A clothing store purchases a sweatshirt for $26 and adds $15 to set the sticker price. The store is having a sale where everything is for 20% off. About How much is the final price of the sweatshirt?

The area rolled during one complete revolution of the lawn roller is approximately  38 square feet (nearest whole number).

To find the area rolled, we need to calculate the lateral surface area of the cylindrical roller. The formula for the lateral surface area of a cylinder is given by A = 2πrh, where π is the mathematical constant pi (approximately 3.14159), r is the radius, and h is the height (length) of the cylinder.

Given that the radius of the roller is 18 inches, we need to convert it to feet by dividing it by 12 since there are 12 inches in a foot. So the radius (r) becomes 18/12 = 1.5 feet.

The height (length) of the roller is given as 4 feet. Therefore, h = 4 feet.

Plugging the values into the formula, we have A = 2π(1.5)(4) = 12π square feet.

Now, to find the area rolled during one complete revolution, we multiply the lateral surface area by the number of revolutions, which is 1. So the total area rolled is 12π square feet.

Using the calculator value of π, which is approximately 3.14159, we can approximate the area rolled as 12(3.14159) = 37.69908 square feet.

Rounding to the nearest whole number, the area rolled during one complete revolution of the lawn roller is approximately 38 square feet.

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There are exists a point ϕ ∈ (a, b) such that f(ϕ) = (1/n) * [f(p1) + f(p2) + ... + f(pn)].

To prove that there exists a point ϕ ∈ (a, b) such that f(ϕ) = (1/n) * [f(p1) + f(p2) + ... + f(pn)], we can utilize the Mean Value Theorem for Integrals.

Let F(x) be the antiderivative of f(x) on the interval [a, b]. By the Mean Value Theorem for Integrals, there exists a point c ∈ (a, b) such that the average value of F(x) on [a, b] is equal to F(c):

1/(b - a) * ∫[a to b] F(x) dx = F(c)

Since F(x) is the antiderivative of f(x), we can rewrite the equation as:

1/(b - a) * ∫[a to b] f(x) dx = F(c)

Taking the definite integral of f(x) from a to b, we have:

1/(b - a) * ∫[a to b] f(x) dx = F(b) - F(a)

Since f(x) is continuous on [a, b], it is also continuous on the closed interval [a, b]. Therefore, by the Extreme Value Theorem, f(x) attains its maximum and minimum values on [a, b]. Let M be the maximum value of f(x) and m be the minimum value of f(x) on [a, b].

Since f(x) is continuous, it satisfies the Intermediate Value Property. Therefore, for any y ∈ [m, M], there exists a point d ∈ [a, b] such that f(d) = y.

Now, consider the points p1, p2, ..., pn ∈ (a, b). Let A = f(p1) + f(p2) + ... + f(pn). Since f(x) satisfies the Intermediate Value Property, there exists a point ϕ ∈ (a, b) such that f(ϕ) = A/n.

Hence, we have proven that there exists a point ϕ ∈ (a, b) such that f(ϕ) = (1/n) * [f(p1) + f(p2) + ... + f(pn)].

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The statement "Choosing a smoothie containing apple and choosing a smoothie containing blueberry are not mutually exclusive events." can be inferred from the calculation.

Firstly, we identify the total number of each type of smoothies available. We have 12 apple smoothies, 7 blueberry, 8 other, and 3 smoothies that are common to both apple and blueberry. This brings our total smoothies to 30.

a) To find the probability that a smoothie contains either apple or blueberry, we need to consider the apple smoothies and smoothies that are common to both apple and blueberry then blueberry smoothies and smoothies that are common to both apple and blueberry. So, we add up the numbers of these smoothies and divide by the total number of smoothies.

