Evaluate function expressions

Answer:

The correct answer:

(A) WZ/WX = XW/YW

The domain of the function f(x) is all real numbers except x = -7. In interval notation, we can express the domain as (-∞, -7) ∪ (-7, +∞).

To find the domain of the function f(x) = (x - 3)/(x + 7), we need to determine the values of x for which the function is defined.

The function is defined for all values of x except those that make the denominator, x + 7, equal to zero.

Division by zero is undefined in mathematics.

So, we set the denominator equal to zero and solve for x:

x + 7 = 0

x = -7

Hence, the only value that makes the denominator zero is x = -7. Therefore, the domain of the function f(x) is all real numbers except x = -7.

In interval notation, we can express the domain as (-∞, -7) ∪ (-7, +∞).

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The point Y best represents 33 1/3% of the distance between zero and one.

We know that if the mixed fraction a b/c can expressed as improper fraction in below way:

a b/c = (a*c + b)/c

Here the given the mixed fraction is = 33 1/3 = (33*3 + 1)/3 = 100/3

The distance between the point 0 and 1 is given by = 1 - 0 = 1

So the required percentage of the distance = 1*(33 1/3%) = 1*(100/3 %) = 100/(3*100) = 1/3.

From the number line given we can see that the distance between 0 and 1 is divided in to 12 equal parts.

So, 33 1/3% stands at = 12*(1/3) = 4 th parts that is point Y.

Hence the point Y best represents 33 1/3% of the distance between zero and one.

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The question is incomplete. The complete question will be -

The two men will charge the same amount to complete the job after 3.5 hours.

Let's start with Charles' fee, which is a combination of an upfront fee and an hourly fee. We can represent this as:

C(x) = 115 + 30x

where x is the number of hours Charles works.

For Peter, his fee is a combination of an upfront fee for paint and an hourly fee. We can represent this as:

P(x) = 80 + 40x

where x is the number of hours Peter works.

To find out when the two men will charge the same amount to complete the job, we need to set the two equations equal to each other and solve for x:

115 + 30x = 80 + 40x

Subtracting 30x from both sides: 115 = 80 + 10x

Subtracting 80 from both sides: 35 = 10x

Dividing both sides by 10:

x = 3.5

So, the two men will charge the same amount to complete the job after 3.5 hours.

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Answer:

The total number of students surveyed is 400.

Step-by-step explanation:

We want to know that 49% of what number is 196

Let x = the total number of students

.49x = 196        (49% as a decimal is .49)   Divide both sides  by .49

x = 400

m/ABC=86° (because of the cyclic quadrilaterals). Hence, Option (B) is correct.

A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a common circle.

The opposite angles of a cyclic quadrilateral have a total of 180°.

m/ABC +  m/ADC  = 180°

m/ABC + 94° = 180°

m/ABC = 180° - 94°

m/ABC = 86°

Similarly, one can find m/DCB

m/DAB +  m/DCB  = 180°

99° + m/DCB = 180°

m/DCB = 180° - 99°

m/DCB = 81°

Hence, m/ABC = 86°

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Answer:

Step-by-step explanation:

To estimate the original volume of the pyramid, we can use the formula for the volume of a frustum of a pyramid:

V = (1/3)h(a^2 + ab + b^2)

Where V is the volume, h is the height of the frustum, a is the side length of the top square, and b is the side length of the bottom square.

In this case, the top square has a side length of 4 m, and the bottom square has a side length of 20 m. The height of the frustum is 12 m.

Plugging these values into the formula, we get:

V = (1/3) * 12 * (4^2 + 4*20 + 20^2)

Simplifying the equation:

V = (1/3) * 12 * (16 + 80 + 400)

V = (1/3) * 12 * 496

V = 12 * 496/3

V = 1984

Therefore, the estimated original volume of the pyramid is 1984 cubic meters.

The angle that is not a solution to the trigonometric equation sin(2θ) = 1 is given as follows:

a) 90º.

The trigonometric equation in the context of this problem is defined as follows:

sin(2θ) = 1

The sine assumes a value of 1 at:

90º.

Hence the solution is obtained as follows:

2θ = 90º.

θ = 45º.

The equivalent angle to 45º on the third quadrant is given as follows:

Hence 225º and -135º are also solutions to the trigonometric equation.

