In an End Centered Bravais lattice, the number of lattice points per unit cell is A) 6 B) 2 C) none of the choices shown D) 4

After 31.6 minutes have passed, approximately 40.3 mg of lead-214 remains in the sample. This can be determined using the decay equation lnA = lnA₀ - kt, where A represents the amount of the substance remaining after time t, A₀ is the original amount of the substance, k is a constant characteristic of the substance, and t is the elapsed time.

The given equation, lnA = lnA₀ - kt, represents the decay of the radioactive isotope lead-214. In this equation, A represents the amount of the substance remaining after time t, A₀ is the original amount of the substance, k is a constant characteristic of the substance, and t is the elapsed time.

To find the amount of lead-214 remaining after 31.6 minutes, we can plug in the given values into the equation. We are given that A₀, the original amount of lead-214 in the sample, is 51.3 mg. The value of k for lead-214 is 2.59×[tex]10^(^-^2^)[/tex][tex]minutes^(^-^1^)[/tex], as mentioned in the question. Finally, t is 31.6 minutes.

Substituting these values into the equation, we have:

lnA = ln(51.3) - (2.59×[tex]10^(^-^2^)[/tex] × 31.6)

Evaluating the right side of the equation, we get:

lnA ≈ 3.937 - (2.59×[tex]10^(^-^2^)[/tex] × 31.6)

    ≈ 3.937 - 0.8164

    ≈ 3.1206

To find A, we need to exponentiate both sides of the equation using the natural logarithm base, e:

[tex]e^(^l^n^A^)[/tex] = [tex]e^(^3^.^1^2^0^6^)[/tex]

A ≈ 22.63 mg

Therefore, after 31.6 minutes have passed, approximately 22.63 mg of lead-214 remains in the sample.

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The velocity function v(t) is 6t + 3. The acceleration function a(t) is constant and equal to 6. When t = 4 sec, the velocity is 27 ft/s and the acceleration is 6 ft/s^2.

To find the velocity and acceleration, we need to differentiate the position function s(t) with respect to time t.a) Velocity (v(t)): The velocity is the derivative of the position function s(t) with respect to time t.v(t) = d/dt [s(t)]

Given s(t) = 3t^2 + 3t, we can differentiate it to find the velocity:

v(t) = d/dt [3t^2 + 3t]

To differentiate, we apply the power rule of differentiation: v(t) = 6t + 3

Therefore, the velocity function v(t) is 6t + 3.

b) Acceleration (a(t)): The acceleration is the derivative of the velocity function v(t) with respect to time t. a(t) = d/dt [v(t)]

Given v(t) = 6t + 3, we can differentiate it to find the acceleration:

a(t) = d/dt [6t + 3]

The derivative of a constant term is zero, so the derivative of 3 is 0:

a(t) = 6

Therefore, the acceleration function a(t) is constant and equal to 6.

c) Velocity and acceleration when t = 4 sec:

To find the velocity and acceleration at t = 4 seconds, we substitute t = 4 into the respective functions: At t = 4 sec: v(4) = 6(4) + 3

v(4) = 24 + 3

v(4) = 27 ft/s ,a(4) = 6

Therefore, when t = 4 sec, the velocity is 27 ft/s and the acceleration is 6 ft/s^2.

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The equation of the circle is:(x - 1)² + (y - 2)² = 17

To find the equation of a circle with center at (1, 2) and passing through (-3, 3), we need to use the formula for the standard form of the equation of a circle.

A circle with center (h, k) and radius r is given by the equation:(x - h)² + (y - k)² = r²

Substituting the given values, we have:(x - 1)² + (y - 2)² = r²

We can now find the value of r using the fact that the circle passes through the point (-3, 3):

(x - 1)² + (y - 2)² = r²(-3 - 1)² + (3 - 2)² = r²16 + 1 = r²17 = r²

So the equation of the circle is:(x - 1)² + (y - 2)² = 17

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The maximum number of grams of CaSO4 that can be produced is calculated by determining the limiting reactant and using stoichiometry to find the corresponding amount of product.

To determine the limiting reactant, we need to compare the moles of each reactant and their stoichiometric ratios in the balanced equation.

