Select all the expressions that are equivalent to the number in the picture

The equivalent expressions of the given expressions are value: ∛-48a⁶b⁹ = ∛- (2)³(6) (a)²b³ = 2a²b³∛-6. [2] √48a⁶b⁹ = √(4)²(3) (a)³b⁴b = 4|a³|b⁴ √3b [3] ∛-48a⁷b⁸ = -2a³b² ∛6ab².

A mathematical expression is a sentence without the equal sign that can include variables, operators, and integers. It can be assessed or simplified, but there isn't a single, perfect answer. An expression is something like 3x + 2y.

On the other hand, an equation is a mathematical statement that uses the equal sign to denote the equivalence of two expressions. You can solve equations to determine the values of the variables that fulfil the equality. An equation is, for instance, 3x + 2y = 7.

The given expressions can be simplified by taking the factors as follows:

∛-48a⁶b⁹ = ∛- (2)³(6) (a)²b³ = 2a²b³∛-6.

√48a⁶b⁹ = √(4)²(3) (a)³b⁴b = 4|a³|b⁴ √3b

∛-48a⁷b⁸ = -2a³b² ∛6ab²

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Answer: 1.5 feet (rounded to the nearest tenth) or 1.4644

Step-by-step explanation:

 For a RIGHT triangle , remember  S-O-H-C-A-H-T-O-A ?

    Cos (Φ) = Adjacent leg / Hypotenuse

  so   cos D =    sqrt (78) / DF =  sqrt (78) / 28

The radius of the spherical boulder is half the diameter, so:

radius = 24 ft / 2 = 12 ft

The formula for the volume of a sphere is:

V = (4/3)πr³

Substituting the given values, we get:

V = (4/3) x 3.14 x (12 ft)³

V = 7238.08 cubic feet

Rounding to the nearest whole number, we get:

V ≈ 7238 cubic feet

Answer:

10c

Step-by-step explanation:

When we have similar terms we can add or subtract them together.

Combine terms is like grouping similar terms together.

Accordingly, let us solve the given question.

5c + 5c

Combine the like terms to simplify the answer.

10c

Unauthorized use of celebrity name and misleading statements are illegal to include in a sales letter.

A sales letter comprises a format that follows a certain set of instruction that should always be implemented to form an ideal sales letter, this set of instruction are as follows

Furthermore, a sales letter provides a basic layout of the needs of the writer and how they can help in providing aid for the needs of the reader.

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a. the sampling distribution is  [tex]\sqrt{[(0.17 \times (1-0.17)) / n]}[/tex]

b. The probability that the sample proportion is within 0.03 of the population proportion is 0.4101.

c. The probability that the sample proportion is within 0.03 of the population proportion for a sample of 200 households is 0.7738.

a. The sampling distribution of the sample proportion, [tex]\bar p[/tex], can be approximated by a normal distribution with mean equal to the population proportion, p, and standard deviation equal to the square root of [tex][(p \times (1-p)) / n][/tex],

where n is the sample size.

Given that p = 0.17 and assuming a large enough sample size, we can use the formula to calculate the standard deviation of the sampling distribution:

Standard deviation = [tex]\sqrt{[(0.17 \times (1-0.17)) / n]}[/tex]

b. To find the probability that the sample proportion will be within a certain range of the population proportion, we need to calculate the z-score for the lower and upper bounds of that range and then find the area under the normal curve between those z-scores.

Let's say we want to find the probability that the sample proportion is within 0.03 of the population proportion. This means we want to find

P([tex]\bar p[/tex]- p ≤ 0.03) = P(([tex]\bar p[/tex]-- p) / [tex]\sqrt{[(p \times (1-p)) / n]}[/tex] ≤ 0.03 / [tex]\sqrt{[(p \times (1-p)) / n][/tex]

We can use the standard normal distribution and z-scores to find this probability:

[tex]z_1[/tex] = (0.03 / [tex]\sqrt{(0.17 \times (1-0.17)/n}[/tex])

[tex]z_2[/tex] = (-0.03 / [tex]\sqrt{(0.17 \times (1-0.17))/n}[/tex])

We can find the probability that the z-score is between [tex]z_1[/tex] and [tex]z_2[/tex]:

P([tex]z_1[/tex]≤ Z ≤ [tex]z_2[/tex]) = P(-0.541 ≤ Z ≤ 0.541) = 0.4101

c. To answer part (c), we need to specify the sample size. Let's say we are taking a sample of 200 households.

