Solve the right triangle ABC, with C=90°. B=36°12′ c=0.6209 m

To find the area of a triangle using trigonometry, you can use the formula A = (1/2)bh, where A represents the area, b is the base of the triangle, and h is the corresponding height.

If you have an acute triangle, you can use trigonometric ratios to find the height. For example, let's say you know the length of one side (a) and the measure of the angle opposite to that side (A). To find the height, you can use the formula h = a * sin(A), where sin(A) represents the sine of angle A.

If you have a right triangle, you can use the lengths of the two legs (a and b) to find the area. The height of the triangle will be one of the legs, and the base will be the other leg. You can then use the formula A = (1/2)ab to calculate the area.

Here's an example to illustrate the process:
Suppose we have a right triangle with a base of 6 units and a height of 8 units. We can use the formula A = (1/2)bh, plugging in the values to get A = (1/2)(6)(8) = 24 square units.

In summary, to find the area of a triangle using trigonometry:
1. If you have an acute triangle, use trigonometric ratios to find the height.
2. If you have a right triangle, use the lengths of the two legs to find the area.

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A check or quadrangle is defined by the intersection of pairs of diagonals of a parallelogram.

A parallelogram is a quadrilateral with two pairs of parallel sides. In a parallelogram, the opposite sides are parallel and have the same length. The opposite angles of a parallelogram are equal. A parallelogram is a unique type of quadrilateral with specific characteristics.

A quadrangle or check is defined as the intersection of pairs of diagonals of a parallelogram. In other words, it is the area inside the parallelogram that is divided into four triangles, each of which shares a common vertex in the center of the parallelogram.

The diagonals of a parallelogram are the line segments that connect the opposite vertices of a parallelogram. When these diagonals intersect, they form four triangles, which are also known as the parallelogram's "sub-triangles." The point where the diagonals intersect is called the center of the parallelogram, and it divides each diagonal into two equal parts.

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13) The electron configuration [Kr]5s²4d¹5p³ corresponds to the element iodine (I).

14) The electron configuration [Xe]6s²4f¹⁴5d¹⁰6p⁶ corresponds to the element lead (Pb).

15) The electron configuration [Rn]7s²5f¹⁴ corresponds to the element lawrencium (Lr).

16) The electron configuration 1s²2s²2p¹3s²3p⁵4s²4d¹⁰4p⁹ is valid.

17) The electron configuration 1s²2s²2p¹3s²3d³ is not valid as it violates the Aufbau principle.

18) The electron configuration [Ra]7s²5f is not valid as it is incomplete.

19) The electron configuration [Kr]5s²4d¹⁰5p³ is valid.

20) The electron configuration [X] is not a valid electron configuration as it does not specify the specific subshells and electrons present.

13) The electron configuration [Kr]5s²4d¹5p³ corresponds to iodine (I) because it has 53 electrons. The [Kr] represents the electron configuration of the noble gas krypton, and the subsequent orbitals represent the additional electrons in the iodine atom.

14) The electron configuration [Xe]6s²4f¹⁴5d¹⁰6p⁶ corresponds to lead (Pb) because it has 82 electrons. The [Xe] represents the electron configuration of the noble gas xenon, and the subsequent orbitals represent the additional electrons in the lead atom.

15) The electron configuration [Rn]7s²5f¹⁴ corresponds to lawrencium (Lr) because it has 103 electrons. The [Rn] represents the electron configuration of the noble gas radon, and the subsequent orbitals represent the additional electrons in the lawrencium atom.

16) The electron configuration 1s²2s²2p¹3s²3p⁵4s²4d¹⁰4p⁹ is a valid electron configuration because it follows the Aufbau principle, filling the orbitals in increasing order of energy.

17) The electron configuration 1s²2s²2p¹3s²3d³ is not valid because it violates the Aufbau principle. According to the Aufbau principle, electrons should fill the orbitals in increasing order of energy.

18) The electron configuration [Ra]7s²5f is not valid because it is incomplete. It should specify the specific subshells and electrons present in the atom.

19) The electron configuration [Kr]5s²4d¹⁰5p³ is a valid electron configuration for an element, following the Aufbau principle.

20) [X] is not a valid electron configuration as it does not specify the specific subshells and electrons present. It is incomplete and does not provide information about any element.

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An expression for m in terms of p is given by:m = p + 5

Given the function B: x + 5, we need to find an expression for m in terms of p.In order to find an expression for m in terms of p, we need to first understand the relation between x and p. Without any context or information given, we cannot directly relate x and p. So, we can assume that there is another function or equation that relates x and p, let's call it function A.Function A: p = 2x + 3 (just an example, not given in the question)Now, we can substitute the expression for x from function A into function B.