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The manufacturer made 1,890 non-defective light bulbs and 210 defective light bulbs.

Let represent number of non-defective light bulbs with x

Let represent number of defective light bulbs with y.

The profit from selling non-defective light bulbs is €0.60x

The loss from selling defective light bulbs is €0.80y.

Given that the manufacturer made a profit of €966, we can form the equation:

0.60x - 0.80y = 966 ---(1)

The total number of light bulbs produced is 2,100, so we have:

x + y = 2,100 ---(2)

To solve this, we will use method of elimination.

We multiply equation (2) by 0.60.

0.60x + 0.60y = 1,260 ---(3)

Now, subtract equation (3) from equation (1):

0.60x - 0.60x - 0.80y - 0.60y = 966 - 1,260

-1.40y = -294

y = -294 / -1.40

y = 210

Substituting value of y into equation (2):

x + 210 = 2,100

x = 2,100 - 210

x = 1,890.

Translated question:

A light bulb manufacturer earns €0.60 for each bulb that leaves the factory but loses €0.80 for each one that is defective. On a certain day when he manufactured 2,100 light bulbs, he made a profit of €966. How many light bulbs of each type did he make?

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The thing that can be interpreted from the scatter plot above is A. There is a negative association between number of hours of daily TV viewing and math quiz scores.

As per given scatter plot, If the number of hour of daily TV viewing increases then the math quiz scores are decreases.

Hence, the number of hour of daily TV viewing and math quiz scores varies inversely.

Therefore, There is a negative ath association between number of hour of daily TV viewing and math quiz scores

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4. RSTU is a trapezoid because the opposite sides RS and UT are parallel.

5. RSTU is not an isosceles trapezoid because the diagonals are not congruent.

In order to verify that RSTU is a trapezoid, we would have to determine slope of the pair of opposite sides and check whether at least one pair of opposite sides are parallel;

RU ║ ST

Slope of side RU = Slope of side ST

Slope of RU = (y₂ - y₁)/(x₂ - x₁)

Slope of RU = (1 + 3)/(5 + 3)

Slope of RU = 4/8

Slope of RU = 0.5.

Slope of RS = (y₂ - y₁)/(x₂ - x₁)

Slope of RS = (-9 + 3)/(-4 + 3)

Slope of RS = -6/-1

Slope of RS = 6.

Slope of ST = (y₂ - y₁)/(x₂ - x₁)

Slope of ST = (-2 - 1)/(10 - 5)

Slope of ST = -3/5

Slope of ST = -0.6.

Slope of UT = (y₂ - y₁)/(x₂ - x₁)

Slope of UT = (-2 + 9)/(10 + 4)

Slope of UT = 7/14

Slope of UT = 0.5.

Therefore, RSTU is a trapezoid because the opposite sides RS and UT are parallel.

Question 5.

In order to determine whether RSTU is an isosceles trapezoid, we would have to determine length of the diagonals by using the distance formula and check whether they are congruent;

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Distance RT = √[(-2 + 3)² + (10 + 3)²]

Distance RT = √170 units.

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Distance US = √[(5 + 4)² + (1 + 9)²]

Distance US = √181 units.

Therefore, RSTU is not an isosceles trapezoid because the diagonals RT and US are not congruent.

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Answer:

  c.  100 m

Step-by-step explanation:

You want the length of the shadow of a 100 m tower if the angle of inclination of the sun is 45°.

The tower and its shadow are modeled as legs of a right triangle. When the acute angle in such a triangle is 45°, the triangle is isosceles and the legs are the same length.

The shadow is the same length as the height of the tower: 100 m.

__

Additional comment

You can check this by using the formula:

  S = 100/tan(45°) = 100/1 = 100

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The distance that describes how long his route was is: 168 miles.

To determine how long the route was, we need to first represent the entire unknown distance covered by this individual with s.

5/7 s + 40 minutes + (0.75s - 118) = s

5/7s + 75/100s + 40 - 118 = s

5/7s + 75/100s - 78 = s

5/7s + 75/100s - s = 78

325s = 54600

s = 168 miles.

So, the entire distance covered by the cyclist is 168 miles.

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The distance that describes how long his route was is: 168 miles.

How to determine the length of the route.

To determine how long the route was, we need to first represent the entire unknown distance covered by this individual with s.

