Assume a > 0, b > 0 and 1 < ax < bx for all x < 0. Which of the following are true?

a. a < b < 1

b. 1 < a < b

c. a < 1 < b

d. b < a < 1

e. b < 1 < a

Explain how you got to the answer.

Boris's gas tank can hold 18 gallons of gas.

Gallons are a unit of volume commonly used to measure liquids, particularly in the United States. One gallon is equivalent to 128 fluid ounces or 3.785 liters. It is often used to measure the amount of gasoline or other fuels used in vehicles, as well as the amount of water in swimming pools or storage tanks.

Let's assume that the tank can hold "x" gallons of gas.

According to the problem, Boris's gas tank is initially half full, which means that it contains:

[tex]1/2 * x = x/2[/tex] gallons of gas

After he buys 3 gallons of gas, the tank is 2/3 full, which means it contains:

[tex]2/3 * x = 2x/3[/tex]gallons of gas

We can set up an equation based on the given information:

[tex]x/2 + 3 = 2x/3[/tex]

Multiplying both sides by 6 to eliminate the fractions:

[tex]3x + 18 = 4x[/tex]

Subtracting 3x from both sides:

[tex]18 = x[/tex]

Therefore, Boris's gas tank can hold 18 gallons of gas.

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Boris's gas tank can hold 18 gallons of gas.

What are gallons?

Gallons are a unit of volume commonly used to measure liquids, particularly in the United States. One gallon is equivalent to 128 fluid ounces or 3.785 liters. It is often used to measure the amount of gasoline or other fuels used in vehicles, as well as the amount of water in swimming pools or storage tanks.

Let's call the total capacity of Boris's gas tank "x". We know that when the tank is half full, it contains 1/2x gallons of gas. After Boris buys 3 gallons of gas, the tank is 2/3 full, which means it contains 2/3x gallons of gas. We can set up an equation to solve for x:

1/2x + 3 = 2/3x

To solve for x, we need to isolate it on one side of the equation. We can do this by first subtracting 1/2*x from both sides:

3 = 2/3x - 1/2x

To simplify the right side, we need to find a common denominator. The smallest common denominator of 2 and 3 is 6, so we can rewrite the fractions as:

3 = (4/6)*x - (3/6)*x

Now we can combine the terms on the right side:

3 = (1/6)*x

To solve for x, we can multiply both sides by 6:

x = 18

Therefore, Boris's gas tank can hold 18 gallons of gas.

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Answer:

Yes, the function f(x) = (x + 2)² is one-to-one.

Answer:

One possible pair of numbers whose product is -10 is 2 and -5. Another possible pair of numbers whose product is -10 is -2 and 5.

The statement can be written as the inequality:

x ≥ 12

We know that x is at least 12, so x can be equal to 12, or larger. We want to write that as an inequality, and it can be written as:

x ≥ 12

The symbol:

"≥"

Means that the thing in the left is larger or equal than the thing in the right, so that inequlity represents the given statement.

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A linear function representing the number of cards as a function of the number of years, t is y = 50 + 50t.

A linear function is a mathematical or algebraic equation of a straight line in the slope-intercept form.

Linear functions are written as f(x) = mx + b, where m is the slope and b is the y-intercept of the line.

Number of Total Cards in Carlos' Collection

Year             Total Cards

Year 1              50

Year 2            100

Year 3            150

Year 4           200

Slope = Rise/Run = (100 - 50)/(2 - 1)

= 50

Initial value in the first year = 50

Let the number of years involved = t

Let the value in year t = y

y = 50 + 50t

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Answer:

The constant of proportionality in the equation 2y = 2x is 1.

Note that this equation is actually the same as y = x, which is a linear equation with a slope of 1 and a y-intercept of 0. In this case, the coefficient of x (which is 2) is not the constant of proportionality because it does not represent a proportional relationship between x and y. Instead, the constant of proportionality is the ratio of the change in y to the change in x, which is 1.

After answering the query, we may state that Therefore, the surface area of the box is 832 in².

An object's surface area is a measure of how much space it takes up overall. The entire amount of space around a three-dimensional form is its surface area. The total surface area of a three-dimensional form is referred to as its surface area. By summing the areas of each face, one may get the surface area of a cuboid with six rectangular faces. Alternatively, you may identify the box's dimensions using the following formula: Surface (SA) equals 2lh, 2lw, and 2hw. A three-dimensional form's surface area is a measurement of how much space it takes up overall on the surface (a three-dimensional shape has a height, breadth, and depth).

