Use the following models to show the equivalence of the fractions 35 and 610 a) Set model

Answer:

f^(-1)(x) = ±√(x - 5).

Step-by-step explanation:

Replace f(x) with y: y = x^2 + 5.

Swap the x and y variables: x = y^2 + 5.

Solve the equation for y. To do this, we'll rearrange the equation:

x - 5 = y^2.

Take the square root of both sides (considering both positive and negative square roots):

±√(x - 5) = y.

Swap y and x again to express the inverse function:

f^(-1)(x) = ±√(x - 5).

The length of the hypotenuse in the isosceles 90-degree triangle is √(2).

In an isosceles 90-degree triangle, two legs are equal in length, and the third side, known as the hypotenuse, is longer. Let's denote the length of the legs as x and the length of the hypotenuse as 4x.

According to the Pythagorean theorem, in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. In this case, we have:

[tex]x^2 + x^2 = (4x)^2.[/tex]

Simplifying the equation:

[tex]2x^2 = 16x^2.[/tex]

Dividing both sides of the equation by [tex]2x^2[/tex]:

[tex]1 = 8x^2.[/tex]

Dividing both sides of the equation by 8:

[tex]1/8 = x^2[/tex].

Taking the square root of both sides of the equation:

x = √(1/8).

Simplifying the square root:

x = √(1)/√(8),

x = 1/(√(2) * 2),

x = 1/(2√(2)).

Therefore, the length of each leg in the isosceles 90-degree triangle is 1/(2√(2)), and the length of the hypotenuse is 4 times the length of each leg, which is:

4 * (1/(2√(2))),

2/√(2).

To simplify the expression further, we can rationalize the denominator:

(2/√(2)) * (√(2)/√(2)),

2√(2)/2,

√(2).

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Statement that is always true about obtuse triangles:

An obtuse triangle always has one angle that measures more than 90 degrees.

In the given tree diagram, the "Obtuse" category represents triangles with at least one obtuse angle.

An obtuse angle is an angle that measures more than 90 degrees. Since an obtuse triangle is defined as having one obtuse angle, it will always have an angle that measures more than 90 degrees.

Therefore, the statement that an obtuse triangle always has one angle that measures more than 90 degrees is always true.

Statement that is sometimes true about obtuse triangles:

An obtuse triangle can have different side lengths.

In the given tree diagram, the "Obtuse" category represents triangles with at least one obtuse angle.

The "Scalene" category represents triangles with different side lengths. Therefore, it is possible for an obtuse triangle to have different side lengths, making the statement "An obtuse triangle can have different side lengths" sometimes true.

However, it is also possible for an obtuse triangle to have two or more sides with the same length, which would make it an isosceles or equilateral triangle.

Hence, the statement is only sometimes true and not always true.

In summary, an always true statement about obtuse triangles is that they always have one angle that measures more than 90 degrees.

A sometimes true statement about obtuse triangles is that they can have different side lengths.

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The mean is the measure that gives the most accurate picture of the data's center. It is an essential measure of central tendency that represents the arithmetic average of a dataset.

It is calculated by summing up all the values in the dataset and dividing the sum by the total number of values. The mean is suitable for datasets that have a normal or symmetrical distribution.

The mean is highly sensitive to outliers, which can significantly influence the average value. When outliers are present, it is appropriate to use other measures of central tendency such as the median or mode to obtain an accurate picture of the data's center.

The median is the middle value in a dataset arranged in ascending or descending order. It is not affected by outliers and is suitable for datasets with skewed distributions.

The mode is the most frequent value in the dataset. It is suitable for categorical data but can also be used for continuous data.

In summary, the mean is the most accurate measure of central tendency, but its accuracy can be improved by using the median or mode in datasets with outliers or skewed distributions.

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Answer:

None of the given options (A, B, C, D) match the correct investment amount.

Explaination:

A = P * e^(rt),

where:

A = the future amount (in this case, $34,000),

P = the principal amount (the initial investment),

e = Euler's number (approximately 2.71828),

r = the interest rate (2.9% expressed as a decimal, so 0.029),

t = the time period (18 years).

We can rearrange the formula to solve for P:

P = A / e^(rt).

Now we can plug in the given values and calculate the investment amount:

P = $34,000 / e^(0.029 * 18).

Using a calculator, we can evaluate e^(0.029 * 18) and divide $34,000 by the result to find the investment amount.