This will give us an expression for B in terms of p.B = (2x + 3) + 5 = 2x + 8Now, we can equate this expression for B to m, and solve for x in terms of m. Then, we can substitute this expression for x in terms of m back into function A to get an expression for m in terms of p.m = 2x + 8 => 2x = m - 8 => x = (m - 8)/2Substituting this expression for x back into function A:p = 2x + 3 = 2[(m - 8)/2] + 3 = m - 5

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The evaluated expressions are:

a) −9−∣9−∣−9∣∣ = -18

b) ∣∣​(−9)2−92∣∣​ = 72

a) To evaluate the expression −9−∣9−∣−9∣∣, we can break it down into smaller steps.

Evaluate the innermost absolute value expression.

∣9−∣9∣∣ = ∣9−9∣ = ∣0∣ = 0

Substitute the result back into the original expression.

−9−∣0−9∣ = −9−∣−9∣

Evaluate the remaining absolute value expression.

∣−9∣ = 9

Substitute the result back into the expression.

−9−9 = -18

Therefore, the value of the expression −9−∣9−∣−9∣∣ is -18.

b) To evaluate the expression ∣∣​(−9)2−92∣∣​, we follow a similar process.

Evaluate the innermost part of the expression.

(−9)2 = 81

Substitute the result back into the absolute value expression.

∣∣81−92∣∣

Evaluate the subtraction inside the absolute value.

81−9 = 72

Substitute the result back into the expression.

∣∣72∣∣ = 72

Therefore, the value of the expression ∣∣​(−9)2−92∣∣​ is 72.

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The response \( y(t) \), when \( y(0)=4 \) and the step change in \( u=2 \) at \( t=0 \  the response \(y(t)\) when \(y(0) = 4\) and a step change \(u = 2\) at \(t = 0\) is \(y(t) = 6e^t + 4\).

To calculate the response \(y(t)\) using the given transfer function \(G(s)\), we need to take the inverse Laplace transform of \(G(s)\) and then solve for the unknown coefficients using the given initial condition and step change.

The Laplace transform of a function \(f(t)\) is defined as:

\[ F(s) = \mathcal{L}\{f(t)\} = \int_0^\infty f(t)e^{-st} dt \]

To take the inverse Laplace transform, we can use partial fraction decomposition and lookup tables. However, in this case, we can directly use the inverse Laplace transform table to simplify the calculation.

According to the inverse Laplace transform table, the inverse Laplace transform of \(e^{-as}\) is \(f(t) = 1\) for \(a > 0\). Applying this to the given transfer function, we have:

\[ Y(s) = G(s)U(s) = \frac{3e^{-s}}{10s+1} \cdot 2 = \frac{6e^{-s}}{10s+1} \]

Now, we can take the inverse Laplace transform of \(Y(s)\):

\[ y(t) = \mathcal{L}^{-1}\{Y(s)\} = \mathcal{L}^{-1}\left\{\frac{6e^{-s}}{10s+1}\right\} \]

Using the inverse Laplace transform table, we find that the inverse Laplace transform of \(\frac{e^{-as}}{s+b}\) is \(f(t) = e^{-(a-b)t}\) for \(a > b\). Applying this to our case, we have:

\[ y(t) = 6e^{-(-1)t} = 6e^t \]

Now, we can apply the initial condition \(y(0) = 4\) to determine the constant of integration:

\[ y(0) = 6e^0 = 6 \cdot 1 = 4 \]

Therefore, the constant of integration is 4. We can write the final expression for \(y(t)\) as:

\[ y(t) = 6e^t + 4 \]

So, the response \(y(t)\) when \(y(0) = 4\) and a step change \(u = 2\) at \(t = 0\) is \(y(t) = 6e^t + 4\).

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The length of the arc intercepted by a central angle of 150° on a circle with a radius of 3 meters is approximately 7.85 meters.

To find the length of the arc intercepted by a central angle θ on a circle of radius r, we can use the formula:

Arc Length = (θ/360) * 2πr

Given:

Radius (r) = 3 meters

Central angle (θ) = 150°

Substituting the values into the formula, we have:

Arc Length = (150/360) * 2π * 3

Calculating the value, we get:

Arc Length = (5/12) * 2π * 3

= (5/12) * 6π

= 2.5π

Rounding to two decimal places, the length of the arc is approximately 7.85 meters.

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The exact values of Sin 60° and Tan 2π/3, as obtained by constructing the relevant diagrams are 0.866 and -1.732 respectively.

For arriving at the correct values of the trigonometric functions, we use the unit circle method.

The unit circle method can be easily applied on the coordinate plane by following some basic steps, which are stated as follows.

1) We first construct a unit circle, which is essentially a circle of unit radius. For the sake of convenience, we center the circle at  (0,0).

2) We mark the points (1,0), (0,1), (-1,0), and (0,-1) for reference.

3) Now, for the angle for which we need the trigonometric function to be applied, we construct a line passing through (0,0), which makes the same angle with the x-axis.

After this is done, we observe that the line intersects the circle at two points. We take that part of the line, which lies in the same quadrant as the angle does.

