#8: Expand the logarithm shown below. *
log981xy

Derek is saving money for the down payment on a used car. So far, he has saved $200. His parents have said they help him put aside money, they will pay him 6% interest each month on the money Derek has saved. It will take Derek approximately 28 months to save at least $1,000 for the down payment on the used car if the only additions to his savings account are his parents' interest payments.

Month Process

0 Derek saves $200 (initial savings)

1 Derek receives 6% interest on $200 (interest = $200 * 0.06 = $12), total savings = $200 + $12 = $212

2 Derek receives 6% interest on $212 (interest = $212 * 0.06 = $12.72), total savings = $212 + $12.72 = $224.72

3 Derek receives 6% interest on $224.72 (interest = $224.72 * 0.06 = $13.48), total savings = $224.72 + $13.48 = $238.20

4 Derek receives 6% interest on $238.20 (interest = $238.20 * 0.06 = $14.29), total savings = $238.20 + $14.29 = $252.49

5 Derek receives 6% interest on $252.49 (interest = $252.49 * 0.06 = $15.15), total savings = $252.49 + $15.15 = $267.64

6 Derek receives 6% interest on $267.64 (interest = $267.64 * 0.06 = $16.06), total savings = $267.64 + $16.06 = $283.70

It will take Derek approximately 28 months to save at least $1,000 for the down payment on the used car if the only additions to his savings account are his parents' interest payments.

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Based on the information, it should be noted that 31.25 subscriptions were left when y=5.

From the information, Saleema was investigating the newspaper subscriptions for her local newspaper. At the beginning of the year, She noted 1000 subscriptions for Subscribed to the newspaper

To find the approximate number of subscriptions left when y=5, we can substitute y=5 into the function N(y) = 1000 * (0.5)^y and evaluate it.

N(5) = 1000 * (0.5)^5

N(5) = 1000 * (0.5)^(5)

N(5) = 1000 * (0.03125)

N(5) ≈ 31.25

Approximately 31.25 subscriptions were left when y=5.

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The statement "Once we give up the full Comprehension Schema, Russell's Paradox is no longer a threat" implies that by leaving out the unrestricted version of the Schema of Comprehension (which gives room for the construction of sets based on any property), one can be able to avoid meeting up Russell's Paradox.

Russell discovered the contradiction that arises when analyzing the collection of sets that lack themselves as components. A contradiction arises due to the paradoxical situation of attempting to ascertain if the set includes itself or not.

By adopting the Subset Schema, which limits the creation of sets based on certain criteria, we can steer clear of the paradox and depart from the comprehensive Comprehension Schema.

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Answer:

[tex]x=\frac{\pi}{4}+\frac{\pi n}{2}[/tex]

Step-by-step explanation:

[tex]\displaystyle \tan^2x-1=0\\\tan^2x=1\\\tan x=\pm 1\\\\x=\frac{\pi}{4}+\frac{\pi n}{2}[/tex]

The total cost of producing the additional units is $767.50.

In order to determine the total cost of producing 10 additional units assuming 2 units are currently being produced, we would have to integrate the marginal cost function for this handbag manufacturer with respect to x, and over the interval [10, 2].

Based on the information provided above, the marginal cost function for this handbag manufacturer is given by this function;

C'(x) = 0.046875x² − x + 100

₂∫¹⁰C'(x)dx = ₂∫¹⁰(0.046875x² − x + 100)dx

₂∫¹⁰C'(x)dx = ₂∫¹⁰(0.046875x²)dx − ₂∫¹⁰(x)dx + ₂∫¹⁰(100)dx

₂∫¹⁰C'(x)dx = 0.046875x³/3|¹⁰₂ - x²/2|¹⁰₂ + 100x|¹⁰₂

₂∫¹⁰C'(x)dx = 0.015625(10³ - 2³) - 1/2(10² - 2²) + 100(10 - 2)

₂∫¹⁰C'(x)dx = 0.015625(1000 - 8) - 0.5(100 - 4) + 100(8)

₂∫¹⁰C'(x)dx = 0.015625(992) - 0.5(96) + 800

₂∫¹⁰C'(x)dx = 15.5 - 48 + 800

₂∫¹⁰C'(x)dx = $767.50

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Answer:  log w²v⁵

Step-by-step explanation:

2log w + 5log v                        >Use exponent log rule

log w² + log v⁵                         >There is addition between, use

                                                   multiplication rule

log w²v⁵

The amplitudes of functions are a) 8 and b) 6.