Using the formula for standard deviation of the sampling distribution from part (a), we get:

Standard deviation = [tex]\sqrt{(0.17 \times(1-0.17)) / 200}[/tex] = 0.034

Now we can repeat the same steps as in part (b) with this standard deviation:

[tex]z_1[/tex] = (0.03 / 0.034)

[tex]z_2[/tex] = (-0.03 / 0.034)

P([tex]z_1[/tex]≤ Z ≤ [tex]z_2[/tex]) = P(-0.882 ≤ Z ≤ 0.882) = 0.7738

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The equation that gives us the height of point P after the windmill blade rotates by an angle of θ is  h = 1.5 * sin(π/4 + θ).

Here, O is the center of the windmill, P is the point at the tip of the blade, and a is the angle through which the blade has rotated. We are given that the center of the windmill is 6 feet off the ground and the blades are 1.5 feet long.

We know that OP is the length of the blade, which is 1.5 feet. To find the angle a, we need to take into account the starting position of P. We are told that P is currently rotated π/4 radians from the point directly to the right of O. This means that if we rotate the blade by an angle of a, P will end up at an angle of π/4 + a from the point directly to the right of O.

where θ is the angle through which the blade has already rotated.

Putting it all together, we have:

h = 1.5 * sin(π/4 + θ)

This equation gives us the height of point P after the windmill blade rotates by an angle of θ.

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The derivative of the sigmoid function is f(x) × (1 - f(x)).Answer: f(x) × (1 - f(x)).

As per the question, we are supposed to determine the derivative of the sigmoid function.

The sigmoid function is given as,

[tex]f(x) = 1 / (1 + e^-x)[/tex]

To determine the derivative of the sigmoid function, we first find out the derivative of f(x) with respect to x using the quotient rule,

[tex]f'(x) = d/dx [ 1 / (1 + e^-x) ][/tex]

[tex]f'(x) = [ (d/dx)[1] × (1 + e^-x) - 1 × (d/dx)[1 + e^-x] ] / [ (1 + e^-x)^2 ][/tex]

[tex]f'(x) = [ 0 × (1 + e^-x) - (-e^-x) ] / [ (1 + e^-x)^2 ][/tex]

[tex]f'(x) = e^-x / (1 + e^-x)^2[/tex]

Now we are supposed to write the answer in terms of f(x),

[tex]f'(x) = e^-x / (1 + e^-x)^2f(x) = 1 / (1 + e^-x)[/tex]

Substituting f(x) in the above equation,

f'(x) = f(x) × (1 - f(x))

Therefore, the derivative of the sigmoid function is f(x) × (1 - f(x)).Answer: f(x) × (1 - f(x)).

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In light of this, it will take roughly 2.25 years for the populace to reach 3050,that shows growth rate. The equation to calculate the number of fishes in a pond after t years would be equal to t = -ln (0.12125) / r

The logistic development model can be used to calculate an equation for the growth rate of fish P(t) after t years:

Where[tex]: P(t) = K / (1 plus A e(-rt))[/tex]

[tex]r[/tex]= growth rate, where[tex]K =[/tex] carrying capacity [tex]= 6100 A =[/tex] starting population[tex]= 500[/tex]

T equals time in years

Using the given data, we can determine the increase rate:

In the first year, the quantity of fish increased from[tex]500 to 740:[/tex]

[tex]740 = 6100 / (1 + A e^(-r1))[/tex]

We arrive at [tex]A e(-r*1) = 8.24324[/tex] after solving for [tex]A e(-r).[/tex]

Given that P(t) must lie half way between A and K in order for there to be a maximal growth rate, the carrying capacity is [tex]6100:[/tex]

[tex]3050 = (6100 + 500) / 2 = 3300[/tex]

As a result, if [tex]P(t) = 3300,[/tex] we have:

[tex]3300 = 6100 / (1 + A e^(-rt))[/tex]

We arrive at [tex]A e(-rt) = 1.84848[/tex] after solving for [tex]A e(-rt).[/tex]

The following is the result of substituting the numbers of [tex]A e(-r*1)[/tex] and [tex]A e(-rt)[/tex] into the equation for P(t):

We can put into an expression and solve for t to determine how long it will take for the population to reach [tex]3050:[/tex]

[tex]3050 = 6100 / (1 + 8.24324 e^(-rt))[/tex]

[tex]e(-rt) = 0.12125 -rt = ln = 1 + 8.24324 e(-rt) = 2(0.12125)[/tex]

[tex]t = -ln(0.12125) / r[/tex]

Although we are unsure of r's precise value, we do know that [tex]A e(-rt) = 1.84848.[/tex]

at[tex]t = 3300, P(t).[/tex] When we enter these numbers into the solution, we obtain:

[tex]1.84848 = A e^(-r1)[/tex]

[tex]A = 1.84848 / e^(-r1)[/tex]

We obtain the following equation for t by substituting this value of A:[tex]t = -ln(0.12125) / r t = -ln(0.12125) / ln(1.84848 / e(-r*1))[/tex]

The precise value of r is not required to provide an answer, but we can solve for it using a numerical approach like trial and error or a graphing method.