5/7 s + 40 minutes + (0.75s - 118) = s

5/7s + 75/100s + 40 - 118 = s

5/7s + 75/100s - 78 = s

5/7s + 75/100s - s = 78

325s = 54600

s = 168 miles.

So, the entire distance covered by the cyclist is 168 miles.

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Answer:

Question 6 :  True

Question 7:  False

Step-by-step explanation:

To find g(f(x)) given f(x) and g(x), substitute the expression for f(x) for every x in g(x)

f(x) = 2x

g(x) = x²

g(f(x) = (2x)² = 4x²

So True

f(x) = x² + x

g(x) = 2x + 1

g(f(x)) = 2(x² + x) + 1 = 2x² + 2x + 1

So False

Answer:

Step-by-step explanation:

To find the difference between the amount of butter and sugar used, we need to subtract the weight of sugar from the weight of butter.

The weight of butter used is 5/8 pounds, and the weight of sugar used is 1/2 pounds. To subtract these fractions, we need to find a common denominator.

The common denominator for 8 and 2 is 8, so we can rewrite the fractions as follows:

5/8 pounds of butter

4/8 pounds of sugar

Now we can subtract the weight of sugar from the weight of butter:

5/8 - 4/8 = 1/8

Therefore, Percy's Desserts used 1/8 pound more butter than sugar.

Answer: Percy's Desserts used 1/8 pounds more butter than sugar in their batch of fresh scones.

Step-by-step explanation:

1. Identify the quantities of butter and sugar used: Percy's Desserts used 5/8 of a pound of butter and 1/2 of a pound of sugar.

2. Convert the quantities to a common denominator: To compare these quantities, we need to express them with a common denominator. The least common denominator of 8 and 2 is 8. So, we express 1/2 as 4/8.

3. Subtract the quantity of sugar from the quantity of butter: Now that both quantities are expressed in eighths of a pound, we can subtract them: 5/8 - 4/8 = 1/8.

4. Convert the result back to a decimal: 1/8 of a pound is equivalent to 0.125 pounds.

So, Percy's Desserts used 0.125 pounds more butter than sugar in their batch of fresh scones.

The phrase "possibly some ui are not free, or even do not occur, in ϕ" refers to a logical formula or expression ϕ that contains variables ui.

When it says "possibly some ui are not free," it means that not all of the variables ui in ϕ are unrestricted or unconstrained. Some of them may have certain conditions or restrictions placed on them within the context of the formula ϕ.

On the other hand, when it says "or even do not occur, in ϕ," it means that some variables ui may not appear at all in the formula ϕ. This implies that those variables are not present or relevant in the expression ϕ and do not play a role in its interpretation or evaluation.

In summary, the phrase indicates that in the formula ϕ, some variables ui may have restrictions or conditions placed on them, while others may not be present or relevant within the context of the formula.

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The property that should be used to simplify the expression 1/t⁻³ is [tex]\frac{1}{a^-m}[/tex] = [tex]a^m[/tex] (where a ≠ 0). Option A

In the given scenario or expression, this property implies that 1/t⁻³ = t³. The negative exponent in the denominator becomes a positive exponent when moved to the numerator.

Here is the step-by-step solution

1/t⁻³ = 1/(t⁻³)

1/(t⁻³) = (t⁻³)⁻¹

(t⁻³)⁻¹ = t⁽⁻³⁾ˣ⁽⁻¹⁾

t⁽⁻³⁾ˣ⁽⁻¹⁾ = t³

Therefore, the simplified expression is t³.

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Answer:

C

Step-by-step explanation:

the expression ! ( factorial ) has the following meaning

n! = n(n - 1)(n - 2) ... × 3 × 2 × 1

Then

4! = 4 × 3 × 2 × 1 = 24

The artist should buy the canvas with dimensions 26" x 16" as it most closely matches the Golden Rectangle.

To determine which canvas size most closely matches the Golden Rectangle, we need to calculate the aspect ratios of the available options and compare them to the Golden Ratio.

The Golden Ratio is approximately 1.618. A rectangle is considered to be a Golden Rectangle if its aspect ratio (width divided by height) is equal to the Golden Ratio.

