50 Points! Multiple choice geometry question. Photo attached. Thank you!

a. Triangle 3 is a right triangle.

b. For any triangles that are not right triangles, we can use any two of the sides to create a right triangle by applying the Pythagorean theorem.

a Let's apply the Pythagorean theorem to each triangle:

Triangle 1:

Side lengths: √519 units, 27 units, √210 units

Checking the squares of the side lengths:

(√519)² = 519

27² = 729

(√210)² = 210

In this case, 519 + 210 is not equal to 729. Therefore, Triangle 1 is not a right triangle.

Triangle 2: Side lengths: 21 units, √109 units, √420 units

Checking the squares of the side lengths:

21² = 441

(√109)² = 109

(√420)² = 420

Similarly, the sum of the squares of the two shorter sides should be equal to the square of the longest side if it is a right triangle. However, 441 + 109 is not equal to 420. Therefore, Triangle 2 is not a right triangle.

Triangle 3: Side lengths: √338 units, 26 units, √338 units

Checking the squares of the side lengths:

(√338)² = 338

26² = 676

(√338)² = 338

Here, 338 + 338 is equal to 676, which satisfies the Pythagorean theorem. Therefore, Triangle 3 is a right triangle.

b. For any triangles that are not right triangles, we can use any two of the sides to create a right triangle by applying the Pythagorean theorem.

Let's take Triangle 1 as an example:

Side lengths: √519 units, 27 units, √210 units

Let's choose the first and third side:

(√519)²+ (√210)² = 519 + 210 = 729

Now, we take the square root of 729 to find the length of the missing side:

√729 = 27

By doing this, we have formed a right triangle with side lengths of 27 units, √519 units, and √210 units.

Similarly, you can apply this process to Triangle 2 or any other triangle that is not initially a right triangle to create a right triangle using two of its sides.

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The steps to solving these differential equations are:

1) x^6 - 2x^5 + x^4/2x^2 = (x^4)(x^2) - 2(x^5) + x^4/(2(x^2)) = x^4 (x^2 - 2x + 1/2)

2) Sec^3x + e^xsecx + 1/sec x = Sec^3x + e^x*cscx + cscx

3) cot^2 x = 1/tan^2 x

4) x^2 - 2x^3 + 7/ cube root x = x^2 - 2x^3 + 7 *x

5) y = x^(1/2) - x^2 +2x is the derivative of the function (factored form)

The correct algebraic expression for the reduced price of the computer is p - 250, as stated in option B. It signifies subtracting $250 from the original price, 'p.' Option B

To determine the algebraic expression that represents the reduced price of a computer, let's consider the given information. It states that the original price of the computer was reduced by $250.

Let's represent the original price of the computer as 'p' in dollars. Since the price was reduced by $250, we need to subtract $250 from the original price.

The expression that represents the reduced price is obtained by subtracting $250 from 'p.' Therefore, the algebraic expression for the reduced price is p - 250, which corresponds to option B.

Option A, 250 + p, suggests adding $250 to the original price, which would result in a higher value, contrary to the information given.

Option C, 250 - p, represents subtracting the original price from $250, which is not correct as we are reducing the price, not subtracting it from a fixed value.

Option D, 250p, represents multiplying the original price by $250, which does not align with the given scenario of reducing the price by $250.

Option B

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Note: The complete question is:

The original price of a computer was reduced by $250.If p = the computer's original price in dollars, which algebraic expression represents the reduced price? A.250 + p B.p – 250 C.250 – p D.250p

To find the area of the combined rectangles, we need the dimensions (length and width) of each rectangle. However, the provided text and numbers do not seem to correspond to a clear description of the rectangles or their dimensions. Could you please provide more specific information or clarify the question?

Answer:

[tex]r'(1)=\dfrac{1}{16}[/tex]

Step-by-step explanation:

Find the derivative of the following function, then evaluate the function at a point.

