3. You get three summer jobs to help you save for college expenses. In your job as a cashier,
you work 20 hours per week and earn $9.50 per hour. Your second and third jobs are at a local
hospital. There, you earn $9.00 per hour as a payroll clerk and $7.00 per hour as an aide. You
always work 10 hours less per week as an aide than you do as a payroll clerk. Your total weekly
salary depends on the number of hours you work at each job.
a. Determine the input and output variables for this situation.
b. Explain how you calculate the total amount earned each week.
c. If x represents the number of hours you work as a payroll clerk, represent the number of
hours you work as an aide in terms of x.
d. Write an equation that describes the total amount you earn each week. Use x to represent
the input variable and y to represent the output variable. Simplify the expression as much
as possible.
e. If you work 12 hours as a payroll clerk, how much will you make in one week?
f. What are the practical replacement values for x? Would 8 hours at your payroll job be a
realistic replacement value? What about 50 hours?
g. When you don't work as an aide, what is your total weekly salary?
Please help!

Answer:

Bus vs Highway: Houston

Let's start by finding out how many people took the bus.

If the number of people who took the highway is x, then the number of people who took the bus is 1.5x (since it's one and a half times the number of people who took the highway).

We know that the total number of people who traveled to Houston is 31,426, so we can write an equation:

x + 1.5x = 31,426

Simplifying the equation:

2.5x = 31,426

Dividing both sides by 2.5:

x = 12,570.4

Since we can't have a fraction of a person, we can round this down to 12,570.

Now we can find the number of people who took the bus:

1.5x = 1.5(12,570) = 18,855

To find out how many more people took the bus, we can subtract the number of people who took the highway from the number of people who took the bus:

18,855 - 12,570 = 6,285

Therefore, 6,285 more people took the bus.

(please mark my answer as brainliest)

In response to the stated question, we may state that As a result, the probability of a randomly picked client at Dillon's salon having blond hair is one-quarter.

Probabilistic theory is a branch of mathematics that calculates the likelihood of an event or proposition occurring or being true. A risk is a number between 0 and 1, with 1 indicating certainty and a probability of around 0 indicating how probable an event appears to be to occur. Probability is a mathematical term for the likelihood or likelihood that a certain event will occur. Probabilities can also be expressed as numbers ranging from 0 to 1 or as percentages ranging from 0% to 100%. In relation to all other outcomes, the ratio of occurrences among equally likely alternatives that result in a certain event.

Dillon's salon has 4 clients with blond hair and 12 clients with different hair colours, for a total of 4+12=16 clients.

The chance of picking a blond-haired customer is equal to the number of blond-haired clients divided by the total number of clients:

P(blond) = number of blond clients / total number of clients = 4 / 16 = 1/4

As a result, the likelihood of a randomly picked client at Dillon's salon having blond hair is one-quarter.

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P(blond) = 4/16 = 1/4 .is the probability that a randomly selected client at Dillon's salon has blond hair.

Probability is a measure of the likelihood that an event or experiment will occur. It is expressed as a number between 0 and 1, with 0 meaning that the event is impossible to occur and 1 meaning that the event is certain to occur. Probability can be calculated using a variety of methods depending on the type of problem. For example, the probability of a coin being heads can be calculated by dividing the number of heads by the total number of coin flips.

Probability theory is an important part of statistics and is used to make predictions about the likelihood of certain events occurring. It is also used to assess risk and make decisions in a wide range of fields, from economics and finance to medicine and engineering.

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Answer:

To find g(f(2)), we first need to find f(2) and then use that result as the input for g.

We are given that f(2) = -3, so we substitute that value into g(x) to get:

g(f(2)) = g(-3)

We are also given that g(-3) = 10, so we substitute that value in:

g(f(2)) = g(-3) = 10

Therefore, g(f(2)) = 10.