Given are the functions we need to determine the amplitude of function,

a) y = 8 Sin (x/2) + 3

b) y = 6 Cos x + 2

So,

To determine the amplitude of a trigonometric function, you can follow these steps:

For a sine function of the form y = A×sin(Bx + C) + D:

The amplitude is equal to the absolute value of the coefficient A.

For a cosine function of the form y = A×cos(Bx + C) + D:

The amplitude is equal to the absolute value of the coefficient A.

Let's apply these steps to the given functions:

a) y = 8×sin(x/2) + 3

The coefficient of sin in this function is 8, so the amplitude is |8| = 8.

Therefore, the amplitude of function a) is 8.

b) y = 6×cos(x) + 2

The coefficient of cos in this function is 6, so the amplitude is |6| = 6.

Therefore, the amplitude of function b) is 6.

Hence the amplitudes of functions are a) 8 and b) 6.

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Answer: B

Step-by-step explanation:

The coefficients 9 & 6, divided by 3, equal 3 & 2.

For the exponents, if you divide the entire equation by xy, you need to subtract the corresponding exponents of x and y. Subtract the denominator's exponent from the numerator's exponent.

So, for the first part x^8/x^1=x^7

y^2/y^1=y

For the second part: x^3/x=x^2

y^4/y=y^3

So, if you put the coefficients and both parts of the equation together, you get 3x^7y - 2x^2y^3 (answer letter B).

Hope this helped!

The missing sides of the triangle are

x = 6.32

y = 7.48

z = 11.83

The missing side x is calculated using similar triangles rules as below

x / 10 = 4 / x

cross multiply gives

x² = 40

x = √40 = 6.32

Other sides will solved by Pythagoras theorem.

the formula of the theorem is

hypotenuse² = opposite² + adjacent²

plugging the values as in the problem

let y be the required distance

y² = 6.32² + 4²

y² = 55.94

y = √55.94

y = 7.48

z² = 6.32² + 10²

z² = 139.94

z = √139.94

z = 11.83

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The area under the curve y = x² + 5x + 4 and the x-axis is [tex]1\frac{2}{3}[/tex] square units.

To find the area under the curve y = x² + 5x + 4 and the x-axis, we need to integrate the given function over a specific interval.

We need to find the definite integral of the function over a suitable interval.

Let's find the definite integral of the function from its roots (where the curve intersects the x-axis).

First, let's find the roots of the function by setting y = 0:

x² + 5x + 4 = 0

Factoring the quadratic equation:

(x + 1)(x + 4) = 0

Setting each factor equal to zero:

x + 1 = 0 => x = -1

x + 4 = 0 => x = -4

The roots of the function are x = -1 and x = -4.

To find the area under the curve, we integrate the function from x = -4 to x = -1:

∫[x=-4 to -1] (x² + 5x + 4) dx

Integrating the function:

∫[x=-4 to -1] (x² + 5x + 4) dx = [1/3x³ + (5/2)x² + 4x] from -4 to -1

Substituting the limits:

[1/3(-1)³ + (5/2)(-1)² + 4(-1)] - [1/3(-4)³ + (5/2)(-4)² + 4(-4)]

Simplifying:

[-1/3 + 5/2 - 4] - [-64/3 + 40/2 - 16]

[tex]1\frac{2}{3}[/tex]

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Answer:

Let’s solve this problem step by step. Since a + b = 28 and a = 2b + 7, we can substitute the value of a in the first equation to get 2b + 7 + b = 28. Solving for b, we get 3b + 7 = 28, which means 3b = 21 and b = 7. So, the second number is 7.