P(contains apple) = (12 apple + 3 common) / 30 total = 15 / 30 which equals 0.5

P(contains blueberry) = (7 blueberries + 3 common) / 30 total = 10 / 30 which equals 0.33

Then, we find P(contains apple) + P(contains blueberry) = 0.5 + 0.33 which equals 0.83.

b) To find the probability that a smoothie contains apple or blueberry, we add up the number of apple smoothies, blueberry smoothies and smoothies common to both, then divide by the total number of smoothies.

P(contains apple or blueberry) = (12 apple + 7 blueberries + 3 common) / 30 total = 22/30 which equals 0.73.

c) The events of choosing a smoothie containing apple and choosing a smoothie containing blueberry are mutually exclusive if P(contains apple) + P(contains blueberry) is equal to P(contains apple or blueberry). As 0.83 is not equal to 0.73, these are not mutually exclusive events.

Therefore, the statement "Choosing a smoothie containing apple and choosing a smoothie containing blueberry are not mutually exclusive events." can be inferred from the calculation.

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Answer:

7.1 in

Step-by-step explanation:

We know that this is an isosceles right triangle because the right triangle's legs are congruent.

The ratio of side lengths in an isosceles right triangle is:

1 : 1 : √2

Therefore, the length of the hypotenuse (the missing side) in the diagrammed triangle is:

5√2 in

This can be approximated as 7.1 in.

The simplified expression is 2x⁵.

To simplify the expression [tex]\left(4x^{9/2}\right)\left(\frac12x^{1/2}\right)[/tex], we can multiply the coefficients and combine the variables with the same base.

Multiplying the coefficients: [tex]4 \times \frac12 = 2[/tex]

Multiplying the variables with the same base:

[tex]$x^{9/2} \times x^{1/2} = x^{\left(\frac92 + \frac12\right)} = x^{10/2} = x^5$[/tex]

Putting it all together, the simplified expression is [tex]2x^5[/tex]

Hence the simplified expression is 2x⁵.

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To rewrite the expression 18a^3b + 9ab^2 using a common factor, we can factor out the common factor from both terms. In this case, the common factor is 9ab.

Taking out the common factor, we have:

18a^3b + 9ab^2 = 9ab(2a^2 + b)

So, the expression 18a^3b + 9ab^2 can be simplified as 9ab(2a^2 + b) by factoring out the common factor 9ab.

This process is known as factoring out the greatest common factor (GCF). By factoring out the GCF, we simplify the expression and make it more manageable and easier to work with.

Factoring out the GCF is a useful technique in algebra to simplify expressions and solve equations. It helps in identifying common factors and allows us to rearrange terms more easily.

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Answer:98

Step-by-step explanation:3/7=42/y

                                            3y=294

                                            3y/3=294/3

                                             y=98

                                             42/98 simplified equals 3/7

The tank height will be 2.3125 feet (2 feet for water height + 0.3125 feet for displacement height + 1 foot for splashing prevention).

To determine the height of the water in the tank, we first need to calculate the volume of the tank. A 4' by 4' square base gives us an area of 16 square feet.

Multiplying this by the height of the water will give us the volume of water in the tank. Let's assume we want the water to be 2 feet deep, so the volume of water will be 32 cubic feet.

Next, we need to calculate the displacement height. When the teacher falls in, they will displace a certain amount of water. Since the teacher is entirely submerged, their volume will be equal to the volume of water displaced.

Assuming the teacher has a volume of 5 cubic feet, this is the amount of water that will be displaced, causing the water level to rise by 5/16 or 0.3125 feet.

To prevent splashing, we need to add an additional foot to the height of the tank beyond the displacement height. This calculation should be labeled "Tank Height" and included on the design sheet.

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The surface area of the given pyramids and cone would be listed below as follows:

1.)576.4in²

2.)71.4yd²

To calculate the surface area the following steps should be taken.

For question 1.)

The formula for surface area of square based pyramid;

= a²+2al

where;

a² = base area = 11² = 121in

l = 20.7in

a = 11

SA = 121+2(11×20.7)

= 121+455.4

= 576.4in²

For question 2.)