Finally, we need only one more statement, which completes the unit-circle method.

The point of intersection of the line with the circle is always (Cosθ, Sinθ).

So, we see how it easily helps us obtain the values of any trigonometric function, as they're all related to Sine and Cosine functions in some way or the other.

Now, we solve the two questions asked.

A) Sin 60°

On referring to the diagram for Sin 60°, we find that the y-coordinate of the point of intersection of the line and the circle is approximately 0.866.

Since the y-coordinate determines the Sine function, we can say

Sin 60° = 0.866

In fractional form, it is equal to √3/2.

B) Tan 2π/3

In the diagram for Tan 2π/3, we need both the x and y coordinates to help us calculate Tan.

2π/3 = 120°

We see that:

x-coordinate = -0.5 = Cos 2π/3

y-coordinate = 0.866 = Sin 2π/3

But we know,

Tan 2π/3 = (Sin 2π/3)/(Cos 2π/3)

               = 0.866/-0.5

               = -1.732

In irrational form, we can write it as -√3.

Thus, we obtain Sin60° = 0.866 and Tan 2π/3  = -√3, through the diagrams.

(Diagrams are given below)

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The lines are neither parallel nor perpendicular.Hence, the answer is neither.

Given lines are, -2x + 5y = 15 and 5x + 2y = 12We need to determine whether the given lines are parallel, perpendicular, or neither.To check if the given lines are parallel or perpendicular, we'll find the slope of each line.The slope of the first line is given by:-2x+5y=15 Rearranging, we get: 5y=2x+15 Dividing by 5 on both sides: y=\frac{2}{5}x+3 Therefore, the slope of the first line is \frac{2}{5}.The slope of the second line is given by: 5x+2y=12 Rearranging, we get: 2y=-5x+12 Dividing by 2 on both sides: y=-\frac{5}{2}x+6 Therefore, the slope of the second line is -\frac{5}{2}. Now, we can use the following rules to determine if the lines are parallel or perpendicular: 1. If two lines have the same slope, then they are parallel. 2. If the slopes of two lines multiply to give -1, then the lines are perpendicular. 3. If neither of the above rules apply, then the lines are neither parallel nor perpendicular.Let's apply these rules to the given lines:Slope of the first line is \frac{2}{5}.Slope of the second line is -\frac{5}{2}.As neither of the above rules apply to the given slopes, we can conclude that the lines are neither parallel nor perpendicular.Hence, the answer is neither.

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To paint the inside of the hemispherical pool, you need to find the surface area of the pool. The formula for the surface area of a hemisphere is 2πr^2, so the surface area of the pool is 2π(40)^2.

Divide this area by the coverage of one gallon of paint (350 square feet) to find the number of gallons needed. To paint the inside of the hemispherical pool, you need to find the surface area of the pool. The formula for the surface area of a hemisphere is 2πr^2, so the surface area of the pool is 2π(40)^2.

Divide this area by the coverage of one gallon of paint (350 square feet) to find the number of gallons needed. The result is approximately 29.01 gallons. If you wanted to fill the pool with paint, you would need to find the volume of the pool. The formula for the volume of a hemisphere is (2/3)πr^3.

Plugging in the radius of 40 feet, the volume is (2/3)π(40)^3. Multiply this volume by the conversion factor of 7.48 gallons per cubic foot to find the number of gallons needed. The result is approximately 952328.48 gallons.

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We have successfully transformed the left side (cot α sin α) into the right side (cos α) using trigonometric identities.

Given the equation, cot α sin α = cos α, we are supposed to use trigonometric identities to transform the left side of the

equation into the right side. We will be using the identity, cot α = cos α / sin α.

To transform the left side of the equation, cot α sin α, into the right side, cos α, we can use the trigonometric identity:

cot α = 1/tan α

Using this identity, we can rewrite cot α sin α as:

cot α sin α = (1/tan α) sin α

Now, let's use another trigonometric identity:

tan α = sin α / cos α

Substituting this in, we get:

cot α sin α = (1/(sin α / cos α)) sin α

Next, simplify the expression by multiplying the numerator and denominator of the fraction by cos α:

cot α sin α = (1 * cos α / (sin α / cos α)) * sin α

Simplifying further, we get:

cot α sin α = (cos α * sin α) / sin α

Canceling out sin α in the numerator and denominator, we have:

cot α sin α = cos α

Therefore, we have successfully transformed the left side (cot α sin α) into the right side (cos α) using trigonometric identities.

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We are given that the first term is 1 and the common differences are 1, 2, and 3, and we are asked to find the true relation between the sum of n terms of three APs. Let us assume the n-th terms of the three APs to be a, b, and c, respectively.We have the first term as 1 and the common differences are 1, 2, and 3 for the three APs, respectively. So the nth terms for the three APs can be found as follows:a = 1 + (n - 1)1 = n b = 1 + (n - 1)2 = 2n - 1 c = 1 + (n - 1)3 = 3n - 2Now we can find the sum of the first n terms of each AP and use that to find the relation between them. 1.