Calculating e^(0.029 * 18) gives us approximately 1.604.

P = $34,000 / 1.604 ≈ $21,179.55

Answer:

Sure. First, we need to find the inverse function of f(x). We can do this by using the following steps:

1. Let y = f(x).

2. Solve the equation y = 4x3 + 6x - 10 for x.

3. Replace x with y in the resulting equation.

This gives us the following inverse function:

```

f^-1(y) = (-1 + sqrt(1 + 12y)) / 2

```

Now, we need to find f^-1′ (0). This is the derivative of the inverse function evaluated at y = 0. We can find this derivative using the following steps:

1. Use the chain rule to differentiate f^-1(y).

2. Evaluate the resulting expression at y = 0.

This gives us the following:

```

f^-1′ (0) = (3 * (1 + 12 * 0) ^ (-2/3)) / 2 = 1.5

```

Therefore, f^-1′ (0) = 1.5.

Step-by-step explanation:

Jacob bought 0.7 yards of ribbon.

To find out how many yards of ribbon Jacob bought, we need to determine 4/5 of the length of ribbon that Cindy bought.

Cindy bought 7/8 yard of ribbon. To find 4/5 of this length, we multiply 7/8 by 4/5:

(7/8) * (4/5) = (7 * 4) / (8 * 5) = 28/40

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 4:

(28/4) / (40/4) = 7/10

Therefore, Jacob bought 7/10 yard of ribbon.

However, we can convert this fraction to a mixed number or decimal to express it in yards.

To convert 7/10 to a mixed number, we divide the numerator (7) by the denominator (10):

7 ÷ 10 = 0 with a remainder of 7

So, 7/10 is equivalent to 0 7/10 or 0.7 yards.

Therefore, Jacob bought 0.7 yards of ribbon.

In summary, Jacob bought 7/10 yard or 0.7 yards of ribbon.

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Step-by-step explanation:

To solve this problem, we need to use the product rule of differentiation and some trigonometric identities. Let's start by finding the derivative of y with respect to x:

y = (sin 2x) √(3+2x)

Using the product rule, we get:

dy/dx = (sin 2x) d/dx(√(3+2x)) + (√(3+2x)) d/dx(sin 2x)

To find these derivatives, we need to use the chain rule and the derivative of sin 2x:

d/dx(√(3+2x)) = (1/2√(3+2x)) d/dx(3+2x) = (1/√(3+2x))

d/dx(sin 2x) = 2cos 2x

Substituting these values, we get:

dy/dx = (sin 2x) / √(3+2x) + 2cos 2x (√(3+2x))

Now, we need to simplify this expression to the desired form. To do that, we can use the trigonometric identity:

sin 2x = 2sin x cos x

Substituting this value, we get:

dy/dx = 2sin x cos x / √(3+2x) + 2cos 2x (√(3+2x))

Now, we can use the trigonometric identity:

cos 2x = 1 - 2sin^2 x

Substituting this value, we get:

dy/dx = 2sin x cos x / √(3+2x) + 2(1 - 2sin^2 x)(√(3+2x))

Simplifying further, we get:

dy/dx = (2cos x - 4cos x sin^2 x) / √(3+2x) + 2√(3+2x) - 4sin^2 x√(3+2x)

Now, we can see that this expression matches the desired form:

dy/dx = sin 2x + (4 + Bx)cos 2x / √(3+2x)

where A = -4 and B = -2. Therefore, we have shown that:

dy/dr = sin 2x + (4 - 2x)cos 2x / √(3+2x)

where A = -4 and B = -2.

The approximate slope of the graph of the linear function is 0.15.

To find the approximate slope of the graph of the linear function, we can choose two points from the table and calculate the slope using the formula:

slope = (change in y) / (change in x)

Let's select the points (50, 7.5) and (250, 37.5) from the table.

Change in y = 37.5 - 7.5 = 30

Change in x = 250 - 50 = 200

slope = (change in y) / (change in x) = 30 / 200 = 0.15

Note: A linear function is a mathematical function that represents a straight line.

It can be written in the form:

f(x) = mx + b

where m is the slope of the line and b is the y-intercept (the point where the line intersects the y-axis).

The slope (m) determines the steepness or slant of the line.

A positive slope indicates an upward-sloping line, while a negative slope indicates a downward-sloping line.

The slope represents the rate of change of the function.