To find the surface area of the rectangular prism, we need to find the area of each of its six faces and then add them up.

The top and bottom faces each have an area of 22 inches by 14 inches = 308 square inches.

The front and back faces each have an area of 22 inches by 3 inches = 66 square inches.

The left and right faces each have an area of 14 inches by 3 inches = 42 square inches.

2(308) + 2(66) + 2(42) = 616 + 132 + 84 = 832 square inches.

Therefore, the surface area of the box is 832 in².

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The angle 9pi/4 is a positive coterminal angle of pi/4.

What is a coterminal angle?

Two angles are called coterminal if they have the same initial and terminal sides. In other words, two angles are coterminal if they differ by a multiple of 360 degrees or 2π radians. For example, 30 degrees and 390 degrees are coterminal angles, as are π/4 and 9π/4.

A coterminal angle of pi/4 can be found by adding or subtracting any multiple of 2pi until we get an angle between 0 and 2pi that is equivalent to pi/4.

One way to find a positive coterminal angle is to add 2pi:

pi/4 + 2pi = 9pi/4

So, The angle 9pi/4 is a positive coterminal angle of pi/4.

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Answer:

Jim would need 3.75 pounds of fruit to make 12 smoothies.

Step-by-step explanation:

Since 1 pound = 16 ounces, 12 smoothies will require:

12 x 5 = 60 ounces of fruit

To convert this to pounds, we divide by 16:

60 / 16 = 3.75 pounds of fruit

Therefore, Jim would need 3.75 pounds of fruit to make 12 smoothies.

The measure of the angle x is 70°, and the measure of its complementary angle is (90 - x) = (90 - 70) = 20°.

When two angles in geometry add up to 90°, they are said to be complimentary.

Right triangles frequently include complementary angles, where one angle is a right angle (measured 90 degrees), while the other two angles are mutually exclusive.

If two angles add up to 90 degrees, they are complimentary.

If one of the angles has a measure of x, then the complementary angle's measure is (90 - x).

We are told that the angle x measures 50° more than its complementary angle, which can be written as:

x = (90 - x) + 50

Simplifying and solving for x, we have:

2x = 140

x = 70

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The equation that represents the relationship between x and y, where y is the number of candy bars and x is the cost in dollars, is x = 0.75y.

An equation is a mathematical statement showing the equality or equivalence of two or more mathematical expressions.

Equations contain mathematical expressions with the equal symbol (=).

                                                     First Order    Second Order

The number of candy bars               36                  24

The total cost of order                   $27                 $18

The unit cost per candy bar =       $0.75           $0.75

                                                ($27 ÷ 36)       ($18 ÷ 24)

Let the number of candy bars bought = y

Let the cost of the candy bars = x

36x = $27

x = $0.75

Similarly, 24x = $18

x = $0.75

Therefore, the equation representing the relationship is:

x = 0.75y

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Answer:

the answer is (HK) Because gj is the only other line that goes across the circle but it is not an option.

Note that all values of y > 0 at x = (0, 1, 2, 3, 4)

As seen from the graph attached in answer area as well, The graph is above  x-axis (i.e. y value of equation) only when x has values {0, 1, 2, 3, 4}.

At x = 0:

y = - | 0-2| + 2
⇒ - 2+ 2

y = 0

At x = 1:

y = - | 0-1| + 2
⇒ 0+ 2

y = 1

At x = 2:

y = - | 2-2| + 2
⇒ - 1+ 2

y = 1

At x = 3:

y = - | 3-2| + 2
⇒ - 1+ 2

y = 1

At x = 4:

y = - | 4-2| + 2
⇒ - 2+ 2

y = 0

Thus,  all values of y > 0 at x = (0, 1, 2, 3, 4)

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Answer:

567

Step-by-step explanation:

n

Answer:

a) 5177 cubic centimeters

b) 3451 cubic centimeters

c) 66.66% (67%)

Step-by-step explanation:

a) V=pi*r2*h

   V=pi*6.5^2*39

   V=5176.5593

b) V=4/3*pi*r^3

   V=4/3*pi*6.5^3

   V=1150.34651

remember we have to multiply by 3 since there are 3 balls:

   V=3451.03953

c) 3451/5177=0.66660*100=66.66%

The Ice Cream container will hold 1.2 liters of Ice Cream.