Sum of n terms of the first AP. The sum of n terms of the first AP is given byS1 = n/2(2a + (n - 1)d1)Putting a = n and d1 = 1, we get S1 = n/2(2n + (n - 1)1)Simplifying this, we get S1 = n².2. Sum of n terms of the second AP. The sum of n terms of the second AP is given byS2 = n/2(2b + (n - 1)d2)Putting b = 2n - 1 and d2 = 2, we get S2 = n/2(2(2n - 1) + (n - 1)2)Simplifying this, we get S2 = n/2(3n - 1).3. Sum of n terms of the third AP. The sum of n terms of the third AP is given byS3 = n/2(2c + (n - 1)d3)Putting c = 3n - 2 and d3 = 3, we get S3 = n/2(2(3n - 2) + (n - 1)3)Simplifying this, we get S3 = n/2(5n - 4).

Now, we can substitute these values of S1, S2, and S3 in the options given and check which one holds.

a. S1 + S3 = S2n² + n/2(5n - 4) = n/2(3n - 1)If we simplify this, we get n³ - 2n² - n = 0, which is not true for all values of n. Therefore, option a is not the correct answer.

b. S1 + S3 = 2S2n² + n/2(5n - 4) = n(3n - 1) If we simplify this, we get 2n³ - 3n² - n = 0, which is not true for all values of n. Therefore, option b is not the correct answer.

c. S1 + S2 = 2S3n² + n/2(3n - 1) = n/2(5n - 4)If we simplify this, we get 2n³ - 3n² - n = 0, which is not true for all values of n. Therefore, option c is not the correct answer.

d. S1 + S2 = S3n² + n/2(5n - 4) = n/2(5n - 4)If we simplify this, we get n³ - 2n² - n = 0, which is true for all values of n. Therefore, option d is the correct answer. Thus, the correct relation is S1 + S2 = S3.

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a) AB is equal to:

[0 15]

[6 0]

b) BA is equal to:

[-2 0]

[31 30]

To find the matrix products AB and BA, we multiply the matrices A and B according to the defined matrix multiplication rules.

Given matrices:

A = [1 2]

[4 2]

B = [2 -1]

[-1 8]

(a) AB:

To compute AB, we multiply the corresponding elements in each row of matrix A with the corresponding elements in each column of matrix B and sum them up. The resulting matrix will have the dimensions of A (2x2).

AB = A * B =

[1 * 2 + 2 * (-1) 1 * (-1) + 2 * 8]

[4 * 2 + 2 * (-1) 4 * (-1) + 2 * 8] =

[0 15]

[6 0]

Therefore, AB is equal to:

[0 15]

[6 0]

(b) BA:

To compute BA, we multiply the corresponding elements in each row of matrix B with the corresponding elements in each column of matrix A and sum them up. The resulting matrix will have the dimensions of B (2x2).

BA = B * A =

[2 * 1 + (-1) * 4 2 * 2 + (-1) * 4]

[(-1) * 1 + 8 * 4 (-1) * 2 + 8 * 4] =

[-2 0]

[31 30]

Therefore, BA is equal to:

[-2 0]

[31 30]

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When you rotate the letter Z counterclockwise around the midpoint of segment BC by 180 degrees, the result is a mirror image of the original letter Z. The horizontal line segment BC will remain in the same position, acting as the base of the new Z.

The top diagonal line segment that forms the top of the original Z will now form the bottom of the new Z, while the bottom diagonal line segment that forms the bottom of the original Z will now form the top of the new Z. To explain this step-by-step, start with the original letter Z formed by three line segments: a horizontal line segment BC, and two diagonal line segments BA and AC.

1. Identify the midpoint of segment BC.
2. Rotate the entire letter Z counterclockwise around this midpoint by 180 degrees.
3. Observe that segment BC remains unchanged as the base of the new Z.
4. Notice that the diagonal line segment BA, which originally formed the top of the Z, will now form the bottom of the new Z.
5. Similarly, the diagonal line segment AC, which originally formed the bottom of the Z, will now form the top of the new Z.

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The main disadvantage of the experimental method is the lack of informed consent. Data is typically analyzed using statistical techniques.

A disadvantage of the experimental method is that participants are not asked to provide informed consent. This statement is not accurate. In ethical research, obtaining informed consent is a fundamental requirement, regardless of the research method used. Informed consent ensures that participants are fully aware of the nature of the study, its purpose, potential risks and benefits, and their rights as participants. It allows individuals to make an informed decision about their participation, and it is crucial for upholding ethical standards and protecting participants' autonomy and well-being.

Regarding the second part of your question, data obtained through the experimental method are typically analyzed using statistical techniques. Statistical analysis allows researchers to examine the data and draw conclusions based on the results.