The logistic development model can be used to calculate an equation for the quantity of fish P(t) after t years:

Where: [tex]P(t) = K / (1 plus A e(-rt))[/tex]

r = growth rate, where K = carrying capacity =[tex]6100 A[/tex] = starting population [tex]= 500[/tex]

T equals time in years

Using the given data, we can determine the increase rate:

In the first year, the quantity of fish increased from 500 to 740:

[tex]740 = 6100 / (1 + A e^(-r1))[/tex]

We arrive at A e(-r*1) = 8.24324 after solving for A e(-r).

Given that P(t) must lie half way between A and K in order for there to be a maximal growth rate, the carrying capacity is 6100:

[tex]3050 = (6100 + 500) / 2 = 3300[/tex]

As a result, if P(t) = 3300, we have:

[tex]3300 = 6100 / (1 + A e^(-rt))[/tex]

We arrive at c[tex]A e(-rt) = 1.84848[/tex] after solving for [tex]A e(-rt).[/tex]

The following is the result of substituting the numbers of A e(-r*1) and A e(-rt) into the equation for P(t):

[tex]P(t) = 6100 / (1 + 8.24324 e^(-rt))[/tex]

We can put P(t) = 3050 into an expression and solve for t to determine how long it will take for the population to reach 3050:

[tex]3050 = 6100 / (1 + 8.24324 e^(-rt))[/tex]

[tex]e(-rt) = 0.12125 -rt = ln = 1 + 8.24324 e(-rt) = 2(0.12125)[/tex]

[tex]t = -ln(0.12125) / r[/tex]

Although we are unsure of r's precise value, we do know that[tex]A e(-rt) = 1.84848.[/tex]

a[tex]t t = 3300, P(t).[/tex] When we enter these numbers into the solution, we obtain:

[tex]1.84848 = A e^(-r1)[/tex]

[tex]A = 1.84848 / e^(-r1)[/tex]

We obtain the following equation for t by substituting this value of A:[tex]t = -ln(0.12125) / r t = -ln(0.12125) / ln(1.84848 / e(-r*1))[/tex]

Assuming r = 0.1 as a sample growth rate, we obtain[tex]t = -ln (0.12125) / ln (1.84848 / e(-0.1*1))[/tex]

The precise value of r is not required to provide an answer, but we can solve for it using a numerical approach like trial and error or a graphing calculator. By assuming a growth rate, we can calculate how long it will take for the populace to reach 3050. Assuming r = 0.1 as a sample growth rate, we obtain[tex]t = -ln(0.12125) / ln(1.84848 / e(-0.1*1))[/tex]

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In a cube, a total of 12 lines can be formed.

It has 8 vertices and 12 edges. In geometry, a line is a straight geometric figure that extends indefinitely in both directions. The lines in a cube are the straight lines that connect two vertices or edges of the cube. When two points are connected with a line, it creates an edge. In a cube, there are 12 edges. Each edge is shared by two faces, and the edges are the line segments that make up the sides of the cube.

Thus, the number of lines that can be formed in a cube is 12. When two or more lines meet at a point, it is called a vertex. A cube has eight vertices or corners. When a line is drawn between any two vertices of a cube, it is called a diagonal. In a cube, there are 12 diagonals. A diagonal is a straight line segment that connects two non-adjacent vertices of a polygon. So, the number of lines that can be formed in a cube is 12, which includes the 12 edges.

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The possible whole-number values of 'a' that satisfy the triangle inequality are 2, 3, 4, 5, and 6.

A triangle is a three-sided polygon with three vertices. The triangle's internal angle, which is 180 degrees, is constructed.