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The indicated operation is the composition of functions. To perform this operation, we substitute the expression for g(x) into f(x). The composition of f(g(x)) is given by f(g(x)) = (4x - 1)^2 + 2.

To compute f(g(x)), we first evaluate g(x) by substituting x into the expression for g(x): g(x) = 4x - 1. Next, we substitute this result into f(x): f(g(x)) = f(4x - 1).

Now, let's expand and simplify f(g(x)):

f(g(x)) = (4x - 1)^2 + 2

        = (4x - 1)(4x - 1) + 2

        = 16x^2 - 8x + 1 + 2

        = 16x^2 - 8x + 3.

The domain of f(g(x)) is the same as the domain of g(x) since the composition involves g(x). In this case, g(x) is defined for all real numbers. Therefore, the domain of f(g(x)) is also all real numbers.

In summary, the composition of f(g(x)) is given by f(g(x)) = 16x^2 - 8x + 3, and the domain of f(g(x)) is all real numbers.

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The circles are similar because you can translate Circle 1 using the transformation rule (9, 5) and then dilate it using a scale factor of 2.

To prove that Circle 1 and Circle 2 are similar, we need to identify the transformations that can be applied to Circle 1 to obtain Circle 2.

First, let's consider the translation of Circle 1. The translation rule is given by (a, b), where a represents the horizontal shift and b represents the vertical shift.

In this case, to translate Circle 1 to align with Circle 2, we need to shift it 9 units to the right and 5 units up. Therefore, the translation rule for Circle 1 is (9, 5).

Next, let's consider the dilation. A dilation is a transformation that changes the size of the figure but preserves its shape. The scale factor, denoted by k, determines the amount of scaling. In this case, Circle 1 needs to be dilated to match the size of Circle 2.

The scale factor can be determined by comparing the radii of the two circles. The radius of Circle 1 is 3 centimeters, while the radius of Circle 2 is 6 centimeters. The scale factor is obtained by dividing the radius of Circle 2 by the radius of Circle 1: 6/3 = 2.

Therefore, the transformation applied to Circle 1 to prove that the circles are similar is a translation by (9, 5) followed by a dilation with a scale factor of 2.

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Answer:

550 gallons

Step-by-step explanation:

Let [tex]x[/tex] be the number of gallons for the 90% antifreeze solution and [tex]x+100[/tex] be the total number of gallons that will contain 80% antifreeze solution:

[tex]\displaystyle \frac{0.90x+0.25(100)}{x+100}=0.80\\\\0.90x+25=0.80x+80\\\\0.10x+25=80\\\\0.10x=55\\\\x=550[/tex]

Therefore, you would need 550 gallons of the 90% antifreeze solution.

Answer:

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Step-by-step explanation:

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Answer:

[tex](x +5)^2+(y-1)^2=25[/tex]

Step-by-step explanation:

To determine the equation of the graphed circle, we need to find the coordinates of its center and the length of its radius.

The center of the circle is a single point that lies at an equal distance from all points on the circumference of the circle.

From inspection of the graphed circle, we can see that its domain is [-10, 0] and its range is [-4, 6].  The x-coordinate of the center is the midpoint of the domain, and the y-coordinate of the center is the midpoint of the range.

[tex]x_{\sf center}=\dfrac{-10+0}{2}=-5[/tex]

[tex]y_{\sf center}=\dfrac{-4+6}{2}=1[/tex]

Therefore, the center of the circle is (-5, 1).

The radius of the circle is the distance from the center to all points on the circumference of the circle. Therefore, to calculate the length of the radius, find the distance between x-coordinate of the center and one of the endpoints of the domain.

[tex]r=0-(-5)=5[/tex]

Therefore, the radius of the circle is r = 5.

To determine the equation of the circle, substitute the center and radius into the standard formula.

[tex]\boxed{\begin{minipage}{4 cm}\underline{Equation of a circle}\\\\$(x-h)^2+(y-k)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(h, k)$ is the center. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

As h = -5, k = 1 and r = 5, then:

[tex](x - (-5)^2+(y-1)^2=5^2[/tex]

[tex](x +5)^2+(y-1)^2=25[/tex]

Therefore, the equation of the graphed circle is:

[tex]\boxed{(x +5)^2+(y-1)^2=25}[/tex]

The ordered pair that is a solution for both inequalities is (-2, 2).

Here we have the system of inequalities:

y < -x + 1

y > x

And we want to see which one of the given points makes both of them true.