1. Calculate the moles of Ca3(PO4)2:

                             = 310.18 g/mol

2. Calculate the moles of H2SO4:

3. Compare the stoichiometric ratios:

  From the balanced equation, the stoichiometric ratio of Ca3(PO4)2 to H2SO4 is 1:3.

  The moles ratio of Ca3(PO4)2 to H2SO4 is 3.22 mol : 10.19 mol.

4. Limiting Reactant:

  Since the stoichiometric ratio is 1:3, we can see that Ca3(PO4)2 is the limiting reactant because it will be completely consumed before H2SO4.

5. Determine the maximum grams of CaSO4:

  The stoichiometric ratio of CaSO4 to Ca3(PO4)2 is 3:1.

  The moles of CaSO4 produced will be equal to the moles of Ca3(PO4)2 used.

  Moles of CaSO4 = 3.22 mol

  Mass of CaSO4 = moles x molar mass = 3.22 mol x (40.08 g/mol) = 129.34 g

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The student pilot would have to fly approximately 125 miles in a straight-line distance to get back to Dallas.

The distance flown due west from Dallas is 100 miles, and the distance flown due north from Wichita Falls is 75 miles. These distances form the two sides of a right-angled triangle.

Using the Pythagorean theorem, we can calculate the hypotenuse (the straight-line distance) as follows:

Hypotenuse² = (Distance due west)² + (Distance due north)²

Hypotenuse² = 100² + 75²

Hypotenuse² = 10000 + 5625

Hypotenuse² = 15625

Taking the square root of both sides gives us:

Hypotenuse = √15625

Hypotenuse ≈ 125

Therefore, the student pilot would have to fly approximately 125 miles in a straight-line distance to get back to Dallas.

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The equation of p(x) in the standard form of a quadratic function after the translation right 4 units and up 5 units is p(x) = 3x² + 6x - y + 65.

The quadratic function is y = 3x² + 30x + 2. To translate it right 4 units, we substitute x with (x - 4). To translate it up 5 units, we substitute y with (y + 5).

So the new equation becomes y + 5 = 3(x - 4)² + 30(x - 4) + 2.

Expanding and simplifying, we get y + 5 = 3x² + 6x - y + 65.

Rearranging the terms, we obtain p(x) = 3x² + 6x - y + 65.

The graph of the quadratic function p(x) will have the same shape as y = 3x², but it will be shifted 4 units to the right and 5 units up compared to the original function. The vertex of the graph will be at the point (-2, 5), and the parabola will open upward.

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1) The slope of y = 6x − 2 and y = 6x + 7 are both 6, thus, making them parallel.

2) The slope of y = 2x + 4 and x + 2y + 10 = 0 is 2 and -1/2, respectively. They are perpendicular.

3) The slope of y = 8x − 1 is 8 and of 7x − y − 1 = 0 is 7 and they are neither parallel nor perpendicular with each other.

1. To find the slope of the given lines, we must write their equations in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.

y = 6x - 2 has a slope of 6

y = 6x + 7 has a slope of 6.

The two equations have the same slopes but different y-intercept. Therefore, the two lines are parallel.

2. y = 2x + 4 has a slope of 2.

Rearranging x + 2y + 10 = 0 into slope-intercept form gives 2y = -x - 10, so y = (-1/2)x - 5, which has a slope of -1/2.

-1/2 is the negative reciprocal of 2. Therefore, the two lines are perpendicular.

3. y = 8x - 1 has a slope of 8.

Rearranging 7x - y - 1 = 0 into slope-intercept form gives y = 7x - 1, which has a slope of 7.

The slopes of the two equation are neither the same nor the negative reciprocal of one another. Therefore, the two lines are neither parallel nor perpendicular.