Answer:

£22

Step-by-step explanation:

50% of 88=88/100 ×50=44

44÷2=25%=22

75% of £88 is deducted, so that 88-66=£22

Answer:16

Step-by-step explanation:

average is similar to mean

you get the Total value and divide it with the number of values added

The equation 32 - 3x + 13 = 0 can be rewritten as 3x = 19. Therefore, x = 19/3. Because x is a real number, the solution to the equation is x = 19/3.

Real numbers are any number that can be expressed as a decimal number or a fraction. This includes the natural numbers (1,2,3,4, etc.), the negative numbers (-1,-2,-3, etc.), the rational numbers (fractions, such as 1/2, 3/4, etc.), and the irrational numbers (numbers that cannot be expressed as a fraction, such as pi, square root of 2, etc.). Real numbers can be manipulated with basic arithmetic operations such as addition, subtraction, multiplication, and division. Real numbers have an infinite number of decimal places, making them useful for measuring physical quantities such as time, distance, and temperature.

Expressing this answer in the form a + bi, where a and b are real numbers, gives the solution a = 19/3 and b = 0. Therefore, the answer is 19/3 + 0i = 19/3.

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Simplify the expression:

[tex]\dfrac{cot^2(a)(1+csc(a))(1-csc(a))}{(1+sec(a))(1-sec(a))}[/tex]

A parabola with a vertex at (1,3) and an opening downhill is depicted by the equation.

A parabola is an equation of a curve with a spot on it that is equally spaced from a fixed point and a fixed line.

In mathematics, a parabola is a roughly U-shaped, mirror-symmetrical plane circle. The same curves can be defined by a number of apparently unrelated mathematical descriptions, which all correspond to it. A point and a line can be used to depict a parabola.

Equation given: 3x² - 6x + 4y - 9 = 0. When the given equation's graph is plotted, it is discovered that the parabola that is created is opened downward and has a vertex at the spot. ( 1,3). The graph and the following response are attached.

The equation that depicts a parabola with a vertex at (1,3) opening downward is option C, making it the right choice.

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Answer:

Parabola with a vertex at (1, 3) opening downward.

Step-by-step explanation:

If the top of Bria's bookcase has a length-to-width ratio of 3:1, the width of the top should be about 10 inches to provide an area of 300 square inches for her soap carving collection.

The top of the bookcase is a rectangle, the area of the top can be expressed as the product of its length and width:

Area of the top = length × width

Area of the top = 300 square inches

length × width = 300 square inches

width = 300 square inches / length

Assuming that the top of the bookcase is with a length-to-width ratio of

3 : 1

length = 3 × width

substituting the value gives

width = 300 / (3 × width)

Simplifying, we have:

width² = 100 square inches (the polynomial)

Taking the square root of both sides, we get:

width = √100 ≈ 10 inches

Then the length will be = 30 inches

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The length of quadrilateral ABCD of AD, which is closest to option A, is [tex]4*sqrt(13)[/tex] (34 units).

A triangle will always have an angle total of 180 degrees. A quadrilateral can be divided in half from corner to corner to form a triangle because the angle total of a quadrilateral is equal to 360°. A triangle is effectively half of a quadrilateral, therefore it makes sense that its angle measurements are also half. 180° is the half of 360°.

The Pythagorean theorem can be used to calculate the duration of AD.

The distance formula can be used to get the length of AB first:

[tex]AB = sqrt((4-(-4))2 + (6-(-2))2 = sqrt(8-(-8-8) + 8-(8) = 8*sqrt (2)[/tex]

The distance formula can then be used to get the length of AC:

[tex]Sqrt((12-(-4)) = AC^2 + (6-(-2))^2)= sqrt(16^2 + 8^2) = 8*sqrt(5) (5)\\[/tex]

Now, we can calculate the length of AD using the Pythagorean theorem:

[tex]AD^2 = AB^2 + BD^2AD^2 = (8sqrt(2))^2 + (4sqrt(5))^2AD^2 = 128 + 80AD^2 = 208AD = sqrt(208)[/tex]

Simplifying the radical, we get:

AD = sqrt(1613) = 4sqrt(13)

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The length of quadrilateral ABCD of AD, which is closest to option A, is  (34 units). Thus, option A is correct.