The formula for surface area of triangular pyramid ;

SA= B+1/2Ps

B = base area = 15.6yd

P = 18yd

Slant height = 6.2 yd

SA = 15.6+1/2×18×6.2

= 71.4yd²

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0.006336 g of this sample is left after 6 half-lives.

Amount remaining = initial amount x (1/2)^number of half-lives

In this case, the initial amount is 0.4055 g and the number of half-lives is 6. So:

Amount remaining = 0.4055 g x (1/2)⁶ = 0.006336 g

if a sample of thulium-171 has a mass of 0.4055 g and undergoes radioactive decay, after 6 half-lives only 0.006336 g of the original sample will remain. This calculation is based on the formula for calculating the amount of a radioactive substance remaining after a certain number of half-lives, which takes into account the decay rate of the substance.

Hence,0.006336 g of this sample is left after 6 half-lives.

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To determine if a(n) (not an event) of flipping a coin is heads is valid, we need to ensure that all probabilities are between 0% and 100% and the sum of all probabilities is 100%.

In probability theory, it is important to ensure that the probabilities associated with an event or outcome are valid. When flipping a coin, if we define the event "a" as getting heads, we need to check that the probability of heads is between 0% and 100% and that the sum of the probabilities of all possible outcomes (heads and tails) is 100%. This ensures that the probabilities are within a valid range and account for all possibilities.

The ratio of the ways to succeed (the favorable outcomes) to all possible ways that something can occur (the total outcomes) is known as the probability. It represents the likelihood of a specific outcome occurring relative to all possible outcomes. By calculating this ratio, we can quantify the probability of an event happening.

When dealing with a multi-step process, the multiplication principle is used to determine the total number of ways the process can be completed. It states that if there are "n" independent steps, and each step has "m" possible outcomes, then the total number of ways the process can be completed is the product of the number of outcomes at each step. This principle is based on the concept that each step's outcomes are independent of the others, allowing us to multiply the possibilities together to determine the overall number of ways the process can unfold.

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The volumes of the solids are 1) 4986 m³, 2) 134 km³ and 3) 4179 in³.

Given are the solids in shapes of spheres, cylinders and cone we need to find the volumes,

So,

Volume of a Sphere:

V = (4/3) × π × r³

Where V is the volume and r is the radius of the sphere.

Volume of a Cylinder:

V = π × r² × h

Where V is the volume, r is the radius of the base, and h is the height of the cylinder.

Volume of a Cone:

V = (1/3) × π × r² × h

Where V is the volume, r is the radius of the base, and h is the height of the cone.

1) Sphere with diameter 21.2 m,

Volume = V = (4/3) × π × (21.2/2)³ = 4986 m³

2) Cone with base diameter and height of 8 km,

Volume = (1/3) × π × (8/2)² × 8 = 134 km³

3) Cylinder with base radius and height of 11 in,

Volume = π × 11² × 11 = 4179 in³

Using the similar process you can find the rest volumes.

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Answer:

(2) - [tex]\vec v= < 15.5885, -9 >[/tex]

(3) - [tex]\vec u= < -59.9371, 148.349 >[/tex]

Step-by-step explanation:

Problem #2:

Given the vector in magnitude-angle form, find it in component form.

Call the vector, vector "v."

[tex]||\vec v||= 18 \ at \ -30 \textdegree\\\\\rightarrow \boxed{\vec v= < ||\vec v||\cos\theta,||\vec v||\sin\theta > }\\\\\Longrightarrow \vec v= < (18)\cos( -30 \textdegree),(18)\sin( -30 \textdegree) > \\\\\therefore \boxed{\boxed{ \vec v= < 15.5885, -9 > }}[/tex]

Problem #3:

Given the vector in magnitude-angle form, find it in component form.

Call the vector, vector "u."