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The exact values of the trigonometric functions are:

Sec(θ) = 2

Csc(θ) = (-2√3)/3

Tan(θ) = -√3

Cot(θ) = (-√3)/3

If θ = -7π/3, then the values of the trigonometric functions are as follows:

Sec(θ) = 1/cos(θ) = 1/cos(-7π/3) = 1/(cos(π/3)) = 1/(1/2) = 2

Csc(θ) = 1/sin(θ) = 1/sin(-7π/3) = 1/(sin(-π/3)) = 1/(-√3/2) = -2/√3 = (-2√3)/3

Tan(θ) = sin(θ)/cos(θ) = sin(-7π/3)/cos(-7π/3) = (sin(-π/3))/(cos(-π/3)) = (-√3/2)/(1/2) = -√3

Cot(θ) = 1/tan(θ) = 1/(-√3) = -1/√3 = (-√3)/3

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(a) The slope of the line parallel to the given line is the same as the slope of the given line.
(b) The slope of the line perpendicular to the given line is the negative reciprocal of the slope of the given line.


(a) To find the slope of the line passing through the points (-3,-9) and (0,0), we use the slope formula: m = (y2 - y1) / (x2 - x1). Plugging in the coordinates, we get m = (0 - (-9)) / (0 - (-3)) = 9/3 = 3. Since parallel lines have the same slope, the slope of the line parallel to the given line is also 3.
(b) The negative reciprocal of a slope is obtained by flipping the fraction and changing its sign. Therefore, the negative reciprocal of 3 is -1/3. So, the slope of the line perpendicular to the given line is -1/3.
These slopes determine the steepness and direction of the lines in relation to the given line passing through the points (-3,-9) and (0,0).

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Answer:any number less than or equal to 0.

Step-by-step explanation:

To determine the value of y for the given inequality, let's solve it step by step:

3(y + 1/4) ≤ 3/4

First, let's simplify the left side of the inequality:

3y + 3/4 ≤ 3/4

Next, let's isolate the term with y by subtracting 3/4 from both sides of the inequality:

3y ≤ 3/4 - 3/4

This simplifies to:

3y ≤ 0

To solve for y, divide both sides of the inequality by 3:

y ≤ 0/3

y ≤ 0

Therefore, the value of y that satisfies the inequality is any number less than or equal to 0.

Answer:

number less than or equal to 0.

Step-by-step explanation:

The shape of the given quadrilateral ABCD can be determined by examining the sides and angles of the quadrilateral. Thus, the correct option is b) Rhombus.  

To identify the shape, we need to consider the properties of different quadrilaterals.

A square has all sides equal in length and all angles equal to 90 degrees.
A rhombus has all sides equal in length, but the angles are not necessarily 90 degrees.

A kite has two pairs of adjacent sides that are equal in length.
A rectangle has opposite sides equal in length and all angles equal to 90 degrees.

By examining the given coordinates, we can calculate the lengths of the sides of the quadrilateral. The distance formula is used to find the lengths between the vertices:

AB = √[(4-0)^2 + (5-0)^2] = √(4^2 + 5^2) = √(16 + 25) = √41
BC = √[(9-4)^2 + (9-5)^2] = √(5^2 + 4^2) = √(25 + 16) = √41
CD = √[(5-9)^2 + (4-9)^2] = √((-4)^2 + (-5)^2) = √(16 + 25) = √41
DA = √[(0-5)^2 + (0-4)^2] = √((-5)^2 + (-4)^2) = √(25 + 16) = √41

As all four sides have the same length, which is √41, we can conclude that the shape of the quadrilateral ABCD is a rhombus.

Thus, the correct option is b) Rhombus.

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The dimensions of the original rectangle are _______ meters (width) and _______ meters (length). [Please provide the calculated values here.]

1. Given:
- The length of the original rectangle is 3 times its width.
- The area of the new rectangle is 68 square meters after increasing the length by 2 meters and decreasing the width by 1 meter. 2. Required:
- The dimensions of the original rectangle. 3. Equation:
Let's represent the width of the original rectangle as 'w' meters. Then, the length of the original rectangle would be '3w' meters.
The area of a rectangle is calculated by multiplying its length by its width:
Area = Length * Width

4. Solution:
Given that the area of the new rectangle is 68 square meters, we can set up the equation:
(3w + 2) * (w - 1) = 68. Expanding the equation, we get: 3w^2 - w - 70 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. After finding the value of 'w', we can substitute it back into the equation to find the length.

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The solutions to the equation 5x^2 - 35 = 0 are x = ±√7.