The sum of the angles in a triangle is always 180 degrees. Because a quadrilateral's total angles are 360°, it can be divided in half from a corner to create a triangle.

The distance formula can be used to get the length of AB first:   [tex]AB = sqrt((4 — (-4))2+ (6— (-2))2 = sqrt(8 — (-8— 8)+8— (8) = 8* sqrt(2)[/tex] The distance formula can then be used to get the length of AC:

[tex]Sqrt((12 — (-4)) = AC2 + (6 — (-2))2) = sqrt(162 + 82) = 8* sqrt(5)(5)[/tex]

Now, we can calculate the length of AD using the Pythagorean theorem:  

[tex]AD2 = AB2 + BD2AD2 = (8sqrt(2))2 + (4sqrt(5))2AD2 = 128 + 80AD2 = 208AD = sqrt(208)[/tex]

Simplifying the radical, we get:  [tex]AD = sqrt(1613) = 4sqrt(13)[/tex]

Therefore, It makes obvious that since a triangle is functionally half of a quadrilateral, its angle measurements are also half. Half of a 360° circle is 180°.

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Answer:

12 dollars

Step-by-step explanation:

480 / 40 = 12

12 dollars

A fraction of the variation in y that can be explained by the variation in the values of x is equal to 0.25189186354.

In Mathematics, the standard form of the equation of a regression line is represented or modeled by the following mathematical expression;

y = bx + c

Where:

In Mathematics and Statistics, the value of slope can be calculated by using the following mathematical expression;

[tex]b=r(\frac{S_y}{S_x})[/tex]

where:

By rearranging, we have:

[tex]r=b(\frac{S_x}{S_y})[/tex]

r = 1.13(1.67/3.76)

r = 0.50188829787

By taking the square of both sides, we have:

r² = 0.25189186354.

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We can factor the polynomial 6z² + 6z as follows:

[tex]6z^2 + 6z = 6z(z + 1)[/tex]

We can use this factorization to express the area of the wall as the product of two factors:

[tex]6z^2 + 6z = 6z(z + 1) = length × width[/tex]

Therefore, the length of the wall is 6z, and the width of the wall is z + 1.

The first limit,  [tex]\lim_{x\to \→-2^- } -3(x+2)/x²+4x+4[/tex]  , evaluates to negative infinity, while the second limit, [tex]\lim_{ x\to \-2^+}-3(x+2)/x²+4x+4[/tex] , evaluates to positive infinity.

Function in maths is a relation between two sets of values. It is a type of mathematical equation in which each input value has a unique output value. In a function, each input value must correspond to only one output value. This means that for any input value, the output must be the same.

This indicates that the function has a vertical asymptote at x=-2.

In order to understand why this is the case, we can first rewrite the function as follows:

f(x) = -3(x+2)/(x+2)(x+2)

The denominator of the function is (x+2)(x+2), which has a double root at x=-2. This means that the denominator is equal to zero when x=-2. As a result, the function f(x) will have a vertical asymptote at x=-2, since the denominator will be equal to zero and the function will approach negative or positive infinity. This is why the two limits mentioned above both evaluate to either negative or positive infinity.

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The limit of the given functions are:

[tex]\lim_{x \to \--2^-}[/tex]-3(x+2)/ x²+4x+4 = positive infinity

[tex]\lim_{x \to \--2^+}[/tex] -3(x+2)/ x²+4x+4 = negative infinity

[tex]\lim_{x \to \--2[/tex] -3(x+2)/ x²+4x+4 = Undefined

Function is a relation between two sets of values. In a function, each input value must correspond to only one output value. This means that for any input value, the output must be the same.

