5 A teacher needs to buy batteries for 32 calculators. ● There are 20 basic calculators that each require 3 batteries. There are 12 advanced calculators that each require 4 batteries. The batteries are sold in packages of 24. The teacher thinks that 6 packages of batteries will be needed and that there will be 12 batteries left over after the calculators are filled. Provide a solution path that shows whether the teacher is correct or incorrect. Explain what each step in the solution path represents in terms of the situation. Enter your answer and your work or explanation in the space provided.​

Answer:

29 blocks

Step-by-step explanation:

You've to count the number of blocks and that's the volume.

The quadratic function that fits the given points is: f(x) = (4/3)x² - (10/3)x + 1

By using the simplification formula

f(x) = ax² + bx + c

where a, b, and c are constants

a(-1)² + b(-1) + c = 1/3

a(0)² + b(0) + c = 1

a(1)² + b(1) + c = 3

a(2)² + b(2) + c = 9

Simplifying each

a - b + c = 1/3

c = 1

a + b + c = 3

4a + 2b + c = 9

We can solve this using any method like substitution, elimination

a - b + c = 1/3

a + b + c = 3

2a + 2c = 9/3

Adding the first two equations 2a + 2c = 10/3

Subtracting the third equation

b = 5a/3 - 2

a - (5a/3 - 2) + 1 = 1/3

a = 4/3

Finally, we can substitute a = 4/3 and b = 5a/3 - 2, and c = 1 into the standard form of the quadratic equation to get: f(x) = (4/3)x² - (10/3)x + 1.

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Answer:

whooper write an equation of the line in slope-intercept form for each of the following write an equation of the line in slope-intercept form for each of the following write an equation of the line in slope-intercept form for each of the following write an equation of the line in slope-intercept form for each of the following

Step-by-step explanation:

The probability that a student who attends the first session also attends the second session is 1/3

We can approach this problem by using conditional probability.

Let's use the notation "S1" to represent the event that a student attends the first session, and "S2" to represent the event that a student attends the second session.

Then we can use the formula for conditional probability:

P(S2 | S1) = P(S1 and S2) / P(S1)

We know that P(S1) = 0.6, since 60% of students attend the first session. We also know that P(S1 and S2) = 0.2, since 20% of students attend both sessions.

So we can plug in these values and solve for P(S2 | S1):

P(S2 | S1) = 0.2 / 0.6

P(S2 | S1) = 1/3

Therefore, the probability that a student who attends is 1/3 or approximately 0.333.

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Answer:

Without knowing the exact length of Ted's pencil, we cannot give an exact decimal representation of its length. However, we do know that Ted's pencil is longer than Jan's pencil, which is 8.5 cm long.

If we assume that Ted's pencil is one centimeter longer than Jan's pencil, then its length would be 9.5 cm. In decimal form, this would be written as 9.5.

If we assume that Ted's pencil is two centimeters longer than Jan's pencil, then its length would be 10.5 cm. In decimal form, this would be written as 10.5.

So, the decimal that could represent the length of Ted's pencil depends on how much longer it is than Jan's pencil.

The total surface area of the given hemispherical scoop is: 30.41 cm²

The formula for the total surface area of a hemisphere is:

TSA = 3πr² square units

Where:

π is a constant whose value is equal to 3.14 approximately.

r is the radius of the hemisphere.

Since the steel is 0.2cm thick and the outside of the scoop has a radius of 2cm, then we can say that:

TSA = 3π(2 + 0.2)²

= 3π(2.2)²

= 30.41 cm²

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The sum of the nth roots of unity is zero.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. Complex numbers are used in mathematics and science to represent quantities that have both a real and imaginary part.

The nth roots of unity are given by:

[tex]\omega ^0, \omega^1, \omega^2, ..., \omega^(n-1)[/tex]

where [tex]\omega = e^{(2\pi i/n)[/tex]is a complex number and i is the imaginary unit.