Let's calculate the aspect ratios for each canvas size:

Canvas 1: 24" x 36"

Aspect ratio = 24 / 36 = 0.67

Canvas 2: 24" x 40"

Aspect ratio = 24 / 40 = 0.6

Canvas 3: 10" x 16"

Aspect ratio = 10 / 16 = 0.625

Canvas 4: 26" x 16"

Aspect ratio = 26 / 16 = 1.625

Canvas 5: 20" x 12"

Aspect ratio = 20 / 12 = 1.667

Now, let's compare the aspect ratios of the available options with the Golden Ratio (1.618) to see which one is closest:

Canvas 1: |0.67 - 1.618| = 0.948

Canvas 2: |0.6 - 1.618| = 1.018

Canvas 3: |0.625 - 1.618| = 0.993

Canvas 4: |1.625 - 1.618| = 0.007

Canvas 5: |1.667 - 1.618| = 0.049

The canvas with the aspect ratio closest to the Golden Ratio is Canvas 4: 26" x 16". Its aspect ratio of 1.625 is only 0.007 away from the Golden Ratio.

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The slope of the graph f(x) = (1/3)x + 4 is less than the slope of g(x) = 4x-1

f(x) = 1/3 x + 4

g(x) = 4x - 1

The equation of line is given by y = mx +c

On comparing both equation with y = mx + c

m is the slope of the line and c is the intercept of the equation

Let the slope of f(x) is m₁

In f(x) =1/3 x + 4

m₁ = 1/3 = 0.33

Let the slope of g(x) is m₂

In g(x)  = 4x - 1

m₂ = 4

m₁ < m₂

slope of f(x) is less than slope of g(x)

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Answer:

Therefore, the y-intercept of the line is (0, 4/3).

Step-by-step explanation:

To find the y-intercept of the line 12x - 18y = -24, we need to set x = 0 and solve for y.

12(0) - 18y = -24

Simplifying, we get:

-18y = -24

Dividing both sides by -18:

y = 4/3

Therefore, the y-intercept of the line is (0, 4/3).

The volume of the rectangular prism in this problem is given as follows:

99 ft³.

The volume of a rectangular prism, with dimensions defined as length, width and height, is given by the multiplication of these three defined dimensions, according to the equation presented as follows:

Volume = length x width x height.

The figure in this problem can be divided into two prisms, with dimensions given as follows:

Hence the volume of the prism is given as follows:

V = 3 x 5 x 1 + 6 x 2 x 7

V = 99 ft³.

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Answer:

Step-by-step explanation:

cos4α + 2 Sin² 2α is a trigonometric expression that involves the cosine and sine functions of the angle α. This expression can be simplified using trigonometric identities to get:

cos4α + 2 Sin² 2α = (cos² 2α)² + 2 (Sin 2α)²

This simplification involves using the double angle identity for sine and the identity for the square of cosine.

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Answer:1

Step-by-step explanation:

cos(4α) + 2sin²(2α)

First, let's expand cos(4α)

cos(4α) = cos²(2α) - sin²(2α)

Now, let's substitute this expression back into the original equation:

cos(4α) + 2sin²(2α) = (cos²(2α) - sin²(2α)) + 2sin²(2α)

= cos²(2α) + sin²(2α)=1 by

cos²(α) + sin²(α)=1

Answer:

details:

54)Write the midpoint formula::

Substitute

into

::

Swap the sides:

Swap the sides:

Multiply both sides of the equation by the common denominator:

Reduce the fractions:

Multiply both sides of the equation by the common denominator:

Reduce the fractions:

Calculate the product or quotient:

Rearrange variables to the left side of the equation:

Calculate the sum or difference:

Express solutions in ordered pairs::

(55)Write the midpoint formula::

Substitute

into

::

Swap the sides:

Swap the sides:

Multiply both sides of the equation by the common denominator:

Reduce the fractions:

Multiply both sides of the equation by the common denominator:

Reduce the fractions:

Calculate the product or quotient:

Rearrange variables to the left side of the equation:

Calculate the sum or difference:

Express solutions in ordered pairs::

(56)Write the midpoint formula::

Substitute

into

::

Swap the sides:

Swap the sides:

Multiply both sides of the equation by the common denominator:

Reduce the fractions:

Multiply both sides of the equation by the common denominator:

Reduce the fractions:

Calculate the product or quotient:

Rearrange variables to the left side of the equation:

Calculate the sum or difference:

Express solutions in ordered pairs::

(57)Write the midpoint formula::

Substitute

into

::

Swap the sides:

Swap the sides:

Multiply both sides of the equation by the common denominator:

Reduce the fractions:

Multiply both sides of the equation by the common denominator:

Reduce the fractions:

Calculate the product or quotient:

Rearrange variables to the left side of the equation:

Calculate the sum or difference:

Express solutions in ordered pairs::

The Ratio of seats to engine cylinders in the Ford F-150 truck that Cody wants to rent is approximately 0.625.