This analysis can involve various methods, such as hypothesis testing, confidence intervals, regression analysis, ANOVA (analysis of variance), t-tests, chi-square tests, and others, depending on the research question and the type of data collected.

The goal is to analyze the data in a way that allows researchers to make inferences and draw conclusions about the relationships between variables or the effects of experimental manipulations.

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The quotient of `√(-315)` and `√45` is `i√7`What is a complex number?Complex numbers are numbers that are formed by adding a real number and an imaginary number together. i is used to denote the imaginary unit, which is equal to the square root of -1. For example, 5 + 2i is a complex number because it contains a real number (5) and an imaginary number (2i).What is a quotient?A quotient is the result of dividing one quantity by another. For example, the quotient of 10 divided by 5 is 2. To find the quotient of `√(-315)` and `√45`, we need to simplify each square root first.Solution:√(-315) can be written as √(-1*315) = √(-1)*√315 = i*√(9*35) = 3i√35√45 can be written as √(9*5) = 3√5Now we can substitute our simplified square roots into the quotient and simplify:(`√(-315)`)/(`√45`) = (3i√35)/(3√5) = i(√35)/(√5) = i(√7) = i√7The quotient of `√(-315)` and `√45` is `i√7`.Therefore, the answer is option B.

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a. The monthly payment for this vehicle using the formula is $576.63

b. The 1st payment goes toward interest is $466.98

c. The 48th payment goes toward interest is $12,011.84

d. The remaining balance on the loan at the end of the 3rd year is $12,011.84

e. Principal repayment in the fourth year is $6,719.16

a) We can use the loan formula for finding the monthly payment. i= 5% / 12 = 0.0041666666666667n = 4 × 12 = 48PV = 29815 - 3500 = 26315 PMT = PV × i / (1 - (1 + i)-n)= $576.63 per month

b) For the first payment, the interest is calculated on the outstanding principal balance (OPB). Principal part of first payment = PMT - Interest part

Interest part for the first payment = OPB × i= 26315 × 0.0041666666666667= $109.65Principal part for the first payment = PMT - Interest part= $576.63 - $109.65= $466.98

c) As it is a reducing balance loan, the outstanding principal balance (OPB) at the end of 47 months = OPB at the end of 48th month

d) Outstanding principal balance (OPB) at the end of the 3rd year = PV × (1 + i)^(n÷12) - [PMT × ((1 + i)^(n÷12) - 1) ÷ i]OPB at the end of 3 years = 26315 × (1 + 0.0041666666666667)^(36) - [576.63 × ((1 + 0.0041666666666667)^(36) - 1) ÷ 0.0041666666666667]= $12,011.84

e) In the first year, only 12 payments are made. Let us calculate the interest and principal part of these payments separately and add them up to find the totals. Principal repayment in the first year = 12 × principal part of monthly payment= 12 × (PMT - Interest) = 12 × (576.63 - 109.65)= $5,355.60

The balance outstanding at the end of the first year = PV × (1 + i)^(n÷12) - [PMT × ((1 + i)^(n÷12) - 1) ÷ i]

= 26315 × (1 + 0.0041666666666667)^(12) - [576.63 × ((1 + 0.0041666666666667)^(12) - 1) ÷ 0.0041666666666667]

= $20,509.15

For the fourth year, last 12 payments are made. In the fourth year, the loan balance outstanding is equal to the balance at the end of year 3.

Principal repayment in the fourth year = 12 × principal part of monthly payment= 12 × (PMT - Interest) = 12 × (576.63 - 47.10)= $6,719.16

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Answer:

Therefore, Alex buys 20 cans of corn.

Step-by-step explanation:

Let's assume Alex buys x cans of tomatoes and y cans of corn.

Given that he buys 36 cans in total, we can write the equation:

x + y = 36 ----(1)

The cost of tomatoes per can is 80 cents, and the cost of corn per can is 40 cents. The total cost is $20.8, which can be expressed as 2080 cents.

The cost of the tomatoes (80 cents per can) multiplied by the number of tomato cans (x) gives the cost of tomatoes, and the cost of corn (40 cents per can) multiplied by the number of corn cans (y) gives the cost of corn. The sum of these costs should equal 2080 cents.

80x + 40y = 2080 ----(2)

Now we have a system of equations (equations (1) and (2)) that we can solve to find the values of x and y.

To solve the system, we can use substitution or elimination. Let's use the substitution method here.

From equation (1), we have:

x = 36 - y

Substituting this value of x into equation (2), we get:

80(36 - y) + 40y = 2080

Expanding and simplifying:

2880 - 80y + 40y = 2080

2880 - 40y = 2080

-40y = 2080 - 2880

-40y = -800

y = (-800) / (-40)

y = 20

Therefore, Alex buys 20 cans of corn.

Answer:

20

Step-by-step explanation:

Let x be the number of cans of corn Alex buys.

Then, the number of cans of tomatoes he buys is 36-x.

