Now that we know this, we can convert the fraction [tex]\frac{75}{100}[/tex] into a percentage.

A percentage is a ratio, or a number expressed in the form of a fraction of 100. Percentages are often used to express a part of a total.

if we know that percentages are expressed as a fraction of 100, we know that [tex]\frac{75}{100}[/tex] is equal to %75.

To convert percentages into decimals, we can just take the percentage (%75) and fit it into 0.00, making it 0.75.

Therefore, the fraction [tex]\frac{3}{4}[/tex] as a percentage is %75, and as a decimal it is 0.75.

Answer:

Step-by-step explanation: We can start the problem by using the principle of Proportional Equivalents. We can write two equations based on the given information:

Let's say that the rate of filling the pool with one large pump is x and the rate of filling the pool with one small pump is y.

Then, we can write the following two equations:

2x + 1y = 1/4 (because two large and one small pump can fill the pool in 4 hours)

1x + 3y = 1/4 (because one large and three small pumps can fill the pool in 4 hours)

Now we have two equations with two unknowns, x and y. We can solve for x and y using either substitution or elimination method. Let's use substitution method.

Solving for x:

From equation (1), we have 2x = 1/4 - 1y

Substituting this value of x in equation (2), we get:

1 (1/4 - 1y) + 3y = 1/4

Expanding and simplifying, we get:

1/4 - 1 + 3y = 1/4

3y = 1/2

y = 1/6

So, the rate of filling the pool with one small pump is 1/6.

Now that we have found the value of y, we can find the value of x.

x = 1/4 - 1y = 1/4 - 1 * 1/6 = 1/4 - 1/6 = 1/12

So, the rate of filling the pool with one large pump is 1/12.

Now that we have found the rate of filling the pool with each large and small pump, we can find the time it will take 4 large and 4 small pumps to fill the pool.

Let's call the rate of filling the pool with 4 large and 4 small pumps as R.

Then, R = 4 * 1/12 + 4 * 1/6 = 1/3

So, it will take 1/R = 3 hours to fill the pool with 4 large and 4 small pumps.

Answer:

Graph D is Greg's graph

From the resting point it goes up and then it straightens out and then increases again

Let's call the smaller integer "x". Then the larger integer would be "x + 3".

The sum of the reciprocal of the smaller integer and twice the reciprocal of the larger integer is 9/10, so we can write that as:

1/x + 2/(x + 3) = 9/10

Expanding and simplifying the right side:

1/x + 2/(x + 3) = 9/10

(x + 6)/(x * (x + 3)) = 9/10

10(x + 6) = 9x * (x + 3)

10x + 60 = 9x^2 + 27x

9x^2 - 83x + 60 = 0

Now we have a quadratic equation that we can solve using the quadratic formula or by factoring. Since the coefficients are integers, we can try factoring:

(3x - 20)(3x - 3) = 0

So either 3x - 20 = 0 or 3x - 3 = 0, which means either x = 20/3 or x = 3. Since x must be a positive integer, the only solution is x = 3.

So the two integers are 3 and 3 + 3 = 6.

Answer:

To make x the subject of the formula, we can isolate x by subtracting 1 from both sides of the equation:

x + 1 = 4y

x + 1 - 1 = 4y - 1

x = 4y - 1

So x is equal to 4y minus 1.

Answer:

[tex]x = 4y - 1[/tex]

Step-by-step explanation:

Currently x is on the left side of the equal sign and is being added to 1. x has to be isolated and be by itself on one side of the equal sign in order to be made the subject of the formula. Therefore the 1 needs to be moved to the other side of the equal sign:

             [tex]x = 4y - 1[/tex]

Answer:

Below

Step-by-step explanation:

If you draw this diagram you will see a right triangle with legs = height

    and   10 miles

for the 6 degree angle  ( in the right triangle)

   tan ( 6) = opposite leg / adjacent leg

      tan (6) = height / 10

       or    10 * tan 6 = height     ( in miles)

                      1.05 miles high

The value of the expression √4x-3 /9-2x =0 will be B. 2.

Rearrange the radical on the left side of the equation:

√4x-3=√9-2x

Power on both sides: (√4x − 3)²=(√9 – 2x)² -

Simplify the radical expression: 4x-3=9-2x Rearrange unknown terms to the left side of the

equation: 4x+2x=9+3

Combine like terms: 6x=9+3

Calculate the sum or difference: 6x=12

Divide both sides of the equation by the coefficient

of variable: x=12÷6

Calculate the product or quotient: x=2

Substitute into one of the equations: √4 x 2 3-√9-2 × 2=0

Calculate the product or quotient: √8

9-2×2

Calculate the product or quotient: √8 3 - √9 4

Calculate the sum or difference: √5-√9

Calculate the sum or difference: √5-√5

Calculate the sum or difference: 0=0

Combine the results: x=2

x=2

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The total amount of money she spends on rent alone would be = $630

The amount of money that Tosha spends on food = X

The amount of money that Tosha spends of rent = 2x

The total amount of money she spent on both = $945

That is,

945 = X+2x

3x = 945

X = 945/3 = 315

Therefore,the amount she spent on rent = 2x= 2× 315 =$630

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The tape diagram helped to understand by that we can subtract the combined $25 at the concession stand from the total and divide it by 5.

A tape diagram is a rectangular visual model resembling a piece of tape, that is used to assist with the calculation of ratios and addition, subtraction, and commonly multiplication. It is also known as a divided bar model, fraction strip, length model or strip diagram.

Given that, a family of 5 went to a matinee movie on a Saturday afternoon. The movie tickets for the matinee were a special price for each person. The family spent a combined $25 at the concession stand on drinks and popcorn. Altogether, the family spent $57.50 at the movies.

By using the tape diagram we get an equation,

5x+25 = 57.50

5x = 32.5

x = 6.5

Therefore, the special price for each person was $6.5

The figure is attached

Hence, the tape diagram helped to understand by that we can subtract the combined $25 at the concession stand from the total and divide it by 5.

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What is a Right Angle Triangle?

A Right Angle Triangle is a triangle where one of the angles is equal to 90 degrees.

To calculate the values using SOHCAHTOA

 SOHCAHTOA calculate the angles and sides of the right triangle, using trigonometric function sine, cosine, and tangent. There are two triangles in which the adjacent and opposite changes based on which angle is in question, the hypotenuse always stays the same.

Where Sin A = Opposite/ Hypotenuse

            Cos A = Adjacent / Hypotenuse

   And  Tan A = Opposite / Adjacent

For question 1,

If Opposite = PR

Adjacent = PQ

Hypotenuse = QR

Sin A = 14/50

Cos A = 48/50

Tan A = 14/48

For question 2,

If Opposite = PQ

Adjacent = PR

Hypotenuse = QR

Sin A = 48/50

Cos A = 14/50

Tan A = 48/14

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The option that is a factor of the function f(x) is C.

(2x - 3)

Because it becomes zero when x = 3/2.

We know that the function f(x) has zeros at x = -1/2 and x = 3/2.

And we want to see which ones of the given expressions can be a factor of the function.

If the function becomes zero at these values, then the factors need to become zero at one of these two values.

With that in mind, we can see that the correct option is C.

(2x - 3)

when x = 3/2 we get:

(2*3/2 - 3) = (3 - 3) = 0

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Answer: 15 pounds

Step-by-step explanation:

10.4(x)+4.3(10)=7.96(x+10)

10.4x+43=7.96x+79.6

2.44x=36.6

x=15 pounds

(check my work. I may have done something wrong, but the equation should be correct.)