What is the justification for each step in the solution of the equation?

23x−13=2(x+2)23x−13=2(x+2)
Select from the drop-down menus to correctly justify each step.

a) [tex]\[ \theta = 67.9^\circ, 112.1^\circ \][/tex], b) [tex]\[ \theta = 133.7^\circ, 226.3^\circ \][/tex], c) [tex]\[ \theta = 122.7^\circ, 302.7^\circ \][/tex], d) [tex]\[ \theta = 35.8^\circ, 215.8^\circ \][/tex], e) [tex]\[ \theta = 45.5^\circ, 314.5^\circ \][/tex] and f) [tex]\[ \theta = 154.4^\circ, 334.4^\circ[/tex].

Let's solve each part:

(a) [tex]\(\sin \theta = 0.9263\)[/tex]

1. Find the reference angle using [tex]\(\theta = \sin^{-1}(0.9263)\)[/tex]:

[tex]\[ \theta \approx 67.9^\circ \][/tex]

2. Since [tex]\(\sin \theta\)[/tex] is positive, the angles are in the first and second quadrants:

[tex]\[ \theta_1 \approx 67.9^\circ \][/tex]

[tex]\[ \theta_2 \approx 180^\circ - 67.9^\circ = 112.1^\circ \][/tex]

So, the solutions are:

[tex]\[ \theta = 67.9^\circ, 112.1^\circ \][/tex]

(b) [tex]\(\cos \theta = -0.6909\)[/tex]

1. Find the reference angle using [tex]\(\theta = \cos^{-1}(-0.6909)\)[/tex]:

[tex]\[ \theta \approx 133.7^\circ \][/tex]

2. Since [tex]\(\cos \theta\)[/tex] is negative, the angles are in the second and third quadrants:

[tex]\[ \theta_1 \approx 133.7^\circ \][/tex]

[tex]\[ \theta_2 \approx 360^\circ - 133.7^\circ = 226.3^\circ \][/tex]

So, the solutions are:

[tex]\[ \theta = 133.7^\circ, 226.3^\circ \][/tex]

(c) [tex]\(\tan \theta = -1.5416\)[/tex]

1. Find the reference angle using [tex]\(\theta = \tan^{-1}(-1.5416)\)[/tex]:

[tex]\[ \theta \approx -57.3^\circ \][/tex]

2. Adjust the reference angle to fall within the interval [tex]\([0^\circ, 360^\circ)\)[/tex]:

[tex]\[ \theta_1 = 360^\circ - 57.3^\circ = 302.7^\circ \][/tex]

3. Since [tex]\(\tan \theta\)[/tex] is negative, the angles are in the second and fourth quadrants:

[tex]\[ \theta_2 = 180^\circ + (-57.3^\circ) = 122.7^\circ \][/tex]

So, the solutions are:

[tex]\[ \theta = 122.7^\circ, 302.7^\circ \][/tex]

(d) [tex]\(\cot \theta = 1.3952\)[/tex]

1. Find the reference angle using [tex]\(\theta = \cot^{-1}(1.3952)\)[/tex]:

[tex]\[ \theta \approx 35.8^\circ \][/tex]

2. Since [tex]\(\cot \theta\)[/tex] is positive, the angles are in the first and third quadrants:

[tex]\[ \theta_1 \approx 35.8^\circ \][/tex]

[tex]\[ \theta_2 \approx 180^\circ + 35.8^\circ = 215.8^\circ \][/tex]

So, the solutions are:

[tex]\[ \theta = 35.8^\circ, 215.8^\circ \][/tex]

(e) [tex]\(\sec \theta = 1.4293\)[/tex]

1. Convert to [tex]\(\cos \theta\)[/tex]:

[tex]\[ \cos \theta = \frac{1}{1.4293} \approx 0.6996 \][/tex]

2. Find the reference angle using [tex]\(\theta = \cos^{-1}(0.6996)\)[/tex]:

[tex]\[ \theta \approx 45.5^\circ \][/tex]

3. Since [tex]\(\cos \theta\)[/tex] is positive, the angles are in the first and fourth quadrants:

[tex]\[ \theta_1 \approx 45.5^\circ \][/tex]

[tex]\[ \theta_2 \approx 360^\circ - 45.5^\circ = 314.5^\circ \][/tex]