[tex]\lim_{x \to \--2^-}[/tex]-3(x+2)/ x²+4x+4

= -3(-2+2)/(-2)²+4(-2)+4

= -3/0 + 8 + 4

= +∞  (infinity)

[tex]\lim_{x \to \--2^+}[/tex] -3(x+2)/ x²+4x+4

= -3(-2+2)/(-2+2)²+4(-2+2)+4

= -3/4 + 0 + 4

= -∞

[tex]\lim_{x \to \--2[/tex] -3(x+2)/ x²+4x+4

= -3(-2+2)/(-2+2)²+4(-2+2)+4

= -3/0 + 0 + 4

= Undefined

Therefore, the limit of the functions given are:

[tex]\lim_{x \to \--2^-}[/tex]-3(x+2)/ x²+4x+4 = +∞

[tex]\lim_{x \to \--2^+}[/tex] -3(x+2)/ x²+4x+4 = -∞

[tex]\lim_{x \to \--2[/tex] -3(x+2)/ x²+4x+4 = Undefined

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1. Graph is attached below. 2. 0.51 3. the percent of tiles with an area between 15.975 and 16.01 square inches is 82.78%. 4. 0.08% 5. 304 tiles 6.  0.4938x tiles.

A uniform density function is a type of probability distribution that assigns equal probability to all values within a specified range. It is also known as a rectangular distribution because the graph of the probability density function appears as a rectangle with a constant height over the interval of possible values.

The probability density function of a uniform distribution is defined by two parameters: a minimum value (a) and a maximum value (b). The function assigns a probability of 1/(b-a) to each value within the range [a, b], and a probability of 0 to any value outside this range.

1. The graph of the uniform density function that takes on values from -2 to 3 is mentioned below

2. To find P(-1.25 < X < 1.3), we need to find the area under the uniform density function between -1.25 and 1.3. Since the density function is uniform, the area is simply the rectangle with base (1.3 - (-1.25)) = 2.55 and height 1/5 (since the range of values is 5). Therefore,

P(-1.25 < X < 1.3) = (2.55 * 1/5) = 0.51

3. To find the percent of tiles that have an area between 15.975 and 16.01 square inches, we need to standardize the values using the z-score formula and then find the area under the normal curve between those two values:

z1 = (15.975 - 16) / 0.008 = -3.125

z2 = (16.01 - 16) / 0.008 = 1.25

Using a standard normal table or calculator, we find the area between these two z-scores to be approximately 0.8278.

Therefore, the percent of tiles with an area between 15.975 and 16.01 square inches is 82.78%.

4. To find the probability that a tile will have an area that is more than 16.025 square inches, we again need to standardize the value using the z-score formula and then find the area under the normal curve to the right of that value:

z = (16.025 - 16) / 0.008 = 3.125

Using a standard normal table or calculator, we find the area to the right of this z-score to be approximately 0.0008.

Therefore, the probability that a tile will have an area that is more than 16.025 square inches is 0.08%.

i. Yes, this is unusual because the probability is very low, indicating that it is highly unlikely for a tile to have an area greater than 16.025 square inches.

5. To find the number of tiles that can be expected to have a surface area less than 15.985 square inches, we first need to standardize the value using the z-score formula:

z = (15.985 - 16) / 0.008 = -1.875

Using a standard normal table or calculator, we find the probability to the left of this z-score to be approximately 0.0304.

Therefore, out of 10,000 tiles, we can expect approximately 0.0304 * 10,000 = 304 tiles to have a surface area less than 15.985 square inches.

6. To find the number of tiles that must be produced to ensure that the manufacturer has 70,000 tiles that are between 15.98 and 16.02 square inches, we need to find the z-scores for the lower and upper bounds of this range:

z1 = (15.98 - 16) / 0.008 = -2.5

z2 = (16.02 - 16) / 0.008 = 2.5

Using a standard normal table or calculator, we find the probability between these two z-scores to be approximately 0.4938.

Therefore, out of x tiles, we can expect approximately 0.4938x tiles to have a surface area between 15.

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1. The graph of the uniform density function that takes on values from -2 to 3 is attached below.