The Volume of Ice Cream can be calculated by Cubical Volume Equation V=l*b*h _______(1). We Have the data : Length = 0.2 meters, Width = 0.1 meters, Height = 0.06 meters

Now Having the Values of Length, Height & Width. Putting these Values in The Equation (1),

Volume = 0.2*0.1*0.06 m³

Volume = 0.0012 m³

We have got the Volume of Ice Cream that can be contained by the Container in m³. To Convert into Liters, we have equation 1 m³=1000 Liters. According to this, The Volume of Ice Cream Contained in The Container will be 1.2 Liters.

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Answer:

If the common ratio is 0.6, then the percent change is a decrease of 40%.

If the common ratio is 0.6 then the percent change is 40%

percentage, a relative value indicating hundredth parts of any quantity.

The percent change for a geometric sequence with common ratio r is given by the formula:

percent change = (r - 1) × 100%

In this case, the common ratio is 0.6, so the percent change is:

percent change = (0.6 - 1) × 100% = -0.4 × 100% = -40%

The percent change is negative, which means the sequence is decreasing.

Hence, If the common ratio is 0.6 then the percent change is 40%

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The image coordinates of L′ are L′(-1, 2).

What are Transformation and Reflection?

Single or multiple changes in a geometrical shape or figure are called Geometrical Transformation.

A geometrical transformation in which a geometrical figure changes his position to his mirror image about some point or line or axis is called Reflection.

To reflect the preimage of point L across the line y = -2, we need to find the distance between L and the line y = -2 and then move that same distance above the line to get the image L′.

The distance between point L and the line y = -2 is 4 units (since the y-coordinate of point L is -6 and the y-coordinate of the line y = -2 is -2).

So, to get the image L′, we move 4 units above the line y = -2 starting from the x-coordinate of L, which is -1. This gives us the coordinates (-1, 2).

Therefore, the image coordinates of L′ are L′(-1, 2).

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Answer:

What is Saving?

Saving means to keep money aside, not using it for any purpose at all, until in reach of desired goal. While some save for emergencies, others save for family or materials.

Example: My father is saving for a really nice vacation trip to Canada. He needs enough money to pay for gas, the hotel rooms, food and other necessities.

What is Investing?

Investing is the practice of putting interest into something. Investing involves paying for something in increments while also having flowy income. You could invest in almost anything.

Example: My mother and I are investing into a large, family-sized, RV. We are making custom arrangements and changes while we are still living in, and keeping our tiny-home.

The volume of the hexagonal prism is 374.13 cubic inches.

The distance from a regular polygon's centre to one of its sides' midpoints is known as the apothem. In other terms, it is the greatest circle whose radius fits inside the polygon. The apothem runs through the middle of the polygon and is perpendicular to the side it touches. All apothems are equivalent in a regular polygon.

For regular polygons, the apothem is a crucial quantity because it is used to determine the polygon's area. The formula: can be used to determine the area of a regular polygon.

A = apothem x perimeter x (1/2)

The volume of the hexagonal prism is given by the formula:

V = B x h

Here, B is the base area thus,

Given the base area = 41.57 sq. in and height = 9 in we have:

V = (41.57)(9) = 374.13 cubic inches.

Hence, the volume of the hexagonal prism is 374.13 cubic inches.

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Therefore, if the insurance provider reimburses damages on a real cash-value basis, the most the insurer could possibly pay Cornell for the destroyed personal goods is $21,333.33.

The fraction or proportion can be expressed as a percentage as a number out of 100. Latin's translation of the term "percent" is "per hundred". The part (numerical number) is divided by the total (numerical value) to determine the percentage, which is then multiplied by 100.

If Ivy's losses are covered on a true cash-value basis, her insurer would be required to pay the following maximum amount:

The depreciated worth of the destroyed personal property must be determined first. Given that the expected 15-year useful life and the average age of the product

Given that its expected usable life is 15 years and that its average age is 5 years, the amount of its usefulness that has already been used up is:

5 years / a fifteen-year period equals a third, or 33.33%

As a result, the depreciated worth of the personal property damaged is:

$32,000 x (100% - 33.33%) = $21,333.33

Therefore, if the insurance provider reimburses damages on a genuine cash-value basis, the most the insurer could possibly pay Harvard for the destroyed personal goods is $21,333.33. This is due to the insurance provider's refusal to pay Ivy the full cost of replacement of what was damaged, but only the diminished worth of it.

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The maximum amount the insurance company would pay Ivy on an actual cash-value basis is $21,338.68.

what is percentage?