To create a triangle with sides a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side. This can be represented by the triangle inequality:

a + b > c

a + 6 > 7 (substituting the given values for b and c)

a > 1

b + c > a

6 + 7 > a

13 > a

a + c > b

a + 7 > 6

a > -1

Since the length of a cannot be negative, the inequality can be simplified to:

a > 1

Therefore, to create a triangle with side lengths of 7, 6, and some whole number value for a, a must be greater than 1. The possible whole-number values of 'a' that satisfy the triangle inequality are 2, 3, 4, 5, and 6.

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According to the information of the graph the correct sentences are: D. Dilate ABC by a scale factor of 1/2 centered at the right angle, and E. Dilate ABC by a scale factor of 2 centered at the right angle.

To show that two triangles are similar, we need to demonstrate that they have the same shape but not necessarily the same size. This can be done using similarity transformations such as dilation, rotation, and reflection. Statements A, B, and C involve rotations which would change the shape of the triangle, therefore they do not demonstrate similarity.

On the other hand, statement D suggests dilating triangle ABC by a scale factor of 1/2 centered at the right angle. This transformation will change the size of the triangle but maintain the same shape, which is a property of similar triangles. Similarly, statement E suggests dilating triangle ABC by a scale factor of 2 centered at the right angle, which again maintains the same shape but changes the size. Therefore, statements D and E demonstrate that triangle ABC and triangle LMN are similar.

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Using multiplication we know that there are 2,00,00,00,00,00,000 red blood cells in the body of a 125-pound person.

One of the four fundamental arithmetic operations, along with addition, subtraction, and division, is multiplication.

A product is the output of a multiplication operation.

When you take a single number and multiply it by several, you are multiplying.

We multiplied the number five by four times.

Due to this, multiplication is occasionally referred to as "times."

So, we have:

= 1.6 * 10¹¹

Solve as follows:

= 1.6 * 10¹¹

= (1.6 * 10¹¹) * 125

= (1.6 * 100000000000) * 125

= 1,60,00,00,00,000 * 125

= 2,00,00,00,00,00,000

Therefore, using multiplication we know that there are 2,00,00,00,00,00,000 red blood cells in the body of a 125-pound person.

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The explaination of the volume of the cylinder calculation is below

To find the volume of a cylinder with a base radius of r units and a height of h units, you can use the formula:

V = πr²h

Where V is the volume of the cylinder, π is a mathematical constant approximately equal to 3.14, r is the radius of the cylinder's base, and h is the height of the cylinder.

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Answer: 7, and it occurs when x = 5 and y = 2

Step-by-step explanation:

1/5 x + 1/3 y = 1 (Equation 1)

x + y = ? (Equation 2)

We want to find the maximum value of x + y, so we need to maximize each of the variables x and y subject to the constraint given in Equation 1.

To solve for one of the variables, we can isolate it in terms of the other variable. Let's solve for y:

1/5 x + 1/3 y = 1

1/3 y = 1 - 1/5 x

y = 3(1 - 1/5 x)

Now we can substitute this expression for y into Equation 2:

x + y = x + 3(1 - 1/5 x)

Expanding the expression, we get:

x + y = x + 3 - 3/5 x

x + y = 3 + 2/5 x

So we can see that x + y is a linear function of x with slope 2/5 and y-intercept 3. To maximize x + y, we want to maximize x. However, x is a non-negative integer, so the maximum value of x occurs when x = 5 (since any larger value would result in a fraction in the expression for y).

Therefore, the maximum value of x + y is:

x + y = 3 + 2/5 x

x + y = 3 + 2/5 (5)

x + y = 5 + 2

x + y = 7

So the maximum value of the sum of these two non-negative integers is 7, and it occurs when x = 5 and y = 2.

The common end point I believe :)

On the basis of similarity of triangles, the triangles in Q9 are similar by AA similarity whereas in Q10 the triangles are similar by SAS similarity.

What is similarity?

Any Two triangles are said to be same or similar if they have the same ratio of the corresponding sides in the triangles and equal pair of corresponding angles in the triangles. If two or more figures or polygons having the same shapes, but their sizes are not same, then those figures or polygons are called similar. To be congruent, the shapes or polygons must have equal angles and equal sides.

Q9)Consider the ΔTUS and ΔUVW

∠STU=∠UVW=(72°)

∠TUS=∠VUW {vertically opposite angles}

The given triangles are similar by AA criteria.