[tex]r=\dfrac{1}{\sqrt{5-\theta} } ; \ r'(1)=??[/tex]

[tex]\hrulefill[/tex]

Taking the derivative of the function, r. Start by applying exponent rules.

[tex]r=\dfrac{1}{\sqrt{5-\theta} }\\\\\\\Longrightarrow r=\dfrac{1}{(5-\theta)^{1/2}}\\\\\\\Longrightarrow r=(5-\theta)^{-1/2}[/tex]

Now we can derive the function. Using the chain rule and power rule:

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{The Chain Rule:}}\\\\\dfrac{d}{dx}\Big[f(g(x))\Big]=f'(g(x))\cdot g'(x) \end{array}\right}[/tex]

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{The Power Rule:}}\\\\\dfrac{d}{dx}\Big[x^n\Big]=nx^{n-1} \end{array}\right}[/tex]

[tex]r=(5-\theta)^{-1/2}\\\\\\\Longrightarrow r'=-\dfrac{1}{2} (5-\theta)^{-1/2-1} \cdot -1\\\\\\\therefore \boxed{\boxed{r'=\dfrac{1}{2} (5-\theta)^{-3/2}}}[/tex]

Thus, the derivative of the function is found.[tex]\hrulefill[/tex]

Now evaluating the function when θ=1.

[tex]r'=\dfrac{1}{2} (5-\theta)^{-3/2}\\\\\\\Longrightarrow r'(1)=\dfrac{1}{2} (5-1)^{-3/2}\\\\\\\Longrightarrow r'(1)=\dfrac{1}{2} (4)^{-3/2}\\\\\\\Longrightarrow r'(1)=\dfrac{1}{2}\Big(\dfrac{1}{8} \Big)\\\\\\\therefore \boxed{\boxed{r'(1)=\frac{1}{16} }}[/tex]

Thus, the problem is solved.

The function f(x) = √3x - 3 could be the one graphed.

From the given information, we can deduce the following about the graph of the square root function:

The graph is a square root function, which means it has the form f(x) = √(ax + b), where a and b are constants.

The graph is shifted horizontally by a certain amount.

The graph is shifted vertically by a certain amount.

Using this information, we can analyze each function option to determine if it matches the graph:

Of(x) = √x - 3: This function is a square root function shifted vertically downward by 3 units. It does not match the given graph, as it is shifted vertically, not horizontally.

Of(x) = √x - 1: This function is a square root function shifted vertically downward by 1 unit. It also does not match the given graph, as it is shifted vertically, not horizontally.

Of(x) = √x + 1: This function is a square root function shifted vertically upward by 1 unit. It does not match the given graph, as it is shifted vertically, not horizontally.

f(x) = √3x - 3: This function is a square root function with a coefficient of 3 in front of x, causing a horizontal compression. It is also shifted vertically downward by 3 units. This function matches the given graph, as it represents a horizontal and vertical shift of the square root function.

Therefore, the function f(x) = √3x - 3 could be the one graphed.

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Cosine of < Q

cos(Q) = (√42) / (√91)

To find the cosine of angle Q, we need to determine the ratio of the adjacent side to the hypotenuse in a right triangle.

Let's consider a right triangle with sides S, √42, and √91, where angle Q is the angle between the side S and the hypotenuse.

- The side adjacent to angle Q is S.

- The hypotenuse of the triangle is √91.

cos(Q) = adjacent side / hypotenuse

cos(Q) = S / √91

To simplify the expression, we multiply both the numerator and denominator by √91:

cos(Q) = (S * √91) / (√91 * √91)

cos(Q) = (S * √91) / 91

In the given problem, the values of S, √42, and √91 were provided. Substitute the corresponding value of S into the expression above to obtain the simplified, rationalized form of the cosine of angle Q.

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The equation W = 32 - 0.05n represents the volume of water used, the glass tank's volume will be equal to W plus the volume occupied by the marbles.