To solve this quadratic equation, we can first isolate the variable by moving the constant term to the other side:

5x^2 = 35

Next, we divide both sides of the equation by 5 to solve for x^2:

x^2 = 7

To find the value of x, we take the square root of both sides:

√(x^2) = ±√7

Since we took the square root, we need to consider both the positive and negative square roots, giving us two solutions:

x = ±√7

Therefore, the solutions to the equation 5x^2 - 35 = 0 are x = ±√7.

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Graph D represents the result. The graph D is obtained by shifting the graph of h(x) = sin(x) 3 pi/2 units to the left and 11 units down, which matches the given translation.

The translation of "3 pi/2 units left and 11 units down" implies that each x-coordinate of the original function h(x) = sin(x) is reduced by 3 pi/2, and each y-coordinate is reduced by 11.

Graph D is obtained by shifting the graph of h(x) = sin(x) 3 pi/2 units to the left and 11 units down. This shift is consistent with the given translation. Therefore, Graph D represents the desired result.

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The complete question is:

Suppose the function h(x) = sinx is translated 3pi/2 units left and 11 units down. Which graph represents the result? (Only one graph from below is the correct answer)

Tangent value of zero.

The tangent (tan) of an angle is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle containing that angle. In this case, we have a point P(-8, 0) on the terminal arm of angle θ in standard position. Since the x-coordinate is negative and the y-coordinate is zero, we can determine that the point is located on the x-axis, specifically to the left of the origin.

When the y-coordinate is zero, it means that the length of the opposite side of the angle is zero. This implies that there is no vertical displacement from the x-axis. Since the tangent of an angle is defined as the ratio of the opposite side to the adjacent side, and the adjacent side is represented by the x-coordinate,

we have y/x = 0/x = 0.

Therefore, "the tangent of the angle θ at the point P(-8, 0) is zero". This indicates that the angle has no vertical displacement relative to the x-axis. The terminal arm lies entirely on the x-axis, resulting in a tangent value of zero.

In summary, tan θ = 0 because the point P(-8, 0) is situated on the x-axis, to the left of the origin, with no vertical displacement. Division by zero is undefined in mathematics, so the tangent value is zero rather than being undefined.

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a) The next 3 terms are aₙ + 3, aₙ + 6, and aₙ + 9, where aₙ is the last term in the sequence.
b) a₅₂ = 154 and a₇₅₈ = 2,272, using the formula aₙ = a₁ + (n - 1)d, with a₁ = 1 and d = 3.

a) To find the next 3 terms in the given sequence, we observe that each term is obtained by adding 3 to the previous term. Therefore, we can continue the pattern by adding 3 to the last term. In general, if the last term is denoted as aₙ, then the next three terms would be aₙ + 3, aₙ + 6, and aₙ + 9.

b) To find a specific term in the sequence, we can use the formula aₙ = a₁ + (n - 1)d, where a₁ is the first term, n is the term number, and d is the common difference. By substituting the given values (a₁ = 1 and d = 3) into the formula, we can find a₅₂ and a₇₅₈. Applying the formula, we find that a₅₂ = 154 and a₇₅₈ = 2,272.

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The answers are:  (I) F, (II) F, (III) T

(I) False. Saltwater is an example of a homogeneous mixture, also known as a solution, where the salt particles are evenly distributed throughout the water. It is not possible to separate salt from water by simple physical means like filtration. To separate the salt from saltwater, a process like evaporation or distillation is required.

(II) False. Coulomb's Law states that the interaction between charges decreases as the distance between them increases. The law describes the electrostatic force between two charged objects, which follows an inverse square relationship. As the distance between charges increases, the force of interaction decreases.

(III) True. The measurement of 101.0 g has three significant figures. In scientific notation, significant figures are the digits that carry meaningful information about the measurement. Non-zero digits and zeros between non-zero digits are considered significant. In this case, all three digits (1, 0, and 1) are non-zero and, therefore, significant.

In summary, saltwater cannot be separated by simple physical means, Coulomb's Law states that the interaction between charges decreases as the distance increases, and the measurement of 101.0 g has three significant figures.

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Given the equation 18 sin²(x) = 9, the values of x in the interval [0, 2] that satisfy the equation are x = π/4 and x = 3π/4.

