Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Austin plans to attend the Perry County Fair and is trying to decide what would be a better deal. He can pay $40 for unlimited rides, or he can pay $16 for admission plus $2 per ride. If Austin goes on a certain number of rides, the two options wind up costing him the same amount. What is that cost? How many rides is that?

Answer:

Step-by-step explanation:

x+9

Let x, be 4

4+9=13

given condition,

x+9<4x

4+9<4(4)

13<16

The answer is:

Work/explanation:

Our inequality is:

\(\sf{x+9 < 4x}\)

Flip it

\(\sf{4x > x+9}\)

Solve

\(\sf{4x-x > 9}\)

Combine like terms

\(\sf{3x > 9}\)

Divide each side by 3

\(\sf{x > 3}\)

hello

to solve this question, we need to sketch this situation to give us a better illustration

to find the angle of elevation, we can use trigonometric ratios here

SOHCAHTOA

since we have the value of hypothenus and opposite, we can use the sine angle to find the angle of elevation

from the calculation above, the angle of elevation is equal to 24 degrees

The probability that the student got a a a given that the student was in the afternoon class is 25/57

In math the term probability is defined as based on the possible chances of something to happen

Here we have given that  a test to a group of students, the grades by the two different classes are summarized.

and it can be written like the following;

                                a             b               c                   Total

Morning class         4             20              8                    32

Evening Class         13              5               7                   25

Total                        17             25              15                  57

Here we know that the total number of students is obtained as

=> 57

And the number of students who have been taking the afternoon class is calculated as,

=> 25

Then the probability the student got a a a given that the student was in the afternoon class is written as,

=> 25/57

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The equation of the Parabola in vertex form that has a vertex of (4, -2) and a y-intercept of (0, -66) is:y = -4(x - 4)^2 - 2

The vertex form of a parabola is given by:y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola.

We are given that the vertex is (4, -2), so we can substitute these values in the equation to get:y = a(x - 4)^2 - 2

Now, we need to find the value of "a". To do this, we can use the fact that the y-intercept is (0, -66). Since the point (0, -66) lies on the parabola, we can substitute x = 0 and y = -66 in the equation above to get:-66 = a(0 - 4)^2 - 2

Simplifying this, we get:-66 = 16a - 2

Adding 2 to both sides, we get:-64 = 16a

Dividing both sides by 16, we get:a = -4

Substituting this value of "a" in the equation above, we get the equation of the parabola in vertex form:y = -4(x - 4)^2 - 2

Therefore, the equation of the parabola in vertex form that has a vertex of (4, -2) and a y-intercept of (0, -66) is:y = -4(x - 4)^2 - 2

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The diameter οf the circle is 28/π inches οr apprοximately 8.89 inches (rοunded tο twο decimal places).

The fοrmula fοr the circumference (C) οf a circle is given by:

C = 2πr

where r is the radius οf the circle.

If the circumference οf the circle is 28 inches, we can sοlve fοr the radius by dividing bοth sides οf the equatiοn by 2π:

C/2π = r

Substituting the given value οf C = 28, we get:

r = 28/2π

r = 14/π

Finally, tο find the diameter (d) οf the circle, we multiply the radius by 2:

d = 2r

Substituting the value οf r = 14/π, we get:

d = 2(14/π) = 28/π

Therefοre, the diameter οf the circle is 28/π inches οr apprοximately 8.89 inches (rοunded tο twο decimal places).

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Answer:

Step-by-step explanation:

a. No, these are not mutually exclusive events because it is possible for a male student at Riverdale High School to be a member of the football team.

Using the Multiplication Rule, we can find the probability of at least one successful surgery out of three surgeries as follows:

P(at least one successful surgery) = 1 - P(no successful surgeries)

To find the probability of no successful surgeries, we use the complement rule and multiply the probabilities of three unsuccessful surgeries:

P(no successful surgeries) = 0.225 x 0.225 x 0.225 = 0.011390625

Therefore, the probability of at least one successful surgery out of three surgeries is:

P(at least one successful surgery) = 1 - 0.011390625 = 0.988609375

So, the answer is d. 0.989 (rounded to three decimal places).