So, the solutions are:

[tex]\[ \theta = 45.5^\circ, 314.5^\circ \][/tex]

(f) [tex]\(\csc \theta = -2.3174\)[/tex]

1. Convert to [tex]\(\sin \theta\)[/tex]:

[tex]\[ \sin \theta = \frac{1}{-2.3174} \approx -0.4316 \][/tex]

2. Find the reference angle using [tex]\(\theta = \sin^{-1}(-0.4316)\)[/tex]:

[tex]\[ \theta \approx -25.6^\circ \][/tex]

3. Adjust the reference angle to fall within the interval [tex]\([0^\circ, 360^\circ)\)[/tex]:

[tex]\[ \theta_1 = 360^\circ - 25.6^\circ = 334.4^\circ \][/tex]

4. Since [tex]\(\sin \theta\)[/tex] is negative, the angles are in the third and fourth quadrants:

[tex]\[ \theta_2 = 180^\circ + (-25.6^\circ) = 154.4^\circ \][/tex]

So, the solutions are:

[tex]\[ \theta = 154.4^\circ, 334.4^\circ[/tex]

The complete question is:

Approximate, to the nearest 0.1°, all angles θ in the interval [0°, 360°) that satisfy the equation. (Enter your answers as a comma-separated list.)

(a) sin θ = 0.9263

θ = °

(b) cos θ = −0.6909

θ = °

(c) tan θ = −1.5416

θ = °

(d) cot θ = 1.3952

θ = °

(e) sec θ = 1.4293

θ = °

(f) csc θ = −2.3174

θ = °

Lucy will be paid $1260 in interest in the first 6 years.

To calculate the interest Lucy will be paid in the first 6 years, we will use the simple interest formula:

Interest = Principal x Rate x Time

Given:

Plugging in the values, we get:

Interest = $7000 x 0.03 x 6 = $1260

Therefore, Lucy will be paid $1260 in interest over the first 6 years.

https://brainly.com/question/22621039

#SPJ2

To find the number of quarters and nickels, we can set up a system of equations using the total number of coins and the total value of the coins. Solving this system of equations, we find that Kevin and Randy have 27 quarters and 14 nickels.

To solve this problem, we can set up a system of equations. Let's use the variables q (number of quarters) and n (number of nickels). We know that there are a total of 41 coins, so we can write the equation q + n = 41. We also know that the total value of the coins is $7.45, so we can write the equation 0.25q + 0.05n = 7.45.

Now we can solve this system of equations using substitution:

Therefore, Kevin and Randy have 27 quarters and 14 nickels in their jar.

The explicit formula for the given sequence is a_n = 3 + 4(n-1).

The given sequence is (3, 7, 11, 15, 19, 23, 27, ...).

To find the explicit formula for this sequence, we can observe that each term is obtained by adding 4 to the previous term. So, the formula can be written as:

an = 3 + 4(n-1)

where n represents the position of the term in the sequence.

The value V of a square glass varies directly as the square of its length X cm. This means that V is proportional to [tex]X^2[/tex], which can be expressed as:

[tex]\[ V = kX^2 \][/tex]

where k is the constant of proportionality.

Given that the value of a square glass with length 3 cm is $243, we can use this information to find the constant k:

[tex]\[ 243 = k(3)^2 \][/tex]

[tex]\[ 243 = 9k \][/tex]

[tex]\[ k = \frac{243}{9} \][/tex]

[tex]\[ k = 27 \][/tex]

Now that we have the value of k, we can find the value of a glass with length 5 cm by substituting X = 5 into the original equation:

[tex]\[ V = 27(5)^2 \][/tex]

[tex]\[ V = 27 \times 25 \][/tex]

[tex]\[ V = 675 \][/tex]

Therefore, the value of a glass with length 5 cm is $675.

The answer is: 675.

To evaluate the expression 4x-3 for x = -3, we substitute -3 into the expression and follow the order of operations, resulting in a value of -15.

To evaluate the expression 4x-3 when x is -3, we simply substitute -3 in place of x and calculate the result as follows:

Therefore, evaluating 4x-3 when x is -3 gives us -15.

The rate of change of the area of the circle when the radius is 39 centimeters is 1976π cm²/min. This uses the concept of related rates in calculus and the area formula A = πr².