To solve the equation, we start by dividing both sides by 18 to isolate the sin²(x):

sin²(x) = 9/18

sin²(x) = 1/2

Next, we use the trigonometric identity sin²θ + cos²θ = 1. By substituting sin²(x) with 1 - cos²(x), we have:

1 - cos²(x) = 1/2

Rearranging the equation, we get:

cos²(x) = 1 - 1/2

cos²(x) = 1/2

Taking the square root of both sides, we have:

cos(x) = ±√(1/2)

cos(x) = ±1/√2

cos(x) = ±√2/2

Since cosine is positive in the first and fourth quadrants, and negative in the second and third quadrants, we have two solutions:

x = π/4 and x = 3π/4

We can confirm that these solutions lie in the interval [0, 2].

Therefore, the values of x in the interval [0, 2] that satisfy the equation are x = π/4 and x = 3π/4.

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The product of the given matrices is not equal to the third matrix.

To compute the matrix product, we need to multiply each entry of the first matrix by the corresponding entry in the second matrix and sum the results. Let's calculate the product of the first entry in the first row of the first matrix (-1) with the first entry in the first column of the second matrix (1). This gives us -1 * 1 = -1.

Similarly, we multiply the second entry in the first row of the first matrix (-1) with the second entry in the second column of the second matrix (-3), which gives us -1 * -3 = 3.

Finally, we multiply the third entry in the first row of the first matrix (3) with the third entry in the second column of the second matrix (0), which gives us 3 * 0 = 0. Combining these results, we have the first entry of the resulting matrix as -1 + 3 + 0 = 2. Following the same procedure, we can compute the other entries of the resulting matrix.

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(a) d(5) = 75. Joe travels 75 miles in 5 hours.

(b) Joe travels 7.5 miles in 30 minutes.

(a) Evaluating d(5) means plugging in the value of 5 for t in the equation d(t) = 15t and calculating the resulting distance.

Substituting t = 5 into the equation, we get d(5) = 15 * 5 = 75.

Therefore, d(5) = 75. This means that Joe travels a distance of 75 miles in 5 hours.

(b) We need to determine the distance Joe travels in 30 minutes.

Since the time is given in hours, we convert 30 minutes to hours by dividing by 60 (since there are 60 minutes in an hour): 30 minutes / 60 = 0.5 hours.

Now, we can substitute t = 0.5 into the distance equation: d(0.5) = 15 * 0.5 = 7.5.

Therefore, Joe travels a distance of 7.5 miles in 30 minutes.

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a. To find the pattern in the sequence 0, 1, 3, 6, 10, we can observe that each term is obtained by adding the next consecutive number starting from 1.

The first term, 0, is obtained by adding 1 + 0.
The second term, 1, is obtained by adding 1 + 0.
The third term, 3, is obtained by adding 1 + 2.
The fourth term, 6, is obtained by adding 1 + 2 + 3.
The fifth term, 10, is obtained by adding 1 + 2 + 3 + 4.

Following the same pattern, we can find the next three terms:

11, 15, 20.

b. In the sequence 52, 47, 42, 37, the pattern is that each term is obtained by subtracting 5 from the previous term.

The first term, 52, is obtained by subtracting 5 from 57.
The second term, 47, is obtained by subtracting 5 from 52.
The third term, 42, is obtained by subtracting 5 from 47.
The fourth term, 37, is obtained by subtracting 5 from 42.

Following the same pattern, we can find the next three terms:
32, 27, 22.

c. In the sequence 6400, 3200, 1600, 800, the pattern is that each term is obtained by dividing the previous term by 2.

The first term, 6400, is obtained by dividing 3200 by 2.
The second term, 3200, is obtained by dividing 1600 by 2.
The third term, 1600, is obtained by dividing 800 by 2.

Following the same pattern, we can find the next three terms:
400, 200, 100.

Remember to choose the correct option based on the pattern observed.

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The given expression in terms of a, b, and c is a - b - c.