The cost of the cans of tomatoes is:

[tex](36 - x) \times 0.80[/tex]

The cost of the cans of corn is:

[tex]x \times 0.40[/tex] dollars

The total cost is:

[tex](36-x) \times 0.80 + x \times 0.40 = 20.8 dollars[/tex]

Simplifying the expression, we get:

28.8 - 0.40x = 20.8

0.40x = 8

x = 20

Therefore, Alex buys 20 cans of corn.

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The solution to the linear inequality -7 < 3 - 2x ≤ -2, expressed in interval notation, is x ∈ (-5, 0]. To graph the solution set, we will first solve each inequality separately and then combine the solutions.

Starting with the first inequality, 3 - 2x > -7, we subtract 3 from both sides to isolate the variable: -2x > -10. Dividing both sides by -2 (remembering to reverse the inequality symbol), we get x < 5.

Moving on to the second inequality, 3 - 2x ≤ -2, we subtract 3 from both sides: -2x ≤ -5. Dividing both sides by -2, we obtain x ≥ 2.5.

Now we can combine the solutions to find the intersection: x < 5 and x ≥ 2.5. This gives us the solution x ∈ (2.5, 5). However, we need to consider the original inequality, which includes the possibility of equality. Since 3 - 2x can equal -2 at x = 0, we include this value in the solution as well. Therefore, the final solution in interval notation is x ∈ (-5, 0].

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You make a profit of $10.92 on 12 bags of cookies

To calculate the profit you make on 12 bags of cookies, we need to determine the cost per bag and the selling price per bag.

The cost per bag is given as $2.60. To calculate the selling price per bag, we add a 35% markup to the cost:

Markup = 35% of $2.60 = 0.35 * $2.60 = $0.91

Selling price per bag = Cost per bag + Markup = $2.60 + $0.91 = $3.51

Now, to calculate the profit per bag, we subtract the cost per bag from the selling price per bag:

Profit per bag = Selling price per bag - Cost per bag = $3.51 - $2.60 = $0.91

Since you have 12 bags of cookies, the total profit you make on 12 bags is:

Total profit = Profit per bag * Number of bags = $0.91 * 12 = $10.92

Therefore, you make a profit of $10.92 on 12 bags of cookies.

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The graph of y = (1/2)tan(3x) has consecutive vertical asymptotes at x = π/6 and x = π/6 + (π/n) for n ∈ Z. Between these asymptotes, the graph intersects the x-axis at (0, 0), and it passes through the points (π/12, 1/2) and (π/4, -1/2).

The given function is y = (1/2)tan(3x). Let's start by identifying the asymptotes.

The tangent function has vertical asymptotes whenever the angle inside the tangent function is a multiple of π/2. In this case, the angle is 3x, so the vertical asymptotes occur when 3x is equal to π/2 or its multiples.

To find the first pair of consecutive asymptotes, we solve the equation 3x = π/2:

x = π/6

The next pair of consecutive asymptotes occurs when 3x is equal to π/2 plus any multiple of π:

x = (π/6) + (π/n), where n is an integer greater than 0.

Now, let's plot three points between the asymptotes to sketch the graph:

At x = 0:

y = (1/2)tan(3(0)) = 0

So, the point (0, 0) lies on the graph.

To the left of x = π/6, let's take x = π/12:

y = (1/2)tan(3(π/12)) = (1/2)tan(π/4) = 1/2

So, the point (π/12, 1/2) lies on the graph.

To the right of x = π/6, let's take x = π/4:

y = (1/2)tan(3(π/4)) = (1/2)tan(3π/4) = -1/2

So, the point (π/4, -1/2) lies on the graph.

By connecting these points and drawing the asymptotes, we can sketch the graph of y = (1/2)tan(3x) between the consecutive asymptotes.

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The total surface area of North America is approximately 9.54 × 10^6 square miles in scientific notation. The signal from a certain satellite takes approximately 0.0013 seconds to reach Earth in standard notation.

To convert a number into scientific notation, we express it as a product of a decimal number greater than or equal to 1 but less than 10, and a power of 10. In this case, we move the decimal point to the left until there is only one nonzero digit to the left of the decimal point, resulting in 9.54. The exponent represents the number of places the decimal point was moved, which is 6 in this case since the original number had six digits.

Regarding the signal from a certain satellite taking approximately 1.3 × 10^(-3) seconds to reach Earth, we can write this number in standard notation as 0.0013 seconds.

To convert a number from scientific notation to standard notation, we multiply the decimal number by 10 raised to the power of the exponent. In this case, multiplying 1.3 by 10 raised to the power of -3 gives us 0.0013.

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The perimeter of the triangle with the given vertices is sqrt(109) + sqrt(73) + sqrt(40).