2. P(-1.25 < X < 1.3) = (2.55 * 1/5) = 0.51

3. The percent of tiles with an area between 15.975 and 16.01 square inches is 82.78%.

4. The probability that a tile will have an area that is more than 16.025 square inches 0.08%.

5. Number of tiles that can be expected to have a surface area less than 15.985 square inches

6.   To find the number of tiles that must be produced to ensure that the manufacturer has 70,000 tiles that are between 15.98 and 16.02 square inches, 0.4938x tiles.

A uniform density function is a type of probability distribution that assigns equal probability to all values within a specified range. It is also known as a rectangular distribution because the graph of the probability density function appears as a rectangle with a constant height over the interval of possible values.

1. The graph of the uniform density function that takes on values from -2 to 3 is mentioned below:

2. To find P(-1.25 < X < 1.3), we need to find the area under the uniform density function between -1.25 and 1.3. Since the density function is uniform, the area is simply the rectangle with base (1.3 - (-1.25)) = 2.55 and height 1/5 (since the range of values is 5). Therefore,

P(-1.25 < X < 1.3) = (2.55 * 1/5) = 0.51

3. To find the percent of tiles that have an area between 15.975 and 16.01 square inches, we need to standardize the values using the z-score formula and then find the area under the normal curve between those two values:

z1 = (15.975 - 16) / 0.008 = -3.125

z2 = (16.01 - 16) / 0.008 = 1.25

Using a standard normal table or calculator, we find the area between these two z-scores to be approximately 0.8278.

Therefore, the percent of tiles with an area between 15.975 and 16.01 square inches is 82.78%.

4. To find the probability that a tile will have an area that is more than 16.025 square inches, we again need to standardize the value using the z-score formula and then find the area under the normal curve to the right of that value: z = (16.025 - 16) / 0.008 = 3.125

Using a standard normal table or calculator, we find the area to the right of this z-score to be approximately 0.0008.

Therefore, the probability that a tile will have an area that is more than 16.025 square inches is 0.08%. Yes, this is unusual because the probability is very low, indicating that it is highly unlikely for a tile to have an area greater than 16.025 square inches.

5. To find the number of tiles that can be expected to have a surface area less than 15.985 square inches, we first need to standardize the value using the z-score formula:

z = (15.985 - 16) / 0.008 = -1.875

Using a standard normal table or calculator, we find the probability to the left of this z-score to be approximately 0.0304.

Therefore, out of 10,000 tiles, we can expect approximately 0.0304 * 10,000 = 304 tiles to have a surface area less than 15.985 square inches.

6. To find the number of tiles that must be produced to ensure that the manufacturer has 70,000 tiles that are between 15.98 and 16.02 square inches, we need to find the z-scores for the lower and upper bounds of this range:

z1 = (15.98 - 16) / 0.008 = -2.5

z2 = (16.02 - 16) / 0.008 = 2.5

Using a standard normal table or calculator, we find the probability between these two z-scores to be approximately 0.4938.

Therefore, out of x tiles, we can expect approximately 0.4938x tiles to have a surface area between 15.

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The joint CDF of X and Y, two independent exponential random variables with parameters 1 and 2, respectively, is Fxy(a,b) = (1 - e^(-a))(1 - e^(-2b)), where a > 0 and b > 0. The correct answer is C).

Since X and Y are independent, the joint CDF of X and Y is the product of their marginal CDFs:

FXY(a,b) = FX(a)FY(b)

where FX and FY are the CDFs of X and Y, respectively.

The CDF of an exponential distribution with parameter λ is given by:

FX(x) = 1 - e^(-λx)

Therefore, the marginal CDFs of X and Y are:

FX(a) = 1 - e^(-a), λ = 1

FY(b) = 1 - e^(-2b), λ = 2

Taking the product, we get:

FXY(a,b) = FX(a)FY(b) = (1 - e^(-a))(1 - e^(-2b))

Therefore, the answer is:

Fxy (a,b) = (1 - e^(-a))(1 - e^(-2b)), a > 0, b > 0.