To find that, just replace the values in both inequalities and see if both become true or not.

For example, for the first point:

(-3, 5)

We will get:

5 < -(-3) + 1 = 4

5 > -3

The first one is false, and the second one is true.

The correct option is the second point:

2 < -(-2) +1 = 3

2 > -2

Both are true.

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Answer:

volume=60cm3, surface area=96cm2

Step-by-step explanation:

volume=1/3×(6×6)×5

=60cm3

surface area= 4(1/2×6×5)+(6×6)

=96cm2

The value of a is 8 ln 9 - 36. Given an arithmetic sequence that has the first term Ina and a common difference In 3. The 13th term in the sequence is 8 ln 9.

We need to find the value of a.

Step 1: Finding the 13th term. Using the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, n is the number of terms, and d is the common difference.

Substituting the given values, we get:an = a1 + (n - 1)d 13th term, a 13 = a1 + (13 - 1)3a13 = a1 + 36 a1 = a13 - 36 ...(1)Given that a13 = 8 ln 9.

Substituting in equation (1), we get: a1 = 8 ln 9 - 36.

Step 2: Finding the value of a. Using the formula for the nth term again, we can write the 13th term in terms of a as: a13 = a + (13 - 1)3a13 = a + 36a = a13 - 36.

Substituting the value of a13 from above, we get:a = 8 ln 9 - 36. Therefore, the value of a is 8 ln 9 - 36.

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Answer:

a. 16

c. 120°

e. 60°

Step-by-step explanation:

Properties of Parallelogram:

For Question:

a.

DC= AB=16 Opposite side is congruent.

c.

m ∡DCB = m ∡DAB=120° Opposite angles are congruent.

e.

m ∡ABC= ​?

m ∡ABC+ m∡DAB =180° Consecutive angles are supplementary.

Substituting value

m ∡ABC + 120°=180°

m ∡ABC =180°-120°=60°

m ∡ABC=60°

Answer:

a)  DC = 16

c)  m∠DCB = 120°

e)  m∠ABC = 60°

Step-by-step explanation:

The opposite sides of a parallelogram are equal in length. Therefore, DC is the same length as AB.

As AB = 16, then DC = 16.

[tex]\hrulefill[/tex]

The opposite angles of a parallelogram are equal in measure. Therefore, m∠DCB is equal to m∠DAB.

As m∠DAB = 120°, then m∠DCB = 120°.

[tex]\hrulefill[/tex]

Adjacent angles of a parallelogram sum to 180°. Therefore:

⇒ m∠ABC + m∠DAB = 180°

⇒ m∠ABC + 120° = 180°

⇒ m∠ABC + 120° - 120° = 180° - 120°

⇒ m∠ABC = 60°

Currency exchange rates are primarily based on each country's economy.

The exchange rate of a currency is influenced by various factors, such as inflation rates, interest rates, political stability, economic performance, trade balances, and market supply and demand.

These factors reflect the overall strength or weakness of a country's economy and play a significant role in determining the value of its currency relative to other currencies in the foreign exchange market.

While historical exchange rate systems, such as the gold standard, had an impact on currency values in the past, the modern exchange rate regime is primarily determined by market forces and economic fundamentals.

Bartering and one-for-one exchange are not directly related to currency exchange rates in the context of global currency markets, as exchange rates involve the relative value of one currency against another.

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Answer:

a+2bc/3a....4+2(--5×--7)/3(4)....4+2(35)/12.....4+70/12...74/12..answer =6⅙..option C

The expression is completed as (x-4)(x -7)

From the information given, we have that the polynomial is given as;

x² - 11x + 28

Using the factorization method, we have;

First, find the product of the coefficient of x squared and the constant value

Then, we have;

1(28) = 28

Now, find the pair factors of the product that adds up to -11, we have;

-7x and -4x

Substitute the values, we have;

x² - 7x - 4x + 28

Group in pairs, we get;

x(x-7) - 4(x - 7)

Then, we have the expressions as;

(x-4)(x - 7)

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The simplified value of the expression given is 3a⁴b/ 7

Given the fraction :

divide the coefficients by 5

From division rule of indices, subtract the powers of values with Equivalent coefficients.

Hence,

Finally we have :

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The y-intercept, b, can be calculated as:

b = (Σy - mΣx) / n

To find the regression line associated with the set of points (4, 6), (6, 10), (10, 14), and (12, 2), we can use the least squares method. The regression line represents the best-fit line that minimizes the sum of the squared differences between the observed y-values and the predicted y-values on the line.