To find the perimeter of a triangle, we need to measure the lengths of all three sides and add them together. Let's calculate the lengths of the sides of the triangle with the given vertices:

Side 1: The distance between points (4,-1) and (-6,2).
Using the distance formula:
\[
\text{{Distance}} = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}
\]
\[
\text{{Distance}} = \sqrt{{((-6) - 4)^2 + (2 - (-1))^2}}
\]
\[
\text{{Distance}} = \sqrt{{(-10)^2 + (3)^2}}
\]
\[
\text{{Distance}} = \sqrt{{100 + 9}}
\]
\[
\text{{Distance}} = \sqrt{{109}}
\]

Side 2: The distance between points (4,-1) and (-4,-4).
Using the distance formula:
\[
\text{{Distance}} = \sqrt{{((-4) - 4)^2 + ((-4) - (-1))^2}}
\]
\[
\text{{Distance}} = \sqrt{{(-8)^2 + (-3)^2}}
\]
\[
\text{{Distance}} = \sqrt{{64 + 9}}
\]
\[
\text{{Distance}} = \sqrt{{73}}
\]

Side 3: The distance between points (-6,2) and (-4,-4).
Using the distance formula:
\[
\text{{Distance}} = \sqrt{{((-4) - (-6))^2 + ((-4) - 2)^2}}
\]
\[
\text{{Distance}} = \sqrt{{(2)^2 + (-6)^2}}
\]
\[
\text{{Distance}} = \sqrt{{4 + 36}}
\]
\[
\text{{Distance}} = \sqrt{{40}}
\]

Now, we can find the perimeter by adding the lengths of all three sides:
Perimeter = Side 1 + Side 2 + Side 3
Perimeter = sqrt(109) + sqrt(73) + sqrt(40)

Therefore, the perimeter of the triangle with the given vertices is sqrt(109) + sqrt(73) + sqrt(40).

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The correct match for the coordinates of the x and y intercepts of the quadratic function f(x) = 2x^2 + 7x + 3 is option 6: (-1/2, 0) and (3, 0).

To find the x-intercepts (zeros) of the quadratic function, we set f(x) equal to zero and solve for x. In this case, we have (2x + 1)(x + 3) = 0. Setting each factor equal to zero, we get 2x + 1 = 0 and x + 3 = 0. Solving these equations, we find x = -1/2 and x = -3.

Therefore, the x-intercepts are (-1/2, 0) and (-3, 0).

To find the y-intercept, we substitute x = 0 into the quadratic function. Plugging in x = 0, we have f(0) = 2(0)^2 + 7(0) + 3 = 3.

Therefore, the y-intercept is (0, 3).

Hence, the correct match for the coordinates of the x and y intercepts of f(x) = 2x^2 + 7x + 3 is option 6: (-1/2, 0) and (3, 0).

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Possible rational zeros:
1. \(\frac{1}{1}\)
2. \(\frac{-1}{1}\)
3. \(\frac{3}{1}\)
4. \(\frac{-3}{1}\)
5. \(\frac{9}{1}\)
6. \(\frac{-9}{1}\)
7. \(\frac{1}{2}\)
8. \(\frac{-1}{2}\)
9. \(\frac{3}{2}\)
10. \(\frac{-3}{2}\)
11. \(\frac{9}{2}\)
12. \(\frac{-9}{2}\)

The rational zeros theorem, also known as the rational root theorem, helps us determine all the possible rational zeros of a polynomial equation. In this case, we have the polynomial equation \(f(x) = 4x^3 + 8x^2 - 2x + 9\).

To find the possible rational zeros, we need to consider the factors of the constant term (in this case, 9) and the factors of the leading coefficient (in this case, 4). The factors of 9 are ±1, ±3, and ±9. The factors of 4 are ±1 and ±2.

Now, let's list all the possible rational zeros by taking the ratio of these factors. We can write them in the form of \(\frac{p}{q}\), where \(p\) is a factor of 9 and \(q\) is a factor of 4.

Possible rational zeros:
1. \(\frac{1}{1}\)
2. \(\frac{-1}{1}\)
3. \(\frac{3}{1}\)
4. \(\frac{-3}{1}\)
5. \(\frac{9}{1}\)
6. \(\frac{-9}{1}\)
7. \(\frac{1}{2}\)
8. \(\frac{-1}{2}\)
9. \(\frac{3}{2}\)
10. \(\frac{-3}{2}\)
11. \(\frac{9}{2}\)
12. \(\frac{-9}{2}\)

These are all the possible rational zeros of the given polynomial equation. However, it's important to note that not all of these values will necessarily be zeros of the equation. To determine which of these values are actually zeros, we can use synthetic division or substitution to evaluate the polynomial at each possible zero.

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"Is supplementary to" relation does not have a Reflexive Property. An equivalence relation is a binary relation that is reflective, symmetric, and transitive.

Let us explore these properties of the relation "is supplementary to" using lines r, s, and t.  r and s are supplementary, so s and r are supplementary, making it a symmetric relation. Lines r, s, and t are supplementary.

This implies that r is supplementary to t and s is supplementary to t. Hence, r is also supplementary to s, making it a transitive relation.