In the given equation W = 32 - 0.05n, W represents the volume of water (in liters) Andrei uses, and n represents the number of marbles used. We can interpret this equation as follows: as Andrei increases the number of marbles used (n), the volume of water required (W) decreases.

To determine the glass tank's volume, we need to consider the relationship between the volume of water used and the number of marbles used. We can rearrange the equation to solve for n:

W = 32 - 0.05n

0.05n = 32 - W

n = (32 - W) / 0.05

The volume of the glass tank can be inferred from the equation, as the total volume will be the sum of the volume occupied by marbles and the volume occupied by water.

If we assume that the volume occupied by the marbles is constant, then the remaining volume in the tank will be filled with water. Therefore, the glass tank's volume can be represented as:

Glass Tank Volume = Volume of Water (W) + Volume of Marbles

The given equation W = 32 - 0.05n represents the volume of water used based on the number of marbles, but it does not directly provide information about the glass tank's volume. Further information about the marbles or the tank's dimensions is required to determine the glass tank's volume accurately.

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According to the information, we can infer that the frog must take 4 hops to travel the same distance as a cat's jump.

Since the cat's jump covers a distance of 1 meter, we need to determine how many hops the frog needs to take to cover the same distance.

The frog's jump covers 1/4 meter, and we want to find the number of hops needed to reach 1 meter (the distance of the cat's jump).So, to do this, we can set up a proportion:

By cross-multiplying, we have:

So, the correct answer is 4 because the frog has to jump 4 times to cover the same distance of the rabbit.

Note: This question is in a different language. Here is in English:

How many hops must the frog take to travel the same distance as a cat's height?

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Answer:

An arithmetic sequence with a common difference of −2 is 20,18,16,14,12..

Step-by-step explanation:

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is same. Here, the common difference is -2, which means that each term in the sequence is obtained by subtracting 2 from the previous term.

To find the arithmetic sequence with a common difference of -2, you can start with an first term and then subtract 2 successively to find the subsequent terms.

Let the initial term is 20. Subtracting 2 from 20, we get 18. Subtracting 2 from 18, we get 16. Continuing this pattern, we subtract 2 from each subsequent term to generate the sequence. The arithmetic sequence with a common difference of -2 starting from 20 is

20,18,16,14,12

In this sequence, each term is obtained by subtracting 2 from the previous term, resulting in a common difference of -2.

The correct step that comes after constructing a circle when constructing an inscribed square is to use a straightedge to draw a diameter of the circle (option C).

When constructing an inscribed square by hand, the step that comes after constructing a circle is to use a straightedge to draw a diameter of the circle. Therefore, the correct answer is C.

To understand why drawing a diameter comes after constructing a circle, let's review the steps involved in constructing an inscribed square:

1. Start by constructing a circle: To do this, you would use a compass and a fixed point as the center to draw a circle.

2. Draw a diameter of the circle: A diameter is a line segment that passes through the center of the circle and divides it into two equal parts. Using a straightedge, you can draw a straight line that passes through the center of the circle.

3. Find the midpoint of the diameter: The midpoint is the point on the diameter that divides it into two equal parts. You can use a compass or measure the distance from each end of the diameter to find the midpoint.

4. Draw a perpendicular bisector: With the midpoint as the center, use a compass to draw an arc that intersects the diameter on both sides. This arc will create two points on the diameter.

5. Connect the points: Use a straightedge to connect the two points on the diameter. This line segment will be one side of the inscribed square.

6. Repeat the process: Repeat steps 2 to 5 to draw the other three sides of the square, using the circle as a guide.

By drawing a diameter of the circle, you establish a reference line that will be the base for constructing the sides of the inscribed square. It allows you to accurately position the square within the circle and ensure that its vertices lie on the circumference.

Therefore, the correct step that comes after constructing a circle when constructing an inscribed square is to use a straightedge to draw a diameter of the circle (option C).

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The linear relationship illustrated in the provided table can be effectively described by the equation Y = -4x + 2. Option D.

To determine the equation that represents the given table with the values of x and y, we can observe the pattern and find the equation of the line that fits these points.