Q10)Consider the ΔKDH and ΔBAD

∠H = ∠A = 38 °

[tex]\frac{KH}{BA} =\frac{DH}{AD}[/tex]

[tex]\frac{20}{28} = \frac{15}{21} = \frac{5}{7}[/tex]

The above triangles are similar by SAS

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Answer:

452.39

Step-by-step explanation:

Used volume formula

Answer:

V = 452.4

Step-by-step explanation:

V = πr²h

(3.14)(6)²(4) = 452.38934211

To better understand your question, it is important to identify the key terms and concepts that are being used.

Title: Understanding Terms in a Question

1. . Look for specific words or phrases that relate to the topic at hand, and try to understand their meaning within the context of the question.

2. It can also be helpful to break the question down into smaller parts or sub-questions, and focus on each one individually.

3. If you are unsure about the meaning of a particular term, do some research or consult a dictionary or other reference source.

4. By taking the time to understand the terms and concepts in a question, you can improve your ability to answer it accurately and effectively.

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Answer:

x ≈ 51°

Step-by-step explanation:

Use trigonometry:

[tex] \sin(x°) = \frac{7}{9} [/tex]

Use the reverse (SHIFT, sin (7/9)) function on your calculator to find x:

x ≈ 51°

Answer:

t=0,  t=4/3

Step-by-step explanation:

can anyone answer my math questions

Answer:

a = 3
b = 8
c = 4

Step-by-step explanation:

We can identify the coefficients a, b, and c by comparing the equation with the standard form of a quadratic equation:

ax^2 + bx + c = 0

(always use this for these kinds of problems)

So, in the given equation:

The coefficient of x^2 is a = 3

The coefficient of x is b = 8

The constant term is c = 4

Options A and B are both correct. Options C, D, and E are not correct, as they do not take into account the discount that James is receiving.

In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. Percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.

The correct calculations to determine how much James will pay in total are:

First, we need to calculate how much 10% of $38.50 is:

10% of $38.50 = 0.1 × $38.50 = $3.85

So, James will receive a discount of $3.85.

To determine how much James will pay in total, we can use either of the following calculations:

A. James' total cost after the 10% discount would be: 38.5 - 0.9(38.5) = 38.5 - 34.65 = $3.85

B. The 10% discount on the total cost can be calculated as 0.1(38.5) = $3.85, which can be subtracted from the total cost to get James' final cost as 38.5 - 3.85 = $34.65

C. James' final cost after the 10% discount would be 1.1(38.5) = $42.35, which is not the correct answer as it does not take into account the discount.

D. James' total cost after the 10% discount would be: 0.9(38.5) = $34.65, which is the correct answer.

E. The calculation 0.1(38.5) = $3.85 represents the amount of the discount, and is not the final cost.

Therefore, options A and B are both correct. Options C, D, and E are not correct, as they do not take into account the discount that James is receiving.

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The upper limit of the confidence interval for the proportion of millennials that care about immigration is 0.20.

In order to find the upper limit of the confidence interval for the proportion of millennials that care about immigration,

we first need to calculate the confidence interval using the margin of error (MOE) and sample proportion.

The formula to find the confidence interval is given by:

Confidence Interval = Sample Proportion ± Margin of Error (MOE)

Where Sample Proportion = Poll Result / 100

Let's first find the Sample Proportion of millennials that care about immigration.

Sample Proportion = Poll Result / 100= 17 / 100= 0.17

Now let's calculate the confidence interval using the MOE.

Confidence Interval = Sample Proportion ± Margin of Error (MOE)

Upper Limit of Confidence Interval = Sample Proportion + MOE= 0.17 + 0.03= 0.20

Therefore, the upper limit of the confidence interval for the proportion of millennials that care about immigration is

0.20.

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Answer: The general equation for a circle with center (h, k) and radius r is:

(x - h)^2 + (y - k)^2 = r^2

To find the equation of the circle that passes through the points (1, 1), (1, 3), and (9, 2), we can use the fact that the perpendicular bisectors of any two chords of a circle intersect at the center of the circle.

The midpoint and slope of the chord connecting (1, 1) and (1, 3) are:

Midpoint: ((1+1)/2, (1+3)/2) = (1, 2)

Slope: undefined (since x values are the same)

The perpendicular bisector of this chord passes through the midpoint (1, 2) and has a slope of 0. Therefore, its equation is:

y - 2 = 0

Simplifying this equation gives:

y = 2

Similarly, the midpoint and slope of the chord connecting (1, 1) and (9, 2) are:

Midpoint: ((1+9)/2, (1+2)/2) = (5, 1.5)