The volumetric flow rate is represented by the symbol Q

Volume of water in the tank after one minute is 10,991 gallons

Volume of water in the tank after two minute is 10,962 gallons

Volume of water in the tank after one minute is 10,730 gallons

Volume of water in the tank after n minute is 11,020 - 29·n

The reason the above values are correct is as follows:

The given parameters are;

The volume of water the large industrial tank holds, V = 11,020 gallons

The rate of flow out of the tank through the drain on the bottom, Q = 29 gallons per minute

The initial level of water in the tank = The tank is initially filled; 11,020 gallons

Required:

The amount of water in gallons left in the tank after one minute

Solution;

The volume, \(V_{out}\), of water that flow out of the tank after a given time, t, is given as follows;

\(V_{out}\) = Q × t

When t = 1 minute, we have;

\(V_{out}\) = 29 gallons/minute × 1 minute = 29 gallons

The volume of water left in the tank, \(V_{in \, tank}\), is given as follows;

\(V_{in \, tank}\) = V - \(V_{out}\)

The volume left in the tank after t = 1 minute, where, \(V_{out}\) = 29 gallons is given as follows;

After one minute, \(V_{in \, tank}\) = 11,020 gal - 29 gal = 10,991 gal

The volume left in the tank after one minute,  \(V_{in \, tank}\) = 10,991 gallons

Similarly, after two minutes, we have;

\(V_{in \, tank}\) = V - \(V_{out}\) = V - Q × t

∴ \(V_{in \, tank}\) = 11,020 - 29 × 2 = 10,962

The number of gallons left in the tank after 2 minutes is \(V_{in \, tank}\) = 10,962 gallons

After, t = 10 minutes, we have;

\(V_{in \, tank}\) = V - Q × t

After ten minutes; \(V_{in \, tank}\) = 11,020 - 29 × 10= 10,730

The number of gallons left in the tank after 10 minutes is \(V_{in \, tank}\) = 10,730 gallons

After n minutes, we have;

t = n, therefore;

\(V_{in \, tank}\) = 11,020 - 29 × n = 11,020 - 29·n

The number of gallons left in the tank after n minutes is \(V_{in \, tank}\) = 11,020 - 29·n

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The numeric values for this problem are given as follows:

The function in this problem is a piecewise function, meaning that it has different definitions based on the input x of the function.

For x between -2 and 2, the function is defined as follows:

g(x) = -(x - 1)² + 2.

Hence the numeric value at x = -1 is given as follows:

g(-1) = -(-1 - 1)² + 2 = -4 + 2 = -2.

For x at x = 2 and greater, the function is given as follows:

g(x) = 0.5x - 1.

Hence the numeric values at x = 2 and x = 3 are given as follows:

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This shows that any increase in x by a certain factor results in an increase in y by the same factor, confirming that y varies directly with x.

What is Linear equation ?

Linear equation can be defined as equation in which highest degree is one.

To determine if y varies directly with x, we need to check if there is a constant ratio between y and x. In other words, if we increase x by a certain factor, does y also increase by the same factor?

The equation 3y = -7x - 18 can be rewritten as y = (-7/3)x - 6. This is in the form of y = kx + b, where k is the constant of variation and represents the ratio between y and x.

Since the equation is in this form, we can say that y varies directly with x, and the constant of variation is k = -7/3.

To verify that y varies directly with x, we can check that any increase in x by a certain factor results in an increase in y by the same factor, as given by the constant of variation. For example, if we increase x by 3, then y will increase by (-7/3)(3) = -7. If we increase x by 6, then y will increase by (-7/3)(6) = -14.

Therefore, This shows that any increase in x by a certain factor results in an increase in y by the same factor, confirming that y varies directly with x.