Using the trigonometric identities, we can express the given expression in terms of a, b, and c:

sin(t+2π) − cos(t+10π) + tan(t+5π)

Using the periodicity of sine and cosine functions, sin(t+2π) is equal to sin(t) and cos(t+10π) is equal to cos(t). We can substitute these values:

sin(t) − cos(t) + tan(t+5π)

Using the trigonometric identity tan(t+π) = -tan(t), we can rewrite tan(t+5π) as -tan(t):

sin(t) − cos(t) - tan(t)

Now, substituting a for sin(t), b for cos(t), and c for tan(t), we have:

a - b - c

So, A - B - C is the provided phrase in terms of a, b, and c.

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As an integer the evaluation comes out to be 5 for the expression +3/24-2%1

Expression is: 5 + 3/24 - 2 % 1.

We will solve this expression step by step:

1) we will solve the modulo operation: 2 % 1 = 0.

2) we will solve the division operation: 3/24 = 0.125.

Now, we will substitute the values in the given expression: 5 + 0.125 - 0 = 5.125.

Since we need to evaluate the expression as an integer, we will round it off to the nearest integer.5.125 is closer to 5 than to 6.

Therefore, we will round it down to 5.Hence, the integer value of the given expression is 5.

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In order to determine whether the project should be accepted or not when the discount rate is 12.5 percent, we can use the profitability index (PI) which is calculated by dividing the present value of cash inflows by the initial investment.

The formula for PI is:PI = (PV of cash inflows) / (initial investment)

A project should be accepted if the profitability index is greater than 1. Therefore, we need to calculate the profitability index (PI) of the project using the given information and determine whether it is greater than 1 or not.

The given answer choices are:

No; because the Pl is 3.3

Yes; because the PI is 3.0

No; because the Pl is 0.8

Yes; because the PI is 2.6

Yes; because the PI is 2.2

However, there is no information given regarding the Pl (present value of cash inflows) for any of these answer choices.

Therefore, we cannot use these answer choices to determine whether the project should be accepted or not when the discount rate is 12.5 percent. So, we need to calculate the PI using the given information and check if it is greater than 1 or not.

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Sally deposits $1,600 into a savings account earning a simple interest rate of 7.00%., then the interest Sally will earn after 180 days is $55.09.

To calculate the amount of interest Sally will earn after 180 days on depositing $1600 in a savings account with a simple interest rate of 7%, the following formula applies;Interest = P × r × t

where;P is the principal (amount deposited),r is the annual interest rate (7%),t is the time in years or fraction of a year.

We can convert 180 days to fraction of a year by dividing it by the total number of days in a year as follows;

180 days ÷ 365 days = 0.49315068 years

Substitute the values of P, r and t to calculate the interest;

Interest = 1600 × 0.07 × 0.49315068

Interest = 55.09

To two decimal places, the interest Sally will earn after 180 days is $55.09.

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Given information:Slope = - 1/3Passing through (1, -5)To write the equation of a line in point-slope form, we use the formula as follows:y - y1 = m(x - x1)Where m is the slope and (x1, y1) are the coordinates of the given point.Substituting the given values in the above formula, we have;y - (-5) = -1/3(x - 1)y + 5 = -1/3x + 1/3y = -1/3x + 1/3 - 5y = -1/3x - 14/3Thus, the equation of the line in point-slope form is y - (-5) = -1/3(x - 1) and in slope-intercept form is y = -1/3x - 14/3. Therefore, the equation of the line in point-slope form is y - (-5) = -1/3(x - 1) and in slope-intercept form is y = -1/3x - 14/3.

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Answer:

FG = 5

Step-by-step explanation:

Helping in the name of Jesus.

(a) The consumer's optimal consumption choice for x and y can be found by taking the partial derivatives of the utility function with respect to x and y, setting them equal to zero, and solving for x and y.

(b) The derivative of the optimal level of utility with respect to m can be computed using the chain rule and the solution obtained in part (a).

(a) To find the consumer's optimal consumption choice for x and y, we need to maximize the utility function u(x, y) = x^a * y^(1-a).

Taking the partial derivative of u(x, y) with respect to x and setting it equal to zero:

∂u/∂x = a * x^(a-1) * y^(1-a) = 0.

Simplifying the equation, we get:

a * x^(a-1) * y^(1-a) = 0.

Since a > 0, x^(a-1) ≠ 0. Therefore, we can divide both sides of the equation by a * x^(a-1) to obtain:

y^(1-a) = 0.