Answer:

The Law of Sines can be used to solve triangles when you have two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA). In this case, triangle A can be solved using the Law of Sines because it has two angles and a side (AAS).

Answer: A

Step-by-step explanation: 32.07 cm^2

Absolute value is a mathematical function that gives the distance of a number from zero on the number line.

Let's find the absolute value of the given numbers:

|17| = 17

|−25| = 25

|−1.23| = 1.23

|0.45| = 0.45

Now, let's identify whether the following numbers are rational or irrational:

234: This is a rational number because it can be expressed as a fraction (e.g., 234/1).

5–√: This is an irrational number because it cannot be expressed as a fraction or a ratio of integers. The square root of 5 is not a perfect square, so it is irrational.

To convert the decimal 0.44444... to a fraction, we can set it as the variable x and solve the equation:

x = 0.44444...

Multiply both sides by 10 to move the decimal point:

10x = 4.44444...

10x - x = 4.44444... - 0.44444...

9x = 4

Divide both sides by 9:

x = 4/9

Therefore, the decimal 0.44444... is equivalent to the fraction 4/9.

Matching the numbers with their classifications:

0.3333...: This is a repeating decimal.

3.48372...: This is a nonrepeating decimal.

3.3: This is a terminating decimal.

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The area of the triangle whose dimensions are given in a vertices form would be = 9

To calculate the area of the triangle, the formula that should be used is the coordinate geometry formula: area = the absolute value of Ax(By - Cy) + Bx(Cy - Ay) + Cx(Ay - By) divided by 2

Where; W=A, X= B, Y = C

Ax= 8

By= -2

Cy= -4

BX= 5

Ay = -8

CX= 3

That is:

= 8(-2--4)+5(-4--8)+3(-8--2)/2

= 16+20-18/2

= 18/2

= 9

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The phrase "it will be tacitly understood that each such formula can be written in a form that only involves ∈ and = as nonlogical symbols" implies that although we may use certain symbols such as defined predicates, operations, and constants in our formulas, the underlying assumption is that these symbols can ultimately be expressed or defined using the fundamental symbols of set membership (∈) and equality (=).

Symbols can always be represented using set membership and equality symbols. We simplify formulas by eliminating unnecessary symbols through this convention.

If we define predicate "P," any formula with "P" can be expressed using set membership and equality. We can translate any formula with "P" into one using only ∈ and =. By adopting this convention, we ensure clear and precise communication, with a well-defined language and logical structure for formulas.

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See text below

In practice, we shall use in formulas other symbols, namely defined pred- icates, operations, and constants, and even use formulas informally; but it will be tacitly understood that each such formula can be written in a form that only involves and as nonlogical symbols.

Concerning formulas with free variables, we adopt the notational convention that all free variables of a formula

(u1,..., Un)

are among u1, ..., un (possibly some u are not free, or even do not occur, in ). A formula without free variables is called a sentence.

What is the meaning of "it will be tacitly understood that each such formula can be written in a form that only involves ∈ and = as nonlogical symbols"?

The area of shaded sector is,

⇒ Area of sector =  15.35 feet²

We have to given that,

In circle O,

⇒ Radius = 4 feet

And,  the central angle a measures 110°.

Since, We know that;

Area of sector = (θ/360) πr²

Where, θ is central angle and r is radius of circle,

Here., r = 4 and θ = 110°

Substitute the given values, we get;

Area of sector = (θ/360) πr²

Area of sector = (110/360) π (4)²

Area of sector = (0.30) π x 16

Area of sector =  4.89 x 3.14

Area of sector = 15.35 feet²

Thus, The area of shaded sector is,

⇒ Area of sector = 36π units²

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By algebra properties, the simplified form of the trigonometric equation cot² x / (csc x - 1) is  csc x + 1.