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The nonzero vector orthogonal to the plane through the points P,Q and R is -7i + 7j -21k and the area of the triangle is 11.6 square units.

(a)Given points are P(-1,3,1) , Q(0,7,2) and R(4,2,-1)

PQ = Q - P

= (0 - (-1))i + (7 - 3)j + (2 -1)k

=i + 4j + k

PR = R - P

= (4 - (-1))i + (2 - 3)j + (-1 - 1)k

5i -j - 2k

The orthogonal vector is PQ x QR

\(\left[\begin{array}{ccc}i&j&k\\1&4&1\\5&-1&-2\end{array}\right]\)

(-8 + 1)i - (-2 - 5)j + (-1 -  20))k

-7i + 7j - 21k

Therefore the nonzero vector orthogonal to the plane through the points P,Q and R is -7i + 7j - 21k

(b) The area of the triangle with the vertices P,Q and R is the half of the length of the cross product of PQ and QR.

Area = 1/2| PQ x QR|

= 1/2 √(7² + (-7)² + (-21)²)

= 1/2 √(49 + 49 + 441)

= 1/2 (23.21)

= 11.60

= 11.60 square units.

Therefore The non zero vector is  -7i + 7j - 21k and the area of the triangle is 11 .6 square units.

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Answer:

The maximum distance between supports for the beam to support a weight of 1600 lb is also 4.5 feet.

Step-by-step explanation:

We are given that weight P is inversely proportional to the distance D between the supports of the beam, and for this certain type of wooden beam, we have:

P = 7200/D

To find the distance between supports needed to carry 1600 lb, we can substitute P = 1600 in the above equation and solve for D:

1600 = 7200/D

D = 7200/1600

D = 4.5 feet

So the distance between supports needed to carry 1600 lb is 4.5 feet.

To find the maximum distance between supports for the beam to support a weight of 1600 lb, we can use the same equation and solve for P = 1600:

1600 = 7200/D

D = 7200/1600

D = 4.5 feet

Therefore, the maximum distance between supports for the beam to support a weight of 1600 lb is also 4.5 feet.

Answer:

Step-by-step explanation:

To get the y values all you need to do is substitute the x value in the equation y=-2/3x+7.

For example:

y=-2/3(-6)=7

-2/3x6=-4

-4+7=3

(-6,3)

You can double check your work by filling the x and y coordinates in the equation and when solved if it it true you know you were correct.

To get the x value, you need to fill in the y in the equation y=-2/3x+7

for example:

5=-2/3x+7

-2=-2/3x

3=x

(3,5)

y=-2/3x+7

y=-2/3(15)+7

y=-10+7

y=-3

(15,-3)

y=-2/3x+7

15=-2/3x+7

8=-2/3x

-12=x

(-12,15)

Answer: A. x <= -3

Step-by-step explanation:

First, let's begin by simplifying the inequality:

-4x >= 12    ← To isolate the x, we need to divide both sides of the inequality by -4 (we must make this change to both sides so that the inequality is still valid). Remember that when dividing an inequality by a negative number, the orientation of the sign switches (greater than → less than, less than → greater then, etc.). So...

-4x / -4 <= 12 / -4    ← Simplify...

x <= -3

Now, we need to find the right number line representing this inequality.

The number line that fits these criteria would be A.

Answer:

The side length, s, of the square is 18

Step-by-step explanation:

The sides can be found by taking the square root of the area.

(Area)^1/2=s, where s = side.