The sum of these roots is:

[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^{(n-1)[/tex]

To simplify this expression, we can use the formula for the sum of a geometric series:

[tex]a + ar + ar^2 + ... + ar^(n-1) = a(1 - r^n)/(1 - r)[/tex]

Let a = 1 and [tex]r = ω.[/tex] Then the sum of the nth roots of unity is:

[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^(n-1) = (1 - \omega^n)/(1 - \omega)[/tex]

Substituting [tex]ω = e^(2πi/n)[/tex], we get:

[tex]\omega^n = (e^(2\pi i/n))^n = e^2 \pi i = 1[/tex]

Therefore, the sum of the nth roots of unity is:[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^(n-1) = (1 - 1)/(1 - \omega) = 0/(1 - \omega) = 0[/tex]

Hence, the sum of the nth roots of unity is zero.

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If the roots of a quadratic equation are given, we can write the equation in factored form as [tex](x - r1)(x - r2) = 0[/tex] , where r1 and r2 are the roots.the quadratic equation with roots 5 and 2 and leading coefficient 4 is: [tex]4x^2 - 28x + 40 = 0.[/tex]

A quadratic equation can be written in the form:

[tex]ax^2 + bx + c = 0[/tex]

where a, b, and c are constants. Since the roots of the equation are 5 and 2, we can write:

[tex](x - 5)(x - 2) = 0[/tex]

Expanding this equation gives:

[tex]x^2 - 7x + 10 = 0[/tex]

To make the leading coefficient of this equation 4, we can multiply both sides by 4/1, which gives:

[tex]4x^2 - 28x + 40 = 0[/tex]

Therefore, the quadratic equation with roots 5 and 2 and leading coefficient 4 is: [tex]4x^2 - 28x + 40 = 0.[/tex]

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a) The total volume equals the sum of the volumes.

[tex]500 = x + y[/tex]

The total octane amount equals the sum of the octane amounts.

[tex]89(500) = 87x + 92y[/tex]

[tex]44500 = 87x + 92y[/tex]

b)

As x increases, y decreases.

c) Use substitution or elimination to solve the system of equations.

[tex]44500 = 87x + 92(500-x)[/tex]

[tex]44500 = 87x + 46000 - 92x[/tex]

[tex]5x = 1500[/tex]

[tex]x = 300[/tex]

[tex]y = 200[/tex]

The required volumes are 300 gallons of 87 gasoline and 200 gallons of 92 gasoline.

Answer:

A: [tex]2x^3+5x^2-8x[/tex]

Step-by-step explanation:

In order to get the answer to this, you have to combine like terms to simplify the answer.

Go through and organize the x's by the size of the exponents. (this step isn't necessary but it can help you visualize it if you are having trouble with that)

[tex]x^3+x^3+2x^2+3x^2-7x-x[/tex]

When the variables are raised to the same degree, the coefficients can be added together.

[tex]2x^3+5x^2-8x[/tex]

Answer:

30

Step-by-step explanation:

its right

order of operations

Answer:

The answer is 30...

Step-by-step explanation:

Apply the rule BODMAS...

3(40)/(4)

120/4

30

Answer:

Step-by-step explanation:

Surface area is calculated as 48 + 120 = 168 square units (area of triangular faces + area of rectangular faces).

A polyhedron with two triangular sides and three rectangles sides is referred to as a triangular prism. It is a three-dimensional shape with two base faces, three side faces, and connections between them at the edges.

Given :

We must calculate the area of each face of the triangular prism and put them together to determine its surface area.

The areas of the triangular faces are equal, so we may calculate one of their areas and multiply it by two:

One triangular face's area is equal to (1/2) the sum of its base and height, or (1/2) 6 x 8 x 6, or 24 square units.

Both triangular faces' surface area is 2 x 24 or 48 square units.

Finding the area of the rectangular faces is now necessary:

One rectangular face's area is given by length x breadth (10 x 6) = 60 square units.

120 square units are the area of both rectangular faces or 2 by 60.