The correct option is C).

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Answer:2

Step-by-step explanation:

Answer:

2 Tons

Step-by-step explanation:

Homer’s car weighs 2 tons because there are 2,000 pounds in a ton and 4,000 divided by 2,000 equals 2

Answer: 77/6 or 12 5/6

Step-by-step explanation:

To solve this subtraction problem, we need to first convert the mixed numbers to improper fractions.

20 1/3 can be written as:

20 + 1/3 = 60/3 + 1/3 = 61/3

7 1/2 can be written as:

7 + 1/2 = 14/2 + 1/2 = 15/2

So, the problem becomes:

61/3 - 15/2

To subtract two fractions, we need to have a common denominator. The smallest number that both 3 and 2 divide into is 6. So we will convert both fractions to have a denominator of 6.

61/3 = (61/3) * (2/2) = 122/6

15/2 = (15/2) * (3/3) = 45/6

Now we can subtract the two fractions:

122/6 - 45/6 = 77/6

So, the final answer is:

20 1/3 - 7 1/2 = 77/6 or 12 5/6 (as a mixed number)

Answer:

30% of the number is 102.

Using the functions, we can predict that the population in the town will grow 't' times as t is directly proportional to the function.

A function in mathematics is a connection between two x and y values that come from different sets. These four types of correspondence are all possible. Yet not every correspondence serves a purpose.

The following are the results of a survey conducted by the National Institute of Standards and Technology (NIST) on the effectiveness of the standardised testing process. So, we state that a function can only have one input and one output. If we are given any x, the only y that can be coupled with that x is the one and only y. Two outputs cannot be connected to the same function.

Here in the given question,

The functions a(t) = 8.75 × (1+0.01) t and

b(t) = 8.75 × e × (0.01t) here are dependent on the variable t, where t is the time in years after 2010.

So, as t will increase after 2010, the value of a(t) and b(t) will increase with t.

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The value of R500 decrease to ratio 9:8 is x = 400.

By using the cross multiplication approach, the denominator of the first term is multiplied by the numerator of the second term, and vice versa. Using the mathematical rule of three, we may determine the answer based on proportions. The best illustration is cross multiplication, where we may write in a percentage to determine the values of unknown variables.

Given that, decrease R450 in the ratio 9:8.

Let 9 = 450

Then 8 will have the value = x.

That is,

9 = 450

8 = x

Using cross multiplication we have:

9x = 450(8)

x = 50(8)

x = 400

Hence, the value of R500 decrease to ratio 9:8 is x = 400.

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Answer: No, Chaz is not correct. Although their balances are both negative, we cannot compare them simply based on their numeric values. The magnitude of the balance does not indicate who owes more to the bank, as it depends on various factors such as account activity, fees, and interest rates. We would need to know more information about their accounts, such as the interest rates and any fees, in order to determine who owes more to the bank.

Excluding the bank fees Chaz would technically be correct.

No, Chaz is not correct.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

No, Chaz is not correct.

Although both Chaz and Will have negative checking account balances, we cannot determine who owes more to the bank based solely on the balance amount.

The balance amount only indicates how much money they owe to the bank, but it does not give any information about the amount they initially deposited or any other financial transactions they may have made.

To determine who owes more to the bank, we would need to know the initial deposit amount, the transaction history, and any fees or interest charges that have been applied to the accounts.

Without this additional information, we cannot accurately compare the two balances or determine who owes more to the bank.

Thus,

No, Chaz is not correct.

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a) The domain and the range of the function are given as follows:

The graph of the function is given by the image presented at the end of the answer.

b) The trip was 5.1875 hours long.

The function for this problem is defined as follows:

g(t) = 23 - 3.2t.

The domain is the set of input values that can be assumed by the function. The time cannot have negative measures, hence the lower bound of the domain is of zero, while the gas cannot be negative, hence the upper bound of the volume is given as follows:

23 - 3.2t = 0

3.2t = 23

t = 23/3.2

t = 7.1875 hours.

