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Using the linear function y = 0.6x + 1.5, he drove 20 miles to be charged $13.50

The parent linear function is f(x) = x, which is a line passing through the origin. In general, a linear function equation is f(x) = mx + c

where m is the slope of the equation and c is the y - intercept.

To solve this problem, we have to write an equation that models y = mx + c

The linear function to model this is given as;

y = 0.6x + 1.5

Assuming a trip cost 13.50, the number of miles can be calculated as;

13.50 = 0.6x + 1.5

0.6x = 13.50 - 1.5

0.6x = 12

x = 12/0.6

x = 20

The drove 20 miles

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The answer is that, save from 13, all real numbers fall into the domain of (fog)(x).

So, {x|x≠13} option 4 is correct.

A function's domain is the collection of all potential inputs. For instance, all real numbers are in the domain of f(x)=x², and all real numbers are in the domain of g(x)=1/x, with the exception of x=0.

Let y = f(x) be a function where x and y are the independent and dependent variables. If a function f offers a means of successfully generating a single value y while using a value for x to that end, then that selected x-value is said to belong to the domain of f.

The result of multiplying the expression for f(x) by the expression for g(x) is (x + 7)(1/x-13).

This can be expressed much more clearly as (x + 7) (x – 13)

All real numbers, with the exception of 13, can be used to replace x in the expression in this condensed form. This is due to the fact that if we substitute 13 for x, our denominator will be equal to 0, making our formula illogical.

The answer is that, save from 13, all real numbers fall into the domain of (fog)(x).

So, {x|x≠13}

Therefore, option 4 is correct.

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Answer:

Step-by-step explanation:

You want to know the prices of jelly beans and almonds when you can buy 7 pounds of jelly beans and 9 pounds of almonds for $46, or 5 pounds of jelly beans and 3 pounds of almonds for $20.

The equations describing each purchase can be written as ...

A graphing calculator plots these equations easily, and shows you the solution is (j, a) = (1.75, 3.75).

Jelly beans are $1.75 per pound; almonds are $3.75 per pound.

If you like, you can solve these equations by elimination. We notice the coefficients of 'a' have a simple ratio, so we can eliminate 'a' easily.

  3(5j +3a) -(7j +9a) = 3(20) -(46) . . . . . subtract the first from 3 times the second

  8j = 14 . . . . . simplify

  j = 1.75 . . . . . divide by 8

Substituting into the second equation, we get ...

  5(1.75) +3a = 20

  3a = 11.25 . . . . . . . . . subtract 8.75

  a = 3.75 . . . . . . . . divide by 3

Jelly beans are $1.75 per pound; almonds are $3.75 per pound.

Answer:

To graph an equation using the slope and y-intercept, 1) Write the equation in the form y = MX + b to find the slope m and the y-intercept (0, b). 2) Next, plot the y-intercept. 3) From the y-intercept, move up or down and left or right, depending on whether the slope is positive or negative.

Step-by-step explanation:

Correct answer of the expression 5+3(x+6) is the sum of 5 and 3 times the difference of a  number by 6

A finite collection of symbols that are well-formed in mathematics according to context-dependent principles is called an expression or mathematical expression.

given

the expression 5+3(x+6)

it is concluded from the expression 5+3(x+6) that the sum of 5 and 3 times the difference of the number which is x by 6

so the correct answer of the expression 5+3(x+6) is the sum of 5 and 3 times the difference of a  number by 6

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Answer:  the correct answer is D

Step-by-step explanation:

The altitude of the balloon now is 324 meters

From the question, we have the following parameters that can be used in our computation:

Initial altitude = 288 meters

Rise = 36 meters

The altitude of the balloon now is calculated as

Current altitude = Initial altitude + Rise in altitude

Substitute the known values in the above equation, so, we have the following representation

Current altitude = 288 + 36

Evaluate the sum

Current altitude = 324

Hence, the current altitude is 324 meters

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Jocelyn buys 41 candies and 25 pretzels by solving system equation using elimination method.

What is a linear equation?

The equations with one, zero, or an infinite number of solutions are known as linear equations with two variables. Each of the two variables in these equations has the largest exponent order of 1. A two-variable linear equation has the conventional form axe + by + c = 0, where x and y are the two variables. The answers can also be expressed as ordered pairs, such as (x, y).

Given that the cost of 1 candy is $4.75 and 1 pretzel is $3.50.

Jocelyn buys 18 candies and pretzels altogether with cost $75.50.