Hence, the triangular prism's total surface area is:

Surface area is calculated as 48 + 120 = 168 square units (area of triangular faces + area of rectangular faces).

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The value of the expression is 142.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. Algebraic expressions are used to represent and solve problems in many areas of mathematics, science, engineering, and finance.

To solve this expression, we need to follow the order of operations (PEMDAS) which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Let's start with the parentheses first:

14 x 0.5 x (4 + 2) + exponent 10*2

= 14 x 0.5 x 6 + exponent 10*2

Next, we can simplify the multiplication and division from left to right:

= 7 x 6 + exponent 10*2

= 42 + exponent 10*2

Now we can evaluate the exponent:

= 42 + 100

= 142

Therefore, the value of the expression is 142.

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Let L be the original length of the rectangle and W be the original width of the rectangle. We know that:

(L - 6) = (W + 3) (1) (since the length is decreased by 6cm and the width is increased by 3cm, the result is a square)

The area of the rectangle is LW, and the area of the square is (L - 6)(W + 3). We also know that the area of the square is 27cm^2 smaller than the area of the rectangle. So we can write:

(L - 6)(W + 3) = LW - 27 (2)

Expanding the left side of equation (2), we get:

LW - 6W + 3L - 18 = LW - 27

Simplifying and rearranging, we get:

3L - 6W = 9

Dividing both sides by 3, we get:

L - 2W = 3 (3)

Now we have two equations with two unknowns, equations (1) and (3). We can solve this system of equations by substitution. Rearranging equation (1), we get:

L = W + 9

Substituting this into equation (3), we get:

(W + 9) - 2W = 3

Simplifying, we get:

W = 6

Substituting this value of W into equation (1), we get:

L - 6 = 9

So:

L = 15

Therefore, the area of the rectangle is:

A = LW = 15 x 6 = 90 cm^2.

Answer:

252

Step-by-step explanation:

lol I take rsm too and I just guessed and checked

By calculating the z-score, we can conclude that, Team A has better eligibility rates than Team B.

What is z-score?

A z-score (also called a standard score) is a measure of how many standard deviations a given data point is away from the mean of its distribution. It is calculated by subtracting the mean of the distribution from the data point, and then dividing the difference by the standard deviation.

To determine this, we can compare the z-scores for each team's eligibility rates.

Let's assume that Team A's eligibility rate is 875 and Team B's eligibility rate is 825.

The mean eligibility rate for all teams is given as 870, with a standard deviation of 50. Therefore, we can calculate the z-scores for each team's eligibility rate as follows:

z-score for Team A's eligibility rate = (875 - 870) / 50 = 0.1

z-score for Team B's eligibility rate = (825 - 870) / 50 = -0.9

Since the z-score for Team A's eligibility rate is positive and greater than the z-score for Team B's eligibility rate, we can conclude that Team A has better eligibility rates than Team B.

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the rectangular equation that is equivalent to the given parametric equations is: [tex]4(x-2)^2 + 9(y+1)^2 = 36[/tex]

The given parametric equations describe a curve in the xy-plane traced by a particle that moves along a circular path centered at (2,-1) with radius 3. To find the rectangular equation, we can use the following trigonometric identity:

[tex]cos^2[/tex]θ + [tex]sin^2[/tex]θ = 1

Multiplying both sides by [tex]3^2[/tex], we get:

9[tex]cos^2[/tex]θ + [tex]9sin^2[/tex]θ = 9

Rearranging and using the fact that cosθ = (x-2)/3 and sinθ = (y+1)/2, we get:

[tex]9((x-2)/3)^2 + 9((y+1)/2)^2 = 9[/tex]

Simplifying, we get:

[tex](x-2)^2/3^2 + (y+1)^2/2^2 = 1[/tex]

Multiplying both sides by 36, we get:

[tex]4(x-2)^2 + 9(y+1)^2 = 36[/tex]

Therefore, the rectangular equation that is equivalent to the given parametric equations is:

[tex]4(x-2)^2 + 9(y+1)^2 = 36[/tex]

This equation represents an ellipse centered at (2,-1) with semi-axes of length 3 and 2 along the x-axis and y-axis, respectively. The parameter θ varies from 0 to 2π, which means the particle completes one full revolution around the ellipse.