Percentage is a way of expressing a number or proportion as a fraction of 100. It is represented by the symbol "%".

To determine the maximum amount the insurance company would pay Ivy on an actual cash-value basis, we need to calculate the depreciated value of the damaged personal property.

First, we can calculate the percentage of the useful life that the damaged personal property had been used up to the time of the fire:

Percentage of useful life used = (age of damaged property / useful life) x 100%

Percentage of useful life used = (5 / 15) x 100%

Percentage of useful life used = 33.33%

Next, we can calculate the depreciation percentage based on the percentage of useful life used:

Depreciation percentage = Percentage of useful life used x 1/100

Depreciation percentage = 33.33% x 1/100

Depreciation percentage = 0.3333

Finally, we can calculate the depreciated value of the damaged personal property:

Depreciated value = Replacement price x (1 - Depreciation percentage)

Depreciated value = $32,000 x (1 - 0.3333)

Depreciated value = $21,338.68

Therefore, the maximum amount the insurance company would pay Ivy on an actual cash-value basis is $21,338.68.

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The probability distribution for the number of spins that land on two is given as follows:

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

Each spinner has a 3/4 probability of not landing on two, hence the probability of landing on two zero times is given as follows:

P(X = 0) = 3/4 x 3/4

P(X = 0) = 9/16

P(X = 0) = 0.5625.

For two times, the probability is given as follows:

P(X = 2) = 1/4 x 1/4

P(X = 2) = 1/16

P(X = 2) = 0.0625.

Hence the probability of one landing is given as follows:

P(X = 1) = 1 - (0.5625 + 0.0625)

P(X = 1) = 0.375.

(considering complementary events)

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The sum of all solutions to the equation 2cos²(x) - 7cos(x) - 4 = 0 for 0 < x < 2π is approximately 9.425 + 2π.

In mathematics, a quadratic is a polynomial expression of the second degree, meaning it contains one or more terms that are squared but no terms that are cubed or higher. Quadratic functions are typically written in the form ax² + bx + c, where a, b, and c are constants and x is the variable.

To find the sum of all solutions to the equation 2cos²(x) - 7cos(x) - 4 = 0 for 0 < x < 2π, we can first use the quadratic formula to solve for cos(x):

cos(x) = [7 ± √(7² - 4(2)(-4))] / (2(2))

cos(x) = [7 ± √89] / 4

The two possible solutions for cos(x) are:

cos(x) = (7 + √89) / 4

cos(x) = (7 - √89) / 4

Note that for 0 < x < 2π, the cosine function has values between -1 and 1. Therefore, the first solution cos(x) = (7 + √89) / 4 is not valid, since it is greater than 1.

The second solution cos(x) = (7 - √89) / 4 is valid, since it is between -1 and 1. Therefore, the sum of all solutions to the equation is:

x = cos⁻¹[(7 - √89) / 4] + 2πn, where n is an integer.

We need to find the sum of all possible values of x for 0 < x < 2π. We can do this by finding the smallest and largest values of n that give valid solutions, and then summing the corresponding values of x.

The smallest value of n that gives a valid solution is n = 0, since cos⁻¹[(7 - √89) / 4] is already between 0 and 2π. The largest value of n that gives a valid solution is n = -1, since adding 2π to cos^(-1)[(7 - √89) / 4] gives a value greater than 2π.

Therefore, the sum of all solutions to the equation is:

89) / 4 - 2π]

x = cos⁻¹[(7 - √89) / 4] + cos⁻¹[(7 - √

x ≈ 6.2832 + 0.8429 + 2π

x ≈ 9.425 + 2π

The sum of all solutions to the equation 2cos²(x) - 7cos(x) - 4 = 0 for 0 < x < 2π is approximately 9.425 + 2π.

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Part A: The equation that could be used to determine the length of each of the five pieces of wood cut by Warren is: x + 0.7 = 4.5

Part B: The equation from Part A is correct because it takes into account the total length of the wood (4.5 feet), as well as the amount of wood that is left over (0.7 feet).

Part C: the length of each of the five pieces of wood cut by Warren is 3.8 feet.

It is made up of two mathematical objects, with an equals sign (=) between them. Equations are used to represent relationships between unknown variables and can be solved to find their values.

Part A:

The equation that could be used to determine the length of each of the five pieces of wood cut by Warren is: x + 0.7 = 4.5, where x is the length of each piece of wood.