Assume that she buys x candies and y pretzels

Therefore,

x + y = 18  ......(i)

The cost of x candies and y pretzels is 4.75x + 3.50y.

4.75x + 3.50y = 75.50  .....(ii)

Solving equation (i) and (ii) by using elimination method.

Multiply equation (i) by 4.75

4.75x + 4.75y = 85.5 ....(iii)

Subtract equation (ii) from (iii)

4.75x + 4.75y = 85.5

4.75x + 3.50y = 75.50

(-)      (-)             (-)

________________

             1.25 y = 10

                    y  = 8

Putting y = 8 in equation (i)

x + 8 = 18

x = 10

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The length of string in feet she will need for her patio is equal to : L{s} = 2(x + y).

Given is Camilla who wants to attach a string of lights to the edges of her patio for a party. She does not want the string to go across the edge with the steps.

Assume the shape of the patio is a rectangle with the dimensions of [x] and [y] units long. The length of string in feet she will need will be equivalent to the perimeter of the rectangle. So, the length of string in feet she will need is equal to L{s} = 2(x + y).

Therefore, the length of string in feet she will need for her patio is equal to : L{s} = 2(x + y).

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The solution to the inequality is (0.4.3.8). The graph of the inequalities is attached with the answer.

When two expressions are connected by a sign like "not equal to," "greater than," or "less than," it is said to be inequitable. The inequality shows the greater than and less than relation between variables and the numbers.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given inequalities are.

y≤2x+3

y> -3x+5

When we plot the graph of the inequalities on the x-y axis the two lines intersect at the point (0.4,3.8). The graph is attached with the answer below.

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To find the value of g(3), we need to substitute the value 3 into the expression for G(x) in place of x and then evaluate the resulting expression. Since G(x) = 4x - 5, we can substitute 3 for x to get G(3) = 4 * 3 - 5 = 12 - 5 = 7. Therefore, g(3) = 7.

Answer:

y = 17.86

Step-by-step explanation:

y = 13.06 + .6x  Substitute in 8 for x

y = 13.06 + .6(8)

y = 13.06 + 4.8

y = 17.86

The derivative of the function is not defined overall on the interval [1, 5]

       [tex]F(x)=\int_{a}^{x}\frac{t +1}{t - 1} dt\\\\F^{'}(x) = \frac{x+1}{x-1}[/tex]

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Answer:

slope = - 2

Step-by-step explanation:

calculate the slope m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (11, - 17) and (x₂, y₂ ) = (2, 1) ← 2 ordered pairs from the table

m = [tex]\frac{1-(-17)}{2-11}[/tex] = [tex]\frac{1+17}{-9}[/tex] = [tex]\frac{18}{-9}[/tex] = - 2

Answer:

Step-by-step explanation:

To solve this problem, we need to add and subtract the fractions given. To do this, we need to find a common denominator for all of the fractions.

The least common multiple of 4, 16, and 12 is 48, so we can rewrite each fraction with a denominator of 48:

1/4 = 3/12

29/16 = 87/48

9735/12 = 4047/48

Then, we can add and subtract the fractions as follows:

3/12 + 87/48 - 4047/48 = (3 - 4047)/48 = -4044/48 = -84/16

Therefore, the final answer is -84/16, or -5 and 1/16 in simplified form.

What is the definition of Bernoulli's principle?

In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.

p+12ρ v2+ρgh=konstant   Bernoulli equation(1)

Two states on a streamline are thus linked by the following equation:

p1+12ρv21+ρgh1=p2+12ρv22+ρgh2p1+12ρv21=p2+12ρv22p2=p1+12ρv21−12ρv2

Due to the constriction of the cross-section to only half the size, the flow speed is doubled. All known values can now be put in equation (5). Note that the pressure is to be used in the basic unit N/m² and the density in the unit kg/m³.

p2=p1+12ρv21−12ρv22p2=4⋅105Nm²+121000kgm³ (4ms)2–121000kgm³ (8ms)2=3.76⋅105Nm²p2=3.76 bar

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The correct statement is "each number in Sabrina’s pattern is 3 times the corresponding number in Martina’s pattern. Each number in Sabrina’s pattern is 3 times the corresponding number in Martina’s pattern."

What is a statement?

A statement in mathematics is a declarative utterance that can only be either true or false. A proposal is another name for a statement. It's important that there be no ambiguity. A sentence can only be true or untrue and not both for it to be a statement.

Given that:

Martina used the rule “Add 2.”

Sabrina used the rule “Add 6.”