In this problem we find the case of a trigonometric equation, whose simplified form must be found. The simplification can be done both by algebra properties and trigonometric formulas. Now we proceed to determine the simplificated form:

cot² x / (csc x - 1)

(csc² x - 1) / (csc x - 1)

[(csc x + 1) · (csc x - 1)] / (csc x - 1)

csc x + 1

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The functions of the coefficients of the quadratic equation indicates that the correct statements are;

a. The y-intercept of the function is (0, c)

c. The coefficient b determines the horizontal shift of the parabola compared to the parent function

d. If a is negative, the parabola opens upwards

The quadratic equation is; y = a·x² + b·x + c

The functions of the coefficients of the above quadratic function are;

a = The scale factor, and the coefficient that determines the direction the projectile faces.

a < 1; Indicates that the graph is wider than the parent function

a > 1; Indicates that the graph is narrower than the parent function

a < 0; Indicates that the parabola opens downward

a > 0; Indicates that the parabola opens upwards

b = The coefficient b together with coefficients a and c determines the coordinate of the vertex of the graph of the quadratic function, which is; (-b/(2·a), -D/(4·a))

Where;

D = The discriminant

Therefore, b determines the x-value, and therefore, the horizontal shift of the parabola compared to the parent function

c = The y-coordinate of the y-intercept of the quadratic function, which is (0, c)

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Answer:

It maybe 12,000/120,000

Step-by-step explanation:

12months a year

12+12=24 so,

500×24=12,000

or 120,000

The value of an automobile company's stock fell 5 points over the last month.  The integer that represents the change in the stock's value is -5.

In the context of stock market performance, positive values typically indicate an increase in the stock's value, while negative values indicate a decrease. In this case, since the stock's value fell by 5 points, we assign a negative sign to represent the change.

By convention, negative numbers are used to represent decreases or losses, while positive numbers represent increases or gains. When the stock's value decreases, it is common to use negative integers to signify the change.

In this scenario, a decrease of 5 points is denoted by the negative sign (-) followed by the numerical value 5. Therefore, the integer -5 represents the change in the stock's value.

Using this notation helps provide clarity and consistency in understanding the direction and magnitude of changes in stock prices. It allows investors, analysts, and market participants to communicate and interpret market movements accurately and effectively.

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Value of an exterior angle of a triangle is equal to the sum of values of two opposite interior angles of a triangle.

[tex]\qquad\displaystyle \tt \dashrightarrow \: 5x + 10 = 40 + 70[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 5x = 110 - 10[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 5x = 100[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = 100 \div 5[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = 20[/tex]

Value of x = 20°

Hey! there . Thanks for your question :)

Answer:

Step-by-step explanation:

In this question we are given with two interior angles of the triangle that are 40° and 70° , also we are given an exterior angle that is (5x + 10)°. And we are asked to find the value of angle x.

Solution :-

For finding the value of angle x , we have to use exterior angle property of triangle which states that sum of opposite interior angles of triangle is equal to the given exterior angle. So :

Step 1: Making equation :

[tex] \longmapsto \: \sf{40 {}^{°} + 70 {}^{°} = (5x + 10) {}^{°} }[/tex]

Solving :

[tex] \longmapsto \: \sf{110 {}{°} = (5x) {}^{°} +10 {}^{°} }[/tex]

Step 2: Subtracting 10 on both sides :

[tex] \longmapsto \sf{ 110 {}^{°} - 10 {}^{°} = 5x + \cancel{10 {}^{°}} - \cancel{10 {}^{°} } }[/tex]

We get ,

[tex] \longmapsto \sf{(5x ){}^{°} = 100 {}^{°} }[/tex]

Step 3: Dividing both sides by 5 :

[tex] \longmapsto \dfrac{ \cancel{5}x {}^{°} }{ \cancel{5}} = \dfrac{ \: \: \: \: \cancel{ 100} {°}^{} }{ \cancel{5} }[/tex]

On cancelling , we get :

[tex] \longmapsto \underline{\boxed{\red{\sf{ \bold{ x = 20 {}^{°} }}}}} \: \: \bigstar[/tex]

Verification :-

For verifying sum of both the interior angles is equal to given exterior angles. As we get the value of x as 20 we need to substitute it's value in place x and then L.H.S must be equal to R.H.S :

Therefore , our answer is correct .

Answer:

Center ( -8, 1 )

Radius ( 13 )

The statement implies that the function f has a surjective property, ensuring full coverage of the target set Y.