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Answer:

581,58 in^3

Step-by-step explanation:

Given:

Cylinder shaped cups

d (diameter) = 4,2 in

r (radius) = 0,5 × 4,2 = 2,1 in

h (height) = 7 in

π = 3.14

.

First, let's find how much juice can he pour into 1 cup:

.

We need to find the base of the cylinder:

.

[tex]a(base) = \pi {r}^{2} = 3.14 \times( {2.1})^{2} = 13.8474[/tex]

.

Now, we can find the volume of one cup:

V = a (base) × h

[tex]v = 13.8474 \times 7 ≈96.93[/tex]

Multiply this number by 6 and we'll get the answer (since there's 6 cups):

96,93 × 6 = 581,58

Answer:

the missing values are 2, 8, and 3.

It can be expressed in interval notation as:[0, 12]

The practical domain for the function g(x) is [0,12], as this is the time during which the football is in the air, from the time it is thrown until it hits the ground.

The practical domain for the function g(x) would be the time interval during which the football is in the air, since it only makes sense to talk about the height of the football while it is airborne.

From the problem statement, we know that the football is thrown in the air at time x = 0, reaches its maximum height of 28 feet at time x = 6, and hits the ground at time x = 12. Therefore, the practical domain for the function g(x) is:
0 <= x <= 12
This means that the function g(x) is defined and meaningful for any value of x between 0 and 12, inclusive. Beyond this domain, the function does not have a practical interpretation because the football is either not yet thrown or has already hit the ground.

The function g(x) is the height of a football x seconds after it is thrown in the air. The football reaches its maximum height of 30 feet in 4 seconds, and hits the ground at 10 seconds.
What is the practical domain for the function f(x)
Write your answer in interval notation.

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The side of the given square is 36W^2 + 12W +1 of is 6W +1 and 81W^2 -72W + 16 is (9W-4).

The area is calculated by multiplying the length of a shape by its width.

and the unit of the square is a square unit.

Given Area of the square :

1) [tex]36W^{2}+12W+1[/tex]

Area of square = [tex]36W^{2}+12W+1[/tex]

[tex]side^{2}[/tex] = [tex]36W^{2} + 6W +6W +1\\6W (6W +1) +1 (6W +1)\\(6W +1)(6W+1)\\(6W +1 )^{2} \\side^{2} = (6W + 1)^{2} \\side = 6W + 1[/tex]

[tex]Area of the square = 81W^{2} - 72W + 16\\(side)^{2} = (9W-4)(9W-4)\\(side)^{2} = (9W-4)^{2} \\side = (9W-4)[/tex]

Therefore the side of the square is 6W+1 and 9x-4.

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Answer:

the answer is A) -2x² + 4x + 30

Step-by-step explanation:

To find the equation of the quadratic function f(x), we can use the standard form of a quadratic function: f(x) = ax^2 + bx + c, where a, b, and c are constants.

We can plug in the values of x and f(x) from the table to get three equations:

a(-1)^2 + b(-1) + c = 24

a(0)^2 + b(0) + c = 30

a(1)^2 + b(1) + c = 32

Simplifying each equation, we get:

a - b + c = 24

c = 30

a + b + c = 32

We can substitute c = 30 into the first and third equations to get:

a - b + 30 = 24

a + b + 30 = 32

Simplifying these equations, we get:

a - b = -6

a + b = 2

Adding these two equations, we get:

2a = -4

Dividing by 2, we get:

a = -2

Substituting a = -2 into one of the equations above, we get:

-2 - b = -6

Solving for b, we get:

b = 4

Therefore, the equation that represents f(x) is:

f(x) = -2x^2 + 4x + 30

So the answer is A) -2x² + 4x + 30

Answer:

[tex] \frac{13y ^{2} - 59y - 90 }{(y + 10)(y - 10)} [/tex]

The desalination plant can produce [tex]9.84*10^6[/tex] gallons of water in four days in scientific notation.