The problem deals with the concept of related rates in calculus. In this problem, we're looking at how the rate of change of the radius of a circle impacts the rate of change of the area of the circle. The formula for the area of a circle is A = πr².

Differentiating both sides with respect to time(t) gives dA/dt = 2πr(dr/dt). In this case, dr/dt (the rate of change of the radius) is given as 8 cm/min. To find dA/dt (the rate of change of the area) when r = 39 cm, we substitute these values into the differentiated equation: dA/dt = 2π(39cm)(8 cm/min) = 1976π cm²/min

So, the rate of change of the area when r = 39 centimeters is 1976π cm²/min. As the radius increases, the area of the circle increases at a rate directly proportional to the radius.

https://brainly.com/question/33509029

#SPJ11

Final answer:

The radius of a circle with a circumference of 201 centimeters is approximately 32 centimeters, found by dividing the circumference by 2 times pi (π), using 3.14 as the value of pi. The correct answer is A) 32 centimeters.

Explanation:

The question asks us to find the radius of a circle when its circumference is given to be 201 centimeters. We use the circumference formula of a circle, which is C = 2πr, where C stands for circumference, π (pi) is a constant approximately equal to 3.14, and r represents the radius of the circle.

To find the radius, we rearrange this formula to solve for r:

Divide both sides of the equation by 2π to isolate r.

Plug in the given circumference value, C = 201 cm, and the approximate value of pi, π = 3.14, into the rearranged formula.

So the calculation would be r = C / (2π) = 201 / (2 * 3.14) = 201 / 6.28. When we compute this, we get r ≈ 32 centimeters to the nearest whole number.

Therefore, the correct answer is A) 32 centimeters.

The correct strategy is (4x275) - 100. So option (3) is correct.

To find 4 multiplied by 275 using the given strategies, let's go through each one:

1. (4x300) + (4x25):

First, calculate (4x300) = 1200. Then, calculate (4x25) = 100. Finally, add these results together: 1200 + 100 = 1300.

2. (4x300) - (4x25):

Calculate (4x300) = 1200. Then, calculate (4x25) = 100. Now, subtract the second result from the first: 1200 - 100 = 1100.

3. (4x275) - 100:

Simply multiply 4 by 275 to get 1100. Then, subtract 100: 1100 - 100 = 1000.

4. (4x200) + 75:

Calculate (4x200) = 800. Then, add 75: 800 + 75 = 875.

The correct strategy is the third one: (4x275) - 100.

Detailed Calculation:

1. [tex]\(4 \times 275 = 1100\)[/tex]

2. [tex]\(1100 - 100 = 1000\)[/tex]

So, using the strategy of (4x275) - 100, the result is 1000.

see picture for the graph

equation: y = 1.50x

Answer:

Step-by-step explanation:

took the test (k12) and its halfway to 50 if you see in the picture

The volume of the cone will be 1526.8 cubic units. The area of the vertical cross-section A at level x = 6 will be 72 square units.

It deals with the size of geometry, region, and density of the different forms both 2D and 3D.

The equation of the base circle is given below.

x² + y² = 81

x² + y² = 9²

Then the radius of the circle is 9 units. Then the height of the cone is the same as the diameter of the circle.

h = 2r

h = 2 x 9 = 18

The volume of the cone will be

V = (1/3)πr²h

V = (1/3) × π × 9² × 18

V = 1526.8 cubic units

The area of the vertical cross-section A at the level x = 6 will be

A = (1/2) x (18 - 6) x (18 - 6)

A = (1/2) x 12²

A = (1/2) x 144

A = 72 square units

More about the geometry link is given below.

https://brainly.com/question/7558603

#SPJ2

On average during their trip, their speed was 3184.8 miles/hr.

A unitary method is a mathematical way of obtaining the value of a single unit and then deriving any no. of given units by multiplying it with the single unit.

We know Distance = Speed×Time.

∴ Speed = Distance/Time.

Given, The distance from Earth to the moon is exactly 238,855 miles and it took astronauts 75 hours.

So, the average speed is (238855/75) miles/hr.

= 3184.8 miles/hr.

learn more about unitary method here :

https://brainly.com/question/22056199

#SPJ2

Answer:

Your answer is B=4

Step-by-step explanation:

hope it helps once again

x = square feet

total cost = 500 + 600x