Given the table:

X: 2 0 4

Y: -4 3 1

We can plot these points on a graph and see that they form a straight line.

Plotting the points (2, -4), (0, 3), and (4, 1), we can see that they lie on a line that has a negative slope.

Based on the given options, we can now evaluate each equation to see which one represents the line:

Y = 1/2x + 5

When we substitute the x-values from the table into this equation, we get the following corresponding y-values: -3, 5, and 6. These values do not match the given table, so this equation does not represent the table.

Y = -1/2x + 3

When we substitute the x-values from the table into this equation, we get the corresponding y-values: 4, 3, and 2. These values also do not match the given table, so this equation does not represent the table.

Y = 2x - 3

When we substitute the x-values from the table into this equation, we get the corresponding y-values: -4, -3, and 5. These values do not match the given table, so this equation does not represent the table.

Y = -4x + 2

When we substitute the x-values from the table into this equation, we get the corresponding y-values: -6, 2, and -14. Interestingly, these values match the y-values in the given table. Therefore, the equation Y = -4x + 2 represents the table.

In conclusion, the equation Y = -4x + 2 represents the linear relationship described by the given table. So Option D is correct.

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The correct answer is: Starting with one, every consecutive line has two more marbles than the previous line. This is the result inferred using deductive reasoning.

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Answer: rotation of 90 counter clockwise about the origin

Step-by-step explanation:

In 12 years, Maxwell will deposit $290 into the savings account each month.

To calculate how much money Maxwell will deposit into the account each month in 12 years, we need to determine the pattern of increasing deposits over time.

Maxwell deposits $125 into the savings account each month this year, which we can consider as Year 1. Starting from Year 2, he plans on increasing the monthly deposit by $15.

In Year 2, the monthly deposit will be $125 + $15 = $140.

In Year 3, the monthly deposit will be $140 + $15 = $155.

This pattern continues, increasing the deposit by $15 each year.

Therefore, in Year 12, the monthly deposit will be $125 + ($15 * 11) = $125 + $165 = $290.

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Rhe point-slope equation of a line with the labeled point (3, -4) is y + 4 = m(x - 3), where 'm' represents the slope of the line.

To find the point-slope equation of a line using the coordinates of a labeled point, you can use the following formula:
y - y₁ = m(x - x₁)
In this formula, (x₁, y₁) represents the coordinates of the labeled point, and m represents the slope of the line.
Given the coordinates (3, -4) of the labeled point, we can substitute these values into the formula:
y - (-4) = m(x - 3)
Simplifying this equation, we get:
y + 4 = m(x - 3)
This is the point-slope equation of the line.
Now, it's important to note that the problem does not provide information about the slope of the line. Therefore, we cannot determine the exact point-slope equation without knowing the slope. The point-slope equation requires the slope value to be defined.
If you have the slope of the line, let's say it is represented by the variable 'm', you can substitute that value into the equation to get the specific point-slope equation. For example, if the slope is 2, the equation becomes:
y + 4 = 2(x - 3)

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To determine the type of correlation between two variables on a graph, we can examine the pattern or relationship exhibited by the data points.

If the data points on the graph form a roughly linear pattern with a positive slope, it indicates a positive correlation. This means that as one variable increases, the other variable also tends to increase.

Conversely, if the data points on the graph form a roughly linear pattern with a negative slope, it indicates a negative correlation. In this case, as one variable increases, the other variable tends to decrease.

Additionally, the strength of the correlation can be assessed by how closely the data points align with the overall trendline. If the points are tightly clustered around the trendline, it suggests a strong correlation, while scattered points indicate a weaker correlation.

By visually inspecting the graph and observing the direction and pattern of the data points in relation to the trendline, we can determine the type of correlation exhibited by the variables.

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The solution to the equation 2(x - 6) + 25 = 49 - 2x is x = 9.

To solve the equation 2(x - 6) + 25 = 49 - 2x for x, we will simplify and isolate the variable x.