The rule that Martina is follow is

M(n) = starting number + 2n

M(n) = 0 + 2n

M(n) = 2n

The rule that Sabrina is follow is

S(n) = starting number + 6n

S(n) = 0 + 6n

S(n) = 6n

S(n) = 3 × 2n

S(n) = 3 × S(n)

Thus Each number in Sabrina’s pattern is 3 times the corresponding number in Martina’s pattern. Each number in Sabrina’s pattern is 3 times the corresponding number in Martina’s pattern.

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The given equations can be simplified as :

a. -2.√15

b.  c ≥ 0

c.   x = ±2√6

d.   y < 0

e.   a > 0

How to solve inequalities ?

When solving an inequality:

a.

Given : √(3-√15)²-√(3+√15)²

   √(3-√15)²-√(3+√15)² = (3-√15)-(3+√15)

   √(3-√15)²-√(3+√15)² = 0 -2√15

   √(3-√15)²-√(3+√15)² = -2.√15

b .  

Given : 3√c² if c≥0

     3√c² if c≥0

    c ≥ 0

c.

Given: √(x²+1)²=5

     √(x²+1)²=5

      on Squaring both sides , we get

      (x²+1) = 25

      x²  = 24

      x = ±2√6

d.

Given : -5√y² if y>0

      -5√y² if y>0

       y < 0 [since , -5 < 0]

e.  

Given : 0.5√16a² if a<0

     0.5√16a² if a<0

      a > 0

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The two true options are A and D

The two options that are true about the graph that represents this real-world scenario are;

All the t values must be positive, and a viable solution is (10, 50). Hence, the correct option for this question is A and D.

Plotting the data in the graph is advantageous because it gives a visual presentation of the data and makes it easy to observe and forecast the large data set in a short time.

In the case of this question, the temperature is directly reliant on the c number of cricket chirps heard in 14 seconds. Therefore, if all the c values must be positive integers, the values of t must also be positive numbers.

t = c + 40

Because 10 is a positive integer, it is a possible c value

If c = 10

t = 40 + 10

t = 50

A viable solution (10, 50) represents the values of c and t, respectively, to predict the data more accurately and with validity.

Therefore, the correct option for this question is A and D.

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Given that they are age 55 or older, it is discovered that there is a (E) 0.8181 = 81.81% probability that the person said no.

Simply put, probability refers to the likelihood that something will occur.

If we don't know how an event will turn out, one can discuss the probability or likelihood of several events.

Statistics is the study of occurrences that match a probability distribution.

So, as 36 out of 44 adults aged 55 or older chose not to answer the question, the probability is given by:

p = 36/44 = 0.8181

Therefore, given that they are age 55 or older, it is discovered that there is a (E) 0.8181 = 81.81% probability that the person said no.

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Answer:

Step-by-step explanation:

x + 2/4 = x + 5/5

x + 1/2 = x + 1

1/2 = 1 or 0.5 = 1

or

x + 2/4 = x + 5/5

x + 1/2 = x + 1

0 = 1/2 or 0 = 0.5

If x = 2, y = 6, and z = 4, which expression is equivalent to 4? à 54+0-3+2=4. D Xtra 4 ... A tree is 12 feet tall and casts a shadow 9 feet long. A building nearby.

The probability of getting a specific number on the first roll of the die is 1/6. The probability of getting that same number on five successive rolls is (1/6)^5 = 1/7776. Because there are six different ways to get the same number on five rolls, multiply by 6 to get the probability of the same number on five successive rolls to be 6/7776 =1/1296.

Determining the probability that all five rolls will be different starts with asking how many ways can you choose 5 numbers from six with replacement and without repetition where order does not matter, then dividing that number by the total number of outcomes in five rolls of the die.

The total number of possible rolls is easy. There are six possible outcomes for each of the five rolls, so the total number of possible combinations after 5 rolls is 6^5 = 7776.

Now if each roll has to be a different number, consider that there are six acceptable outcomes on the first roll of the die. There are only five acceptable outcomes on the second roll, four on the third, three on the fourth, and two on the fifth. Multiplying all these together gives 6x5x4x3x2 = 6! = 720 ways to get a different number on five successive rolls of the die. This probability is 720/7776 ( 0.0926 or 9.26% or 120 times greater than getting the same number on each of five rolls.

Consider a game in which six true dice are rolled. What is the probability of obtaining exactly one ace, at least one ace and exactly two aces?