The equation for the regression line, y(x), can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.

Using the given points, we can calculate the slope, m, and the y-intercept, b, to obtain the equation of the regression line.

The slope, m, is calculated as:

m = (nΣxy - ΣxΣy) / (nΣ[tex]x^2[/tex] - (Σ[tex]x)^2[/tex])

where n is the number of points, Σxy is the sum of the product of x and y values, Σx is the sum of the x-values, and Σy is the sum of the y-values.

Similarly, the y-intercept, b, can be calculated as:

b = (Σy - mΣx) / n

By substituting the given values into the formulas and performing the calculations, the equation for the regression line can be obtained.

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Answer:

C-351

Step-by-step explanation:

The actual length corresponding to the 3-inch length on the model is 10.5 feet.

Given that the scale of the model is 1 inch to 3.5 feet, we can use this information to find the actual length corresponding to a given length on the model.

Let's denote:

Model's length = 3 inches

Actual length = ?

According to the given scale, 1 inch on the model represents 3.5 feet in reality. We can set up a proportion to find the actual length:

(1 inch) / (3.5 feet) = (3 inches) / (x feet)

Cross-multiplying, we get:

1 inch * x feet = 3 inches * 3.5 feet

Simplifying the equation:

x feet = 10.5 feet

Therefore, the actual length corresponding to the 3-inch length on the model is 10.5 feet.

In summary, the actual length is 10.5 feet.

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The result of the row operation on the matrix is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

The matrix in this problem is defined as follows:

[tex]\left[\begin{array}{cccc}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

The row operation is given as follows:

[tex]R_1 \rightarrow \frac{1}{2}R_1[/tex]

The first row of the matrix is given as follows:

[2 0 0 16]

The meaning of the operation is that every element of the first row of the matrix is divided by two.

Hence the resulting matrix is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

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Answer: 28

Step-by-step explanation:

If [tex]a = 7[/tex] and [tex]b = 2[/tex], then [tex]2ab[/tex] can be worked out as follows:

[tex]\Large 2ab = 2 \times a \times b[/tex]

Substituting the values of [tex]a[/tex] and [tex]b[/tex], we get:

[tex]2 \times 7 \times 2 = 28[/tex]

Therefore, [tex]2ab[/tex] is equal to 28 when [tex]a = 7[/tex] and [tex]b = 2[/tex].

________________________________________________________

The answer is:

Work/explanation:

To evaluate the expression [tex]\sf{2ab}[/tex], I begin by plugging in 7 for a and 2 for b:

[tex]\large\pmb{2(7)(2)}[/tex]

Simplify by multiplying.

[tex]\large\pmb{2*14}[/tex]

[tex]\large\pmb{28}[/tex]

Answer:  the correct answer is 5/2

Step-by-step explanation:

To find the slope of a line perpendicular to a given line, we can use the property that the product of the slopes of two perpendicular lines is equal to -1.

The given line has an equation of y = -2/5x + 4/5.

The slope of this line can be determined by comparing it to the slope-intercept form (y = mx + b), where "m" represents the slope. In this case, the slope of the given line is -2/5.

To find the slope of the line perpendicular to this line, we take the negative reciprocal of the given slope. The negative reciprocal of -2/5 is 5/2.

The coordinates of the reflected quadrilateral A'B'C'D' are:

A' = (6, 4)

B' = (-3, 2)

C' = (-1, 23)

D' = (-x, 12)

To find the coordinates of the reflected quadrilateral A'B'C'D', we need to apply the transformation Ty = x to each vertex of the original quadrilateral ABCD. The transformation Ty = x reflects each point across the y-axis.

Given the coordinates of the original quadrilateral ABCD as:

A = (-6, 4)

B = (3, 2)

C = (+1, 23)

D = (x, 12)

Applying the transformation Ty = x to each vertex, we can determine the coordinates of the reflected quadrilateral A'B'C'D':

A' = (-(-6), 4) = (6, 4)

B' = (-3, 2)

C' = (-1, 23)

D' = (-x, 12)

The reflected quadrilateral A'B'C'D' thus has the following coordinates:

A' = (6, 4)

B' = (-3, 2)

C' = (-1, 23)

D' = (-x, 12)

Therefore, the x-coordinate for point D' will be represented as -x in the reflected quadrilateral.

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