We must divide the daily production rate by the number of days in order to get the total volume of water that a desalination plant can generate in four days:

4. days at a rate of [tex]2.46*10^6[/tex] gallons equals [tex]9.84*10^6[/tex] gallons.

Hence, in four days, the desalination plant can produce [tex]9.84 * 10^6[/tex]gallons of water.

Large or small numbers can be conveniently represented using scientific notation, especially when working with measurements in science and engineering. A number is written in scientific notation as a coefficient multiplied by 10 and raised to a power of some exponent. For example, [tex]2.46*10^6[/tex] denotes 2,460,000, which is 2.46 multiplied by 10 to the power of 6.

The solution in this case is [tex]9.84*10^6[/tex], or 9,840,000, which is 9.84 multiplied by 10 to the power of 6. Large numbers can be written and compared more easily using this format, and scientific notation rules can be used to conduct computations with them.

In conclusion, the desalination plant has a four-day capacity of [tex]9.84 * 10^6[/tex] gallons of water production.

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The rate that yields maximum revenue is $9 per month, and the maximum revenue is $1,296,000.

To find the rate that yields maximum revenue, we need to find the price that maximizes the revenue. Let x be the number of $0.50 decreases in the monthly fee, and let y be the number of subscribers who sign up at the new rate. Then we have the following equations:

y = 4800 + 120x (number of subscribers)

p = 24 - 0.5x (price per month)

r = xy(p) = (4800 + 120x)(24 - 0.5x) (revenue)

To find the rate that yields maximum revenue, we need to take the derivative of the revenue function concerning x, set it equal to zero, and solve for x:

r' = 120(24 - x) - (4800 + 120x)(0.5) = 0

x = 60

Therefore, the optimal number of $0.50 decreases is 60, and the corresponding price per month is $24 - 0.5(60) = $9. The number of subscribers at this rate is 4800 + 120(60) = 12000.

Finally, the maximum revenue is given by r = xy(p) = (4800 + 120(60))(24 - 0.5(60)) = $1,296,000.

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so we're assuming there are 360 days in a year, so 83 days is really just 83/360 of a year, so

[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$586.21\\ r=rate\to 6.3\%\to \frac{6.3}{100}\dotfill &0.063\\ t=years\dotfill &\frac{83}{360} \end{cases} \\\\\\ I = (586.21)(0.063)(\frac{83}{360}) \implies I \approx 8.51[/tex]

The number of non-fiction novels among the following 100 books that are released is anticipated to be 36%.

Probability refers to the likelihood that an occurrence will occur.

In actual life, we frequently have to make predictions about the future. We might or might not be conscious of how an event will turn out.

When this occurs, we proclaim that there is a possibility that the event will occur. In conclusion, probability has a broad range of amazing uses in both business and this rapidly developing area of artificial intelligence.

Simply dividing the favorable number of possibilities by the total number of possible outcomes using the probability formula will yield the chance of an event.

According to our question-

Total number of books that arrived that day = 23 + 41 = 64.

Let E be the event of arriving at a non-fictional book.

The event of arriving at a non-fictional book is n(E) = 23.

Total sample space, n(S) = 64.

The probability of a non-fictional book to arrive is = n(E) / n(S)

= 23 / 64 = 0.359375

= 0.359375 × 100

= 35.9375 ≈ 36.

Hence, The number of non-fiction novels among the following 100 books that are released is anticipated to be 36%.

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Answer:

Step-by-step explanation:

Solution:

   [tex](x-2)^2+(y+3)^2=25[/tex]

Equation of general circle:

    [tex](x-a)^2+(y-b)^2=r^2[/tex]      where (a,b) is circle center and r is the radius.

Answer:

Step-by-step explanation:

500/0.4 = 1250