Let's start by simplifying both sides of the equation:

2(x - 6) + 25 = 49 - 2x

Expanding the parentheses:

2x - 12 + 25 = 49 - 2x

Combining like terms:

2x + 13 = 49 - 2x

Now, let's isolate the variable x by moving the terms involving x to one side of the equation. We can do this by adding 2x to both sides:

2x + 2x + 13 = 49 - 2x + 2x

Simplifying:

4x + 13 = 49

Next, we'll get rid of the constant term on the left side by subtracting 13 from both sides:

4x + 13 - 13 = 49 - 13

Simplifying:

4x = 36

To solve for x, we'll divide both sides of the equation by 4:

4x/4 = 36/4

Simplifying:

x = 9

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The correct answer is: OD. (x+7)(x-7)

The correct answer is (C) SSS Similarity. The ratio of AA' to AB will be constant and equal to the scale factor. Similarly, the ratio of A'B' to AB will also be equal to the scale factor.

The correct answer is (C) SSS Similarity.

In a dilation, the image of a figure is created by enlarging or reducing it while maintaining the same shape. The scale factor determines the ratio of the lengths of corresponding sides between the original figure and its image.

In the given problem, we have AABC as the original figure, and AA'B'C' as its image under a dilation with the center at (0,0). To determine the scale factor, we need to compare the corresponding side lengths of the two figures.

In SSS (Side-Side-Side) similarity, we compare the ratios of the corresponding side lengths. If all corresponding side lengths have the same ratio, the figures are similar. In this case, we can compare the lengths of AA', AB, and A'B' to determine the scale factor.

Since the center of dilation is at (0,0), AA' and AB are radii of the same circle centered at (0,0). Therefore, the ratio of AA' to AB will be constant and equal to the scale factor. Similarly, the ratio of A'B' to AB will also be equal to the scale factor.

Hence, by comparing the lengths of corresponding sides AA', AB, and A'B', we can determine the scale factor. If all three ratios are equal, which is the case for SSS similarity, then the scale factor remains constant.

Therefore, the correct answer is (C) SSS Similarity.The correct answer is (C) SSS Similarity.

In a dilation, the image of a figure is created by enlarging or reducing it while maintaining the same shape. The scale factor determines the ratio of the lengths of corresponding sides between the original figure and its image.

In the given problem, we have AABC as the original figure, and AA'B'C' as its image under a dilation with the center at (0,0). To determine the scale factor, we need to compare the corresponding side lengths of the two figures.

In SSS (Side-Side-Side) similarity, we compare the ratios of the corresponding side lengths. If all corresponding side lengths have the same ratio, the figures are similar. In this case, we can compare the lengths of AA', AB, and A'B' to determine the scale factor.

Since the center of dilation is at (0,0), AA' and AB are radii of the same circle centered at (0,0). Therefore, the ratio of AA' to AB will be constant and equal to the scale factor. Similarly, the ratio of A'B' to AB will also be equal to the scale factor.

Hence, by comparing the lengths of corresponding sides AA', AB, and A'B', we can determine the scale factor. If all three ratios are equal, which is the case for SSS similarity, then the scale factor remains constant.

Therefore, the correct answer is (C) SSS Similarity.

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The correct answer is: d. x → -1 Simplifying further, we get: [tex]\frac{1}{2}+7[/tex] Which equals [tex]\frac{15}{2}[/tex]. d. x → -1

To find the limit of the expression [tex](\frac{x^2}{2} ) - (\frac{7}{x} )[/tex] as x approaches a specific value, we substitute that value into the expression.

a. x → 0:

[tex]lim (\frac{x^2}{2} ) - (\frac{7}{x} ) x \rightarrow 0[/tex]

Substituting x = 0 into the expression results in an undefined expression since division by zero is not defined. Therefore, the limit in this case is undefined.

b. x → 0:

[tex]lim (\frac{x^2}{2} ) - (\frac{7}{x} ) x \rightarrow 0[/tex]

Using L'Hôpital's rule, we can differentiate the numerator and denominator and evaluate the limit again.