Exactly 1 ace(one) =6C1 * 5^5 = 18750 / 6^6 = 0.401877572

At Least 1 ace(one) =6C1 *5^5 +6C2 * 5^4 + 6C3 * 5^3......6C6 * 5^0= 31031/6^6 = 0.6651020233

Exactly 2 aces(ones) =6C2 * 5^4 = 9375 / 6^6 = 0.200938786

If you require two specific times(e.g. first time and second time) to be 5(probability 1/6), others to be non-5(probability 1−1/6=5/6). The probabiliy is (1/6)2(5/6)3, you have C52=10 different ways to roll two 5. Therefore the answer is 10(1/6)2(5/6)3, which is approximately 0.1608.

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The  GCF of each expression that was given above are: 3, 2 , 4 , and 6 which can be written as:

The GCF which isthe “greatest common factor”.  can be  defined as the largest number that is a factor of two or more numbers.

Intance of this is that the GCF of 24 and 36 is 12, and this is due to the fact that the largest factor that is shared by 24 and 36 is 12.

Option 1:

The GCF of 12y-3=3 because the common factor which is the highest factor of 12 and 3 is 3

Option2

The GCF of 4y+10=2  because the common factor which is the highest factor of 4 and 10 is 2 and so on.

Option 3:

The GCF of 28 − 8= 4 because the common factor which is the highest factor of 28 and 8 is 4

Option 4:

The GCF of 30y+18=6 because the common factor which is the highest factor of 30 and 18 is 6

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An essential part of a business' operations is its fixed and variable expenses.

It is an expenditure that does not fluctuate, or rather, does not alter over time.

An illustration of a fixed expense is:

1. Mortgage,

2. Rent payments,

3. Utility bills etc

Let see about Variable expense:

It is a cost that typically varies based on how the business uses the goods or services.

Examples of variable costs are as follows:

1. Sales commissions

2. Direct labor costs

3. Cost of raw materials etc

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[tex]6\frac{2}{5} - 4\frac{4}{5}[/tex]  can be solved using subtraction of simple fraction and the final result is 8/5 .

what are simple fraction ?

A fraction in which both the numerator and the denominator consist of whole numbers.

Simplest form of a fraction:

A fraction is said to be in its simplest form if 1 is the only common factor of its numerator and denominator. For example, 8/9 ,because 1 is the only common factor of 8 and 9 in this fraction.

Simplifying proper and improper fraction

We simplify fractions because it is always to work or calculate when the fractions are in the simplest form.

To solve :  [tex]6\frac{2}{5} - 4\frac{4}{5}[/tex]

We know that , in simple fraction  [tex]6\frac{2}{5} - 4\frac{4}{5}[/tex] can be written as ,

[tex]6\frac{2}{5} - 4\frac{4}{5} = \frac{32}{5} - \frac{24}{5} = \frac{8}{5}[/tex]

Hence , 8/5 is the final answer .

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The answers for the following trigonometric functions are:

a) Sin(A)=√15/4

b) Cos(A)=1/4

c) Cos(A+B)=(√3-√15)/8

Cos (A- B)=(√3+15)/8

d) Sin(A+ B) =(3√5+1)/8

Sin (A- B)==(3√5-1)/8

e) The quadrant of A+B and A-B is 4th quadrant

f)  Sin(2A + 2B)= (√15-6√5+4√3)/8

The trigonometric functions in mathematics are real functions that link the angle of a right-angled triangle to the ratios of two side lengths. They are also known as circular functions, angle functions, or goniometric functions. They are widely employed in all fields of geometry-related study, including geodesy, solid mechanics, celestial mechanics, and many more. Because they are some of the most basic periodic functions, Fourier analysis is frequently employed to examine periodic events.

The sine, cosine, and tangent are the trigonometric functions that are most frequently utilized in contemporary mathematics.

Given,

Cos(A) = 1/4 and Sin(B) = 1/2

4=√1+x²

1+x²=16

x²=15

x=√15

4=1+x²

x²=3

x=√3

a) Sin(A)=√15/4

Sin(B)=1/2

b) Cos(A)=1/4

Cos(B)=√3/2

c) Cos(A+B)=CosACosB-SinASinB

=(1/4)(1/2)-(√15/4)(1/2)

=(√3-√15)/8

Cos(A-B)=CosACosB+SinASinB

=(1/4)(√3/2)+(√15/4)(1/2)

=(√3+15)/8

d) Sin(A+B)+SinACosB+CosASinB

=(√15/4)(√3/2)+(1/4)(1/2)