Slope: (2 - 1)/(9 - 1) = 1/8

The perpendicular bisector of this chord passes through the midpoint (5, 1.5) and has a slope of -8 (the negative reciprocal of the slope of the chord). Therefore, its equation is:

y - 1.5 = -8(x - 5)

Simplifying this equation gives:

y = -8x + 41.5

The intersection of the two perpendicular bisectors (y = 2 and y = -8x + 41.5) is the center of the circle. Solving for x gives:

2 = -8x + 41.5

8x = 39.5

x = 4.9375

Substituting x = 4.9375 into either equation for the perpendicular bisectors gives:

y = 2

Therefore, the center of the circle is (4.9375, 2) and the radius is the distance from the center to any of the three given points. For example, the distance from (4.9375, 2) to (1, 1) is:

r = sqrt((4.9375 - 1)^2 + (2 - 1)^2) = sqrt(24.9219) = 4.992

So the equation of the circle is:

(x - 4.9375)^2 + (y - 2)^2 = (4.992)^2

Expanding and simplifying this equation gives:

x^2 - 9.875x + 24.526 + y^2 - 4y + 0.076 = 24.916

Rearranging terms gives:

x^2 + y^2 - 9.875x - 4y - 0.34 = 0

So the general equation for the circle is:

x^2 + y^2 - 9.875x - 4y - 0.34 = 0

Step-by-step explanation:

The answer of the given question based on compass and Straight edge construction is , option (b) False.

A pentagon is a geometric shape that has five straight sides and five angles. It is a polygon, which means it is a closed shape made up of straight line segments. The word "pentagon" come from Greek word "penta" meaning "five" and "gonia" meaning "angle".

There are several types of pentagons, including regular and irregular pentagons. A regular pentagon has five equal sides and angles, while an irregular pentagon has sides and angles of different lengths and measures.

False.

The Greeks believed that all constructions could be made with only a compass and straightedge, but some constructions were more difficult than others. In fact, they were able to make remarkable geometric constructions using only these two tools, such as trisecting an angle, duplicating a cube, and constructing a regular pentagon. However, in the 19th century, the discovery of new mathematical methods showed that some constructions, like squaring the circle, were actually impossible to complete with just a compass and straightedge.

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Answer:

(C) (x)^2 + (y)^2 = 49.

Step-by-step explanation:

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

(x - h)^2 + (y - k)^2 = r^2

In this case, the center of the circle is (0, 0), so h = 0 and k = 0. The radius of the circle is not given, but we can see that the equation must have a radius of 7 because the only terms involving x and y are squared and have coefficients of 1, which means they represent the distance from the center squared.

Therefore, the equation of the circle is:

(x - 0)^2 + (y - 0)^2 = 7^2

Simplifying, we get:

x^2 + y^2 = 49

So the correct answer is (C) (x)^2 + (y)^2 = 49.

Triangle PQR is translated to triangle P'Q'R'. (-3,3) and (-1,1) are the co-ordinates of Q' and R'.

The co-ordinates of Q' and R' when triangle PQR is translated to triangle P'Q'R' are (-3,3) and (-1,1), respectively.

A translation is a type of transformation where a shape is moved from one position to another.

However, the shape is not rotated, resized, or reflected in a translation.

Instead, it is moved in a straight line from one position to another.

A translation in a two-dimensional plane is specified by a vector, which specifies the distance and direction of the translation.

The co-ordinates of Q' and R' when triangle PQR is translated to triangle P'Q'R' are (-3,3) and (-1,1), respectively.

To determine these co-ordinates, we can use the following steps:

Determine the vector of the translation.

To do this, we find the difference between the corresponding vertices of the two triangles.

Here, we can take PQ as the base.

So, P'Q' = QRQR

= PQP' = P + (-1,2)

= (-2,3)Q'

= Q + (-1,2)

= (-3,3)R'

= R + (-1,2)

= (-1,1).

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The values of the missing sides a and c are 3√6/2 and 3√2 respectively using the trigonometric ratio of tan 45° for the right triangle.

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.

The basic trigonometric ratios includes;

sine, cosine and tangent.

using the trigonometric ratio of tan 60°

{recall that cos 45° and sin 45° are equal to √2/2}

cos 45° = 3/c {adjacent/hypotenuse}

c = 3 × 2/√2 {cross multiplication}

c = 3√2

sin 45° = a/3√2 {opposite/hypotenuse}

a = 3√2 × √3/2

a = 3√6/2

Therefore, the values of the missing sides a and c are 3√6/2 and 3√2 respectively using the trigonometric ratio of tan 45° for the right triangle.

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