The range is given by the set of all output values assumed the function, which are the values of the gas, hence it is [0,23].

The graph is a linear function between points (0, 23) and (7.1875, 0).

At the end of the trip there were 6.4 gallons left, hence the length of the trip is given as follows:

23 - 3.2t = 6.4

t = (23 - 6.4)/3.2

t = 5.1875 hours.

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Answer:

1

Step-by-step explanation:

We must solve the absolute values first

An absolute value is the numbers distance from zero

So a simplified version of your problem would be

2-1

So your answer is 1

The absolute value of a number is its distance from 0, so |-2| is 2 and |-1| is 1. Substituting these back in, the expression 2 - 1 equals 1.

To evaluate the expression |-2| – |-1|, you first need to understand what the absolute value symbol does. The absolute value of a number is its distance from 0 on the number line, regardless of direction. So, both |-2| and |-1| become positive numbers, as distance is always positive.

Therefore, |-2| equals 2 and |-1| equals 1. If we substitute these values back into the original expression, 2 - 1, the result is 1.

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The slope of the sole sequence's perpendicular line is [tex]-\frac{3}{2}[/tex].

As two lines meet at right angles, the word "perpendicular" refers to an angle. Every direction, including up and down, across, and side to side, can be faced by a pair of perpendicular lines.

A perpendicular is a straight line in mathematics that forms a correct angle (90 °) with another line. In other words, two lines are parallel to one another if they connect at a right angle.

[tex]y = mx + b[/tex], where [tex]m[/tex] is the slope:

[tex]4x - 6y = -24[/tex]

[tex]-6y = -4x - 24[/tex]

[tex]y = (4/6)x + 4[/tex]

[tex]y = (2/3)x + 4[/tex]

So the slope of the given line is [tex]2/3[/tex].

To find the slope of a line perpendicular to this line, we need to take the negative reciprocal of [tex]2/3[/tex]:

[tex]-1/(2/3) = -3/2[/tex]

Therefore, the slope of a line perpendicular to the given line is [tex]-\frac{3}{2}[/tex].

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An exponential function can model California's population growth from 1950 to 1980. The predicted population in 2020 is around 43.3 million (43,301,300), assuming exponential growth.

To find an exponential function that models the population, we can use the formula:

P(t) = P0 * e^(rt)

where P0 is the initial population, r is the growth rate, and t is the time in years. We can use the information given to find P0, r, and then plug in t = 0 to find the exponential function.

P0 = 10,586,223 (initial population in 1950)

P(30) = 23,668,562 (population in 1980, 30 years later)

t = 30

P(0) = P0 * e^(0*r) = P0 (population in 1950)

So we have:

23,668,562 = 10,586,223 * e^(30r)

Dividing both sides by 10,586,223:

e^(30r) = 2.234

Taking the natural logarithm of both sides:

30r = ln(2.234)

r = ln(2.234)/30

Now we can use this value of r to find the exponential function:

P(t) = 10,586,223 * e^(t*ln(2.234)/30)

To predict the population in 2020, we can plug in t = 70 (since 2020 is 70 years after 1950) into the function we just found:

P(70) = 10,586,223 * e^(70*ln(2.234)/30) ≈ 43,301,300

Therefore, the predicted population of California in 2020 is approximately 43,301,300.

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The congruence x + 13 ≡ 55 (mod 34) simplifies to x ≡ 12 (mod 34). There are three solutions for x less than 100 that satisfy this congruence.

The given congruence is x + 13 ≡ 55 (mod 34). Simplifying this, we get x ≡ 12 (mod 34).

We need to find the number of solutions for x that are less than 100 and satisfy this congruence.

The general solution for the congruence x ≡ 12 (mod 34) is x = 12 + 34k, where k is an integer.

The solutions that are less than 100 are obtained when k = 0, 1, or 2.

Thus, the number of solutions is 3.

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