Part B:

The equation from Part A is correct because it takes into account the total length of the wood (4.5 feet), as well as the amount of wood that is left over (0.7 feet).

This equation can be used to calculate how much wood Warren has cut off from the original 4.5 feet of wood, which is the length of each of the five pieces of wood.

Part C:

Solving the equation from Part A:

x + 0.7 = 4.5

Subtract 0.7 from both sides:

x + 0.7 - 0.7 = 4.5 - 0.7

x = 3.8

Therefore, the length of each of the five pieces of wood cut by Warren is 3.8 feet.

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Answer:

Step-by-step explanation:

well 2.20 (ronded) pounds is  one kilogram so multiply 258 by 2.20 then divide by two to get 117.03

The side length of the dilated side would be 3 cm.

The area of the original square is 16 cm ², so its side length is √ 16 = 4 cm.

When the square is dilated by a factor of 0. 75, its area will be reduced to ( 0. 75 ) ² times the original area , which would then be:

= ( 0.75 ) ² x 16 cm ² = 9 cm²

Therefore, the area of the scaled square is 9 cm². The area of a square equals to the size of its side duration, thus we can locate the side length of the magnified square by extracting the square root of its area:

√ 9 cm = 3 cm

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Based on the data, the estimated number of lemon candies in the bag of 100 assorted candies is about 19.

In mathematics, proportion refers to the relationship between two quantities that are equivalent or have a constant ratio. It is usually expressed as a fraction or a ratio. For example, if a recipe calls for 2 cups of flour and 1 cup of water, the proportion of flour to water is 2:1 or 2/1. Proportions are used in a variety of mathematical and real-world situations, such as solving problems involving ratios, rates, and percentages.

In the given question,

We can use a simple proportion to estimate the number of lemon candies in the bag. We know that Ridgette handed out a total of 6 lemon candies out of a total of candies. Therefore, the proportion of lemon candies in the bag is:

6/total candies = number of lemon candies/100

We can cross-multiply to solve for the number of lemon candies:

number of lemon candies = 6 x 100 / total candies

We can also use the information about the other flavors to estimate the total number of candies in the bag. The total number of candies is:

total candies = lemon + orange + grape + cherry

Using the information from the table, we can estimate the total number of candies:

total candies = 6 + 5 + 8 + 13 = 32

Therefore, the estimated number of lemon candies in the bag is:

number of lemon candies = 6 x 100 / 32 = 18.75

We can round this to the nearest whole number to estimate that there were about 19 lemon candies in the bag of 100 assorted candies.

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The value of x for which sin(x+16)°=cos(2x+14)° is 20

Trigonometry is a branch of mathematics concerned with relationships between angles and ratios of lengths

The given equation says:

sin(x+16)°=cos(2x+14)°

Remember that sin(90-x) = cos x

Applying this formula we have

So comparing we have,

sin(x+16)°=cos(2x+14)°

90 - (x + 16) = 2x +14

distributing - over the bracket

90 -x - 16 = 2x + 14

bringing like terms together,

90-16-14 = 2x +x

60 = 3x

Dividing both sides by 3,

x =20

Therefore the value of sin(x+16)°=cos(2x+14)° is 20⁰

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The height of the air plane is obtained as 10503 ft.

We know that we can be able able to use the principles that can apply to the right triangle so as to be able to obtain the height that the triangle has been able to attain.

We know that in working with the right triangle, we have to look out for the ratio that we can be able to use to obtain the height that the problem in this case requires that we should find and as such we have that;

Tan 35 = H/15000

H = height of the airplane

H = 15000 Tan35

H = 10503 ft

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Answer:

65° + 127° + 140° = 332°

360° - 332° = 28° (measure of missing interior angle)

28° + c = 180°, so c = 152°.

The value of angle C is 152°

A quadrilateral is a four-sided polygon, which means it is a closed shape with four straight sides. The sum of the angles in a quadrilateral is always 360 degrees.

To solve the given problem, find the missing angle using the sum of angles and find the angle C with the 4th angle of the Quadrilateral.

Here we have

A quadrilateral with angles 65°, 140°, 127°

Let 'x' be the missing angle

As we know the sum of angles in a quadrilateral is 360°

=>  65° + 140° + 127° + x = 360°

=>  332° + x = 360°

=>  x = 28°  

Here ∠C + 28° = 180°   [ Sum of Linear angles ]  

=> ∠C = 180° - 28°

=> ∠C = 152°

Therefore,

The value of angle C is 152°

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