However, y^(1-a) ≠ 0 because y > 0 and 1-a ≠ 0. Therefore, the equation y^(1-a) = 0 has no solution.

Next, we take the partial derivative of u(x, y) with respect to y and set it equal to zero:

∂u/∂y = (1-a) * x^a * y^(-a) = 0.

Simplifying the equation, we get:

(1-a) * x^a * y^(-a) = 0.

Since 1-a ≠ 0, x^a ≠ 0, and y^(-a) ≠ 0, we can divide both sides of the equation by (1-a) * x^a * y^(-a) to obtain:

1 = 0.

However, 1 ≠ 0, so the equation 1 = 0 has no solution.

Therefore, the consumer's optimal consumption choice for x and y cannot be determined using the partial derivatives of the utility function. Additional information or constraints are needed to find the optimal solution.

(b) Since the optimal consumption choice for x and y cannot be determined, we cannot compute the derivative of the optimal level of utility with respect to m.

This completes the explanation.

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The distance between City A and City C, when an automobile driven at speed of 65 mph takes 1.8 hours to drive from City A to City B and takes 1.2hours to drive from City B to City C, is approximately 287.87 miles.

Given that the bearing from City A to City B is N 56° E and the bearing from City B to City C is S 34° E. Also, the time taken by the automobile to drive from A to B and from B to C is 1.8 hours and 1.2 hours, respectively.

Let's calculate the distance between City A and City B. Let CB = x, then AB = x cosec 56° = x / sin 56°.

Now, let's calculate the distance between City B and City C.BC = x cosec 34° = x / sin 34°

Thus, the distance between City A and City C is AB + BC

Now, AB = x / sin 56° and BC = x / sin 34°. Therefore, the distance between City A and City C = x / sin 56° + x / sin 34° = 110.53 + 177.34 ≈ 287.87 miles (approximately). Therefore, the required distance between City A and City C is approximately 287.87 miles.

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The solution to the given Bernoulli's differential equation xy - dy/dx = y³ e⁻ˣ^² is  y = -1/(3Ei(-x^2) + C).

To solve the given Bernoulli's differential equation, we can use a substitution. Let's substitute y = u^(1-n) where n is not equal to 0 and 1.

Here's how we solve it step-by-step:

1. Start by differentiating both sides of the equation with respect to x:
  d/dx (xy - dy/dx) = d/dx (y^3 e^(-x^2))

2. Simplify the left side using the product rule:
  y + x(dy/dx) - dy/dx = 3y^2 e^(-x^2) * d/dx (e^(-x^2))

3. Differentiate the right side using the chain rule:
  y + x(dy/dx) - dy/dx = 3y^2 e^(-x^2) * (-2x)

4. Rearrange the equation to isolate dy/dx terms on one side:
  x(dy/dx) - dy/dx = 3y^2 e^(-x^2) * (-2x) - y

5. Multiply both sides of the equation by dx:
  xdy - ydx = -2x * 3y^2 e^(-x^2) dx - ydx

6. Simplify the equation by canceling out the common terms:
  xdy = -6xy^2 e^(-x^2) dx

7. Divide both sides of the equation by x * y^2 to separate variables:
  (1/y^2) dy = -6e^(-x^2) dx/x

8. Integrate both sides of the equation:
  ∫(1/y^2) dy = ∫-6e^(-x^2) dx/x

9. The left side of the equation can be integrated as follows:
  ∫(1/y^2) dy = -1/y

10. The right side of the equation requires a substitution. Let's substitute u = -x^2, then du/dx = -2x:
  ∫-6e^(-x^2) dx/x = -6 ∫e^u du/-2u
                     = 3 ∫e^u du/u

11. The integral on the right side is a special function called the exponential integral, Ei(u). So we have:
  -1/y = 3Ei(u) + C

12. Substitute back u = -x^2:
  -1/y = 3Ei(-x^2) + C

13. Rearrange the equation to solve for y:
  y = -1/(3Ei(-x^2) + C)

That's the solution to the given Bernoulli's differential equation. Remember to consider any initial conditions or constraints to determine the value of the constant C.

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