[tex]lim[/tex]([tex]\frac{2x}{2}[/tex]) - ([tex]\frac{7}{1}[/tex]) as x → 0

Simplifying further, we get:

lim x - 7 as x → 0

Substituting x = 0, we find that the limit is -7.

c. x → 3/14:

lim ([tex](\frac{x^2}{2} ) - (\frac{7}{x} ) x\rightarrow \frac{3}{14}[/tex]

Substituting x = 3/14 into the expression gives:

[tex]\frac{\frac{3}{14}^2 }{2}-\frac{ 7}{\frac{3}{14} }[/tex]

Simplifying this expression, we find that the limit evaluates to [tex]\frac{-77}{6}[/tex]

d. x → -1:

[tex]lim (\frac{x^2}{2} ) - (\frac{7}{x} ) x \rightarrow 0[/tex]

Substituting x = -1 into the expression gives:

[tex]\frac{-1^2}{2} - (\frac{7}{-1} )[/tex]

Simplifying further, we get:

[tex]\frac{1}{2}+7[/tex]

Which equals [tex]\frac{15}{2}[/tex]

Therefore, the correct answer is:

d. x → -1

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The important variables are the two types of test questions which can be represented as :

Variables are used to represent unknown values which could be worked out in a mathematical expression or problem.

The variables or unknown in this case are the type of test questions. which are : m for multiple choice, s for short answer

Therefore, the correct option is C. m for multiple choice, s for short answer

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We need to apply a translation of 3 units up.-

We can see that both of them have almost the same slope, but the line d starts at y = 0, while the data starts at y = 3.

Then we could apply a translation up of 3 units, so if the line is:

d = f(x)

Then the translated line is:

d = f(x) + 3

And that translates the line 3 units upwards.

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The cost of packaging per weight unit is x + 1.

To derive this answer, we first need to understand the given information. The cost of packing a box of chocolates is given by [tex]x^2[/tex], where x is the number of chocolates. However, we also know that a box can never have fewer than 3 chocolates.

Now, let's calculate the weight of a box of chocolates. It is given by x + 2.

To find the cost of packaging per weight unit, we need to divide the cost of packing by the weight of the box. Therefore, the cost of packaging per weight unit can be calculated as ([tex]x^2[/tex]) / (x + 2).

Simplifying this expression, we can rewrite it as ([tex]x^2[/tex]) / (x + 2) = (x + 1) - 1 / (x + 2).

Hence, the cost of packaging per weight unit is x + 1, which means that for every unit of weight, the cost of packaging is x + 1.

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Answer:

The area of this triangle is (1/2)(9)(8) = 36.

Answer:

The 2nd option is correct, a 90 degree counterclockwise rotation about the origin and then a reflection across the y-axis

Step-by-step explanation:

Answer:

V = 4(5)(7) + 6(5)(15) = 140 + 450 = 590 cm³

Answer:

Minimum=60

First Quartile(Q1)=79

Median=86

Third Quartile (Q3)=89

Interquartile range (IQR)=10

The range of the exponential function is B. y > 0.

The range of the exponential function f(x) = 11 × (1/3)ˣ can be determined by analyzing the behavior of the function as x approaches positive and negative infinity.

As x approaches positive infinity, (1/3) becomes smaller and smaller, tending towards zero.

f(x) approaches 11 × 0, which is equal to 0.

As a result, the function approaches 0 as x goes to infinity.

On the other hand, as x approaches negative infinity, (1/3)ˣ becomes larger and larger.

Since 1/3 is between 0 and 1, raising it to a negative power causes it to grow exponentially.

f(x) approaches infinity as x goes to negative infinity.

Combining these two behaviors, we can conclude that the range of the function f(x) = 11 × (1/3)ˣ is all positive real numbers, excluding zero.

In other words, the range is y > 0.

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