The statement "If Y = ran(f), then f is a function onto Y" can be understood in the context of mathematical functions and their range.

In mathematics, a function is considered "onto" or "surjective" if every element in the target set (in this case, Y) has a pre-image in the domain set. In other words, for every element y in Y, there exists at least one element x in the domain set such that f(x) = y.

The given statement is saying that if Y is equal to the range of the function f, then f is a function that covers or maps onto every element in Y. In this case, the function f has a mapping for each element in the set Y, ensuring that no element in Y is left without a pre-image in the domain set.

Thus, the statement implies that the function f has a surjective property, ensuring full coverage of the target set Y.

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The meaning of "The field of a relation R" is a constraint on the values that can be stored in that attribute.

We are given that;

The statement The field of a relation R

Now,

According to the web results, the field of a relation R is the set of all values that appear in the relation R. In other words, it is the union of all the attributes (columns) of R. For example, if R is a relation with attributes A, B, and C, then the field of R is the set of all values that appear in A, B, or C. The field of R can also be denoted by F.

The field of a relation is different from the domain of an attribute, which is the set of all possible values that an attribute can take. For example, if A is an attribute that can only take integer values, then the domain of A is the set of all integers. The domain of A may be larger than the field of A, which is the set of all values that actually appear in A. The domain of an attribute defines a constraint on the values that can be stored in that attribute.

Therefore, by the domain the answer will be a constraint on the values that can be stored in that attribute.

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The new height above sea level would be Option (C) 3980m.

New sea-level = Previously given sea level + 20m

Given,

Height above sea previously = 4000m

Increase in sea level = 20m

Now, since there has been a rise in sea level due to global warming the distance between the previous 4000m mark and the sea level would decrease. This shall now be less than 4000m. The distance/height would reduce by 20m. It can also be observed that the distances below the sea level would increase due to a rise in the sea level mark.

∴ New height above sea level = 4000m - 20m

                                                 = 3980m.

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Answer:

0.15 flowers/ ft²

Step-by-step explanation:

area = 12 X 18 = 216.

population density = population/area

= 32/216

= 0.148

= 0.15 flowers/ ft²

Answer:

A. 0.15 flowers/ft²

Step-by-step explanation:

The population density of flowers is the number of flowers per square foot.

To find the population density, we need to know the area of the garden and the number of flowers.

The area of the garden is 12 feet*18 feet =216 square feet.

The number of flowers is 32.

The population density is 32flowers/216square feet = 0.15 flowers/ft².

Therefore, the answer is (A).

The new coordinates of the triangle after reflecting in the x-axis are:

(3, -1), (3, -3), and (-2, -3).

We have,

The triangle has the coordinates:

(3, 1), (3, 3), and (-2, 3)

Now,

To reflect a point or a shape in the x-axis, we simply change the sign of the y-coordinate while keeping the x-coordinate the same.

Let's reflect each of the three given points in the x-axis:

Point (3, 1):

The reflected point in the x-axis will have the same x-coordinate of 3 but with the y-coordinate sign flipped:

Reflected point: (3, -1)

Point (3, 3):

The reflected point in the x-axis will have the same x-coordinate of 3 but with the y-coordinate sign flipped:

Reflected point: (3, -3)

Point (-2, 3):

The reflected point in the x-axis will have the same x-coordinate of -2 but with the y-coordinate sign flipped:

Reflected point: (-2, -3)

Therefore,

The new coordinates of the triangle after reflecting in the x-axis are:

(3, -1), (3, -3), and (-2, -3).

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Answer:

Step-by-step explanation:

ll hypothesis The critical value for this test is 9.488. Based on the test statistic of 1.358, we fail to reject the null hypothesis that the amount of smoking is independent of seat belt use. In other words, there is not enough evidence to support the claim that seat belt use and smoking habits are related.

Personal Note:

You really need to study, or you will end taking summer courses. trust me school isn't that easy and one time I failed I took summer courses but then I passed these summer courses with god's help. so please focus on your studys.