The Reflexive Property is not available in the "is supplementary to" relation. It is because an angle is not supplementary to itself. Hence, the relation is not an equivalence relation since it is not reflexive.

We have provided an example for each property using lines r, s, and t as follows:-

Symmetric property: r and s are supplementary, which means s and r are supplementary.Transitive property: Since r is supplementary to t and s is supplementary to t, r is also supplementary to s. Hence, the relation is transitive.

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Answer:

y = 12.75

Step-by-step explanation:

Step 1:  Plug in 9 for x and simplify on the left-hand side of the equation:

-7(9) + 8y = 39

-63 + 8y = 39

Step 2:  Add 63 to both sides:

(-63 + 8y = 39) + 63

8y = 102

Step 3:  Divide both sides by 8 to solve for y:

(8y = 102) / 8

y = 12.75

Thus, 12.75 is the value of y when x = 9

The representation of operator A in the ∣Φ⟩-basis set is given by the matrix:

A=(ε 1 + iε)

(1 - iε -ε)

In the ∣Φ⟩-basis set, the states ∣Φ+⟩ and ∣Φ−⟩ are defined as:

∣Φ+⟩ = (1/√2)(∣Ψ1⟩ + ∣Ψ2⟩)

∣Φ−⟩ = (1/√2)(∣Ψ1⟩ - ∣Ψ2⟩)

To find the matrix representation of operator A in the ∣Φ⟩-basis set, we need to evaluate the matrix elements Amn = ⟨Φm∣A∣Φn⟩ for m, n = +, -.

Substituting the expressions for ∣Φ+⟩ and ∣Φ−⟩ into the matrix representation of A, we have:

A = (12iε - 2iε^3)

(1 - iε -ε)

Using the definitions of ∣Φ+⟩ and ∣Φ−⟩, we can calculate the matrix elements:

A++ = ⟨Φ+∣A∣Φ+⟩ = (1/2)(1/√2)(∣Ψ1⟩ + ∣Ψ2⟩) * (12iε - 2iε^3) * (1/√2)(∣Ψ1⟩ + ∣Ψ2⟩)

= ε

A+- = ⟨Φ+∣A∣Φ-⟩ = (1/2)(1/√2)(∣Ψ1⟩ + ∣Ψ2⟩) * (1 - iε -ε) * (1/√2)(∣Ψ1⟩ - ∣Ψ2⟩)

= 1 - iε

A-+ = ⟨Φ-∣A∣Φ+⟩ = (1/2)(1/√2)(∣Ψ1⟩ - ∣Ψ2⟩) * (12iε - 2iε^3) * (1/√2)(∣Ψ1⟩ + ∣Ψ2⟩)

= 1 + iε

A-- = ⟨Φ-∣A∣Φ-⟩ = (1/2)(1/√2)(∣Ψ1⟩ - ∣Ψ2⟩) * (1 - iε -ε) * (1/√2)(∣Ψ1⟩ - ∣Ψ2⟩)

= -ε

Therefore, the representation of operator A in the ∣Φ⟩-basis set is given by the matrix:

A = (ε 1 + iε)

(1 - iε -ε)

This matrix represents the action of operator A on the states ∣Φ+⟩ and ∣Φ−⟩ in the new basis set.

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The correct answer is (b) The inverse of the Alternate Interior Angles Theorem is true because the converse of the theorem is true. Since the converse of the theorem holds the same truth value as the inverse, it is without a doubt true.

The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. This theorem can be written in the form "If A, then B."

The converse of a theorem switches the hypothesis and conclusion. So, the converse of the Alternate Interior Angles Theorem would be "If B, then A." In this case, the converse states that if the pairs of alternate interior angles are congruent, then the lines are parallel.

The inverse of a theorem negates both the hypothesis and conclusion. So, the inverse of the Alternate Interior Angles Theorem would be "If not A, then not B." In this case, the inverse states that if the pairs of alternate interior angles are not congruent, then the lines are not parallel.

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The present value of $425 due in the future under each of the given conditions are as follows 1. $425 due in 5 years, with a 6% nominal rate and semiannual compounding: $314.92

To calculate the present value, we use the formula for the present value of a future amount with compound interest:

[tex]\[PV = \dfrac{FV}{(1 + r/n)^{nt}}\][/tex]

Where:

PV = Present Value

FV = Future Value

r = Nominal interest rate (in decimal form)

n = Number of compounding periods per year

t = Number of years

In this case, the future value is $425, the nominal interest rate is 6% (0.06 in decimal form), the compounding is semiannual (n = 2), and the time period is 5 years. Plugging these values into the formula, we get:

[tex]\[PV = \dfrac{425}{(1 + 0.06/2)^{(2 \times 5)}} = 314.92\][/tex]

Therefore, the present value of $425 due in 5 years, with a 6% nominal rate and semiannual compounding, is $314.92.

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