The ratio of seats to engine cylinders in the Ford F-150 truck that Cody wants to rent can be calculated by dividing the number of seats by the number of engine cylinders. In this case, the Ford F-150 has a seating capacity of 5 passengers and a V8 engine configuration.

To calculate the ratio, we divide the number of seats (5) by the number of engine cylinders (8):

Ratio of seats to engine cylinders = Number of seats / Number of engine cylinders

Ratio of seats to engine cylinders = 5 / 8

Simplifying the ratio, we get:

Ratio of seats to engine cylinders = 0.625

Therefore, the ratio of seats to engine cylinders in the Ford F-150 truck that Cody wants to rent is approximately 0.625.

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Note the full question may be :

Cody works at a vehicle rental company. The vehicles available to rent are listed in the table below:

Vehicle Type  |  Seats  |  Engine Cylinders

----------------------------------------------

Car           |    5    |         4

SUV           |    7    |         6

Truck         |    3    |         8

What is the ratio of seats to engine cylinders in the truck?

A. 1:3

B. 3:8

C. 8:3

D. 3:1

Choose the correct option that represents the ratio of seats to engine cylinders in the truck.

The answer is:

( - 179 – 78 ) + 41 - ( - 165 )

=-51

Answer:

(-257) +41+165

-257+206

-51

The total amount in the account when turning 60 years is A = $ 11,13,706.08

Given data ,

A savings account with an interest rate of 6% per year and you aren't adding any additional funds in the future

Now , you make $80,000 within the year you turn 60

So , the number of years = 60 - 16 = 44 years

And , from the compound interest , we get

A = P ( 1 + r/n )ⁿᵇ

On simplifying , we get

Where A is the final amount, P is the initial amount (which is 0 in this case), r is the annual interest rate (6% or 0.06), n is the number of times the interest is compounded per year (let's assume it is compounded monthly, so n=12), t is the time in years (44 years from age 16 to age 60).

A = 80,000 ( 1 + 0.06/12 )¹²ˣ⁴⁴

A = $ 11,13,706.08

Hence , the amount in account is A = $ 11,13,706.08

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Using the mean value theorem in  f(x) = 1/2x² + 7, c =  2√3

The mean value theorem states that let f be continuous over the closed interval  [a,b] and differentiable over the open interval  (a,b). Then, there exists at least one point c ∈(a,b) such that f'(c) = [f(b) - f(a)]/(b - a)

To find c using the mean value theoren for f(x) = 1/2x² + 7 over the interval [0, 6], we proceed as follows.

Uisng the mean value theorem, we know that

f'(c) = [f(b) - f(a)]/(b - a)

⇒ f(c) = 1/(b - a)∫ₐᵇ[f(x)

Now over the interval [a, b] = [0,6]

f(c) = 1/(6 - 0)∫₀⁶[f(x)

Now, since f(x) = 1/2x² + 7

Substituting this into the equation, we have that

f(c) = 1/(6 - 0)∫₀⁶[f(x)

f(c) = 1/(6 - 0)∫₀⁶(1/2x² + 7)

f(c) = 1/6∫₀⁶(1/2x² + ∫₀⁶7)

f(c) = 1/6[1/2x³/3 + 7x]₀⁶

f(c) = 1/6[x³/6 + 7x]₀⁶

f(c) = 1/6[6³/6 + 7(6)] - [0³/6 + 7(0)]

f(c) = 1/6[6²+ 42] - [0 + 7(0)]

f(c) = 1/6([36 + 42] - [0 + 0])

f(c) = 1/6(78 - 0)

f(c) = 1/6(78)

f(c) = 13

Now, since f(x) = 1/2x² + 7

f(c) = 1/2c² + 7

1/2c² + 7 = 13

1/2c² = 13 - 7

1/2c² = 6

c² = 2 × 6

c² = 12

c = √12

c = 2√3

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