(324)^1/2=18

The area of the given figure is 27.5 square centimeter which has a rectangle and triangles

The given figure has a rectangle and triangles

The area of rectangle is length times width

Area of rectangle =5×4

=20 square centimeter

Area of triangle =1/2×base×height

=1/2×2.5×3

=7.5/2

=3.75 square centimeter

As there are two triangle = 2(3.75)

=7.5 square centimeter

Total area = 7.5+ 20

=27.5 square centimeter

Hence, the area of the given figure is 27.5 square centimeter

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Answer:

$31.04

Step-by-step explanation:

well, the interest is going to be the increase factor on both cases, so we can simply check how much each will accumulate to in those 4 years

\(~~~~~~ \stackrel{\textit{\LARGE University Bank}}{\textit{Compound Interest Earned Amount}} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$4000\\ r=rate\to 5\%\to \frac{5}{100}\dotfill &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &4 \end{cases}\)

\(A = 4000\left(1+\frac{0.05}{4}\right)^{4\cdot 4} \implies A=4000(1.0125)^{16}\implies \boxed{A \approx 4879.56} \\\\[-0.35em] ~\dotfill\)

\(~~~~~~ \stackrel{\textit{\LARGE Rosemont Savings Bank}}{\textit{Compound Interest Earned Amount}} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$4000\\ r=rate\to 5.5\%\to \frac{5.5}{100}\dotfill &0.055\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{semiannually, thus two} \end{array}\dotfill &2\\ t=years\dotfill &4 \end{cases}\)

\(A = 4000\left(1+\frac{0.055}{2}\right)^{2\cdot 4} \implies A=4000(1.0275)^8\implies \boxed{A \approx 4969.52} ~~ \textit{\LARGE \checkmark}\)

Answer:

The rate is 5 dollars per student.

Step-by-step explanation:

The rate per student can be found with \(\frac{dollars}{student}\):

\(\frac{150dollars}{30student}=5\frac{dollars}{student}\\\\\frac{350dollars}{70student}=5\frac{dollars}{student}\)

The rate is 5 dollars per student.

A. The total value of the loan with quarterly compounded interest is $52,400.

B. The total value of the loan with monthly compounded interest is $52,010.04

Part A: Total value of the loan with the quarterly compounded interest

To find the total value of the loan, we'll use the formula for compound interest:

A = P * (1 + r/n)^(nt)

Where:

A = final amount (loan + interest)

P = principal amount (initial loan) = $20,000

r = interest rate = 7% = 0.07

n = number of times compounded per year = 4 (quarterly)

t = number of years = 10

Plugging in the values, we get:

A = $20,000 * (1 + 0.07/4)^(4 * 10)

A = $20,000 * (1.0175)^40

A = $20,000 * 2.620075

A = $52,401.51

Rounding to the nearest hundred, the total value of the loan with quarterly compounded interest is $52,400.

Part B: Total value of the loan with the monthly compounded interest

To find the total value of the loan with monthly compounded interest, we'll use the same formula with a different value of n:

A = P * (1 + r/n)^(nt)

Where:

n = number of times compounded per year = 12 (monthly)

Plugging in the values, we get:

A = $20,000 * (1 + 0.07/12)^(12 * 10)

A = $20,000 * (1.005833)^120

A = $20,000 * 2.600502

A = $52,010.04

Rounding to the nearest hundredth, the total value of the loan with monthly compounded interest is $52,010.04

The difference between the two loans is $52,401.51 - $52,010.04 = $391.47

The difference between the two loans is due to the higher frequency of compounding in the monthly compounded interest loan. The interest is compounded more times per year, so the interest is accumulating on the interest more quickly, leading to a higher overall interest charge.

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Answer:

C  69

Step-by-step explanation:

The correct option for Which confidence level would produce the widest interval when estimating the mean of a population based on the mean and standard deviation of a sample of that population is C. 69%

A population is the number of living people that live together in the same place. A city's population is the number of people living in that city. These people are called inhabitants or residents. The population includes all individuals that live in that certain area.

Conclusion: It can be a group of people, objects, events corporations, and so forth. you use populations to attract conclusions. determine 1: population. An example of a population would be the complete scholar body at a faculty. it'd consist of all of the college students who examine in that university at the time of statistics collection.

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The missing value in the table include the following: B. 4 1/2 in.