=(3√5+1)/8

Sin(A-B)=SinACosB-SinBCosA

=(√15/4)(√3/2)-(1/4)(1/2)

=(3√5-1)/8

e) The quadrant of A+B and A -B is 4th quadrant.

f) Sin(2A+2B)=Sin2ACos2B+Cos2ASin2B

=(√15/8)(1-√3)+((4-√15)/4)(√3/2)

=(√15-3√5+4√3-3√5)/8

=(√15-6√5+4√3)/8

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Answer:

351.5625

1,440,000

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{7 cm}\underline{General form of an Exponential Function}\\\\$y=Ae^{kt}$\\\\where:\\\phantom{ww}$\bullet$ $A$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $k$ is a constant.\\ \phantom{ww}$\bullet$ $t$ is time.\\\end{minipage}}[/tex]

Given:

(Let t = time in minutes).

If the doubling period is 15 minutes, then at t = 135 minutes, y = 180,000:

[tex]\implies 90000=Ae^{120k}[/tex]

[tex]\implies 180000=Ae^{135k}[/tex]

Divide the second equation by the first to eliminate A, and solve for k:

[tex]\implies \dfrac{180000}{90000}=\dfrac{Ae^{135k}}{Ae^{120k}}[/tex]

[tex]\implies 2=\dfrac{e^{135k}}{e^{120k}}[/tex]

[tex]\implies 2=e^{135k} \cdot e^{-120k}[/tex]

[tex]\implies 2=e^{15k}[/tex]

[tex]\implies \ln 2 = \ln e^{15k}[/tex]

[tex]\implies \ln 2 =15k \ln e[/tex]

[tex]\implies \ln 2 =15k[/tex]

[tex]\implies k=\dfrac{1}{15}\ln 2[/tex]

Substitute t = 120, y = 90000 and the found value of k into the formula and solve for A:

[tex]\implies 90000=Ae^{\left(120 \cdot \frac{1}{15}\ln 2\right)}[/tex]

[tex]\implies 90000=Ae^{\left(8\ln 2\right)}[/tex]

[tex]\implies 90000=Ae^{\ln256}[/tex]

[tex]\implies 90000=256A[/tex]

[tex]\implies A=\dfrac{90000}{256}[/tex]

[tex]\implies A=351.5625[/tex]

Therefore, the function that models the scenario is:

[tex]\large\boxed{y=351.5625e^{\left(\frac{1}{15}t \ln 2\right)}}[/tex]

So the initial population at time t = 0 was:

To find the size of the bacteria population after 3 hours, substitute t = 180 into the found formula:

[tex]\implies y=351.5625e^{\left(\frac{1}{15}(180) \ln 2\right)}[/tex]

[tex]\implies y=351.5625e^{\left(12 \ln 2\right)}[/tex]

[tex]\implies y=351.5625e^{\left(\ln 4096\right)}[/tex]

[tex]\implies y=351.5625 \cdot 4096[/tex]

[tex]\implies y=1440000[/tex]

Therefore, the size of the bacterial population after 3 hours was:

The initial population at the time t = 0 is 351.5625. And the size of the bacterial population after 3 hours is 1,440,000.

An exponential function's curve is created by a pattern of data called exponential growth, which exhibits higher increases over time.

If n₀ is the initial size of a population experiencing exponential growth, then the population n(t) at time t is modeled by the function:

n(t) = n₀(e[tex])^{rt}[/tex]

Where r is the relative rate of growth expressed as a fraction of the population.

Given:

Doubling period = 15 minutes

At t = 120 minutes, n(t) = 90,000

If the doubling period is 15 minutes, then at t = 120+15 = 135 minutes,

90000 = n₀(e[tex])^{120r}[/tex]

18000 =  n₀(e[tex])^{135r}[/tex]]

To find the r:

Take ratio of both of the equations,

90000 / 18000 = n₀(e[tex])^{120r}[/tex]  /   n₀(e[tex])^{135r}[/tex]

2 = (e[tex])^{135r}[/tex] . (e[tex])^{-120r}[/tex]

r = 1/15 ln2

Substitute the value of r, t and y.

90000 = n₀(e[tex])^{120r}[/tex]

90000 = 256n₀

n₀ = 351.5652

Now, the function

n(t) = n₀(e[tex])^{rt}[/tex]

n(t) = (351.5652)(e[tex])^{(1/15)(180)(ln2)}[/tex]

n(t) = 1440000

Therefore, the initial population at the time t = 0 is 351.5625. And the size of the bacterial population after 3 hours is 1,440,000.

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