In Mathematics and Geometry, the volume of a rectangular prism can be calculated by using the following formula:

Volume of a rectangular prism = L × W × H

Where:

By substituting the given dimensions (parameters) into the formula for the volume of a rectangular prism, we have;

50 5/8 = 5 × W × 2 1/4

Width, W = 4 1/2 inches.

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Complete Question:

The volume of a rectangular prism is 50 5/8 cubic inches.

The dimensions are given below.

What is the missing value in the table?

Length Width Height

5 in. 214 in.

A. 79in.

B. 412in.

C. 134in.

D. 1018in.

Height: 5ft.

Legnth: 6ft.

Width: 1.2 ft.

1% of 500 is 5ft

1% of 600 is 6ft

1% of 120 is 1.2ft

Answer:

The range of the function is: -∞≤y≤∞.

Consider the provided function.

The range of the function is the set of all values which a function can produce or the set of y values which a function can produce after substitute the possible values of x.

The range of a cubic root function is all real numbers.

Now consider the provided function.

The above function can be written as:

Taking cube on both sides.

The graph of the function is shown in figure 1:

For any value of x we can find different value of y.

Here, the cube root function can process negative values. Since, the function can produce any values, the range of the given function is  -∞≤y≤∞ .

Therefore, the range of the function is: -∞≤y≤∞ (A).

The solution to the equation \( {6}^{(4x - 3)} = - 4\) is x = 0.3019.

To solve the equation \({6}^{(4x - 3)} = - 4\), we need to isolate the variable x. Here are the steps to solve this equation:

First, we need to take the logarithm base 6 of both sides of the equation. This will help us to isolate the exponent.

log6 \({6}^{(4x - 3)}\)= log6 -4

Use the logarithmic property that \( log(b) {b}^{x} = x\) to simplify the left-hand side.

4x - 3 = log6 -4

Use a logarithmic table or calculator to find the value of the right-hand side.

4x - 3 = log6 -4

4x - 3 = -1.7924

Finally, we can isolate the variable x by adding 3 to both sides of the equation and then dividing both sides by 4.

4x = 3 - 1.7924

4x = 1.2076

x = 1.2076 ÷ 4

x = 0.3019

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Answer:

3047.52

Step-by-step explanation:

If you do the amount of pieces that they need is 112, and it costs 27.21 per piece, then what you have to do is 112*27.21

112 * 27.21 is 3047.52

Explanation:

A = set of odd numbers = {1,3,5}

B = set of factors of 12 = {1,2,3,4,6}

A U B = union of set A and set B

A U B = {1,3,5} union {1,2,3,4,6}

A U B = {1,3,5,       1,2,3,4,6}

A U B = {1,2,3,4,5,6}

The set union operation combines two sets into one bigger set. Duplicates are tossed out.

There are 6 elements in the set A U B = {1,2,3,4,5,6} out of 6 faces of the number cube.

Therefore, the probability event  A U B happens is 6/6 = 1 = 100%; i.e. it is guaranteed to happen. Each face of the number cube is either odd, a factor of 12, or both.

Side notes:

Answer:

the probability of flipping n times is 12n, since that's the probability of flipping exactly n−1 tails followed by 1 heads.

Step-by-step explanation:

Answer:

f(2) = 1

x = 0

Step-by-step explanation:

f(2) = 1 since y=1 when x=2.

f(0) = -3 since x=0 when y=-3

The expressions that are equivalent are;

4a²b(4ab - 2a³b⁴ + 3a²)

2a²b(8ab - 4a³b⁴ + 63a²)

Options 1, 2, 4, 5, 6

Algebraic expression are defined as expressions that are made up of terms, variables, coefficients, constants and factors.

These algebraic expressions are also made up of mathematical operations , such as;

From the information given, we have the expression given as;

16a³b² - 8a⁵b⁵ + 12a⁴b

Factorize the expression, we get;

4a²b(4ab - 2a³b⁴ + 3a²)

2a²b(8ab - 4a³b⁴ + 63a²)

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