[tex]||\vec u||= 160 \ at \ 112 \textdegree\\\\\rightarrow \boxed{\vec u= < ||\vec u||\cos\theta,||\vec u||\sin\theta > }\\\\\Longrightarrow \vec u= < (160)\cos( 112 \textdegree),(160)\sin( 112 \textdegree) > \\\\\therefore \boxed{\boxed{ \vec u= < -59.9371, 148.349 > }}[/tex]

Answer:

The volume is about 5275.2 m^3

Step-by-step explanation:

The formula for volume of a cone is given by:

V = 1/3πr^2h, where

Step 1:  We're not given the radius, but we see that the slant height and the regular height (altitude) are parts of a right triangle inside the cone, where

Since we're working with a right triangle, we can find the other leg (our radius) using the Pythagorean theorem, which is:

a^2 + b^2 = c^2, where

Thus, we can plug in 35 for a and 37 for c, allowing us to solve for b, the measure of our radius:

1.1 Plug in 35 for a and 37 for c.  Then simplify:

35^2 + b^2 = 37^2

1225 + b^2 = 1369

1.2 Subtract 1225 from both sides:

(1225 + b^2 = 1369) - 1225

b^2 = 144

1.3 Take the square root of both sides to isolate and solve for b, the measure of the radius:

√b^2 = ± √144

b = ± 12

Although taking the square root of a number gives us both a positive and negative answer, you can't have a negative measure, so b = 12 and thus the radius, r, = 12 m

Step 2:

Plug in 3.14 for π, 12 for r, and 35 for h in the volume formula.  Then simplify and round to find the volume of the cone:

V = 1/3(3.14)(12)^2(35)

V = 157/150 * 144 * 35

V = 150.72 * 35

V = 5275.2 m^3

Thus, the volume of the cone is 5275.2 m^3

The coordinates of T" include the following: D. (8, 10).

In Mathematics and Geometry, the translation of a geometric figure upward means adding a digit to the value on the positive y-coordinate (y-axis) of the pre-image.

(x, y)                                                    →                  (x - 1, y + 3)

Coordinate T (5, 2)                             →                  T' (5 - 1, 2 + 3) = T' (4, 5).

Next, we would dilate the coordinates of the vertices by applying a scale factor of 2 that is centered at the origin as follows:

Coordinate T' (4, 5) → (4 × 2, 5 × 2) = Coordinate T" (8, 10).

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The area of the shaded region is 336 ft².

We have,

From the diagram,

The area of the shaded region.

= Area of the swimming pool along with the walkway - Area of the swimming pool _______ (A)

Now,

Area of the swimming pool along with the walkway.

= 22 x 40

= 880 ft² _____(1)

Area of the swimming pool.

= 16 x 34

= 544 ft² ______(2)

Now,

Substitute (1) and (2) in (A)

The area of the shaded region.

= 880 - 544

= 336 ft²

Thus,

The area of the shaded region is 336 ft².

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The actual error ≈ 0.00002082 and Error Bound E ≈ 0.00002083. Comparing the actual error to the error bound E, we can see that they are very close in magnitude.

To determine a formula of the form √(1+x) = 1 + 0.5x ± E using the degree 1 Taylor polynomial of f(x) = √(1+x) with remainder, we start by finding the degree 1 Taylor polynomial:

P1(x) = f(a) + f'(a)(x - a)

where a = 0 (the point of expansion). Let's calculate the derivatives:

f(x) = √(1+x)

f'(x) = 1/(2√(1+x))

Substituting a = 0 and f(a) = f(0) = √1 = 1, we have:

P1(x) = 1 + f'(0)(x - 0)

     = 1 + (1/2)(x)

     = 1 + 0.5x

The remainder term R1(x) is given by:

R1(x) = (x - a)²/2! * f''(c)

To find the error bound E, we need to evaluate the second derivative f''(c) for some value c between 0 and x. Taking the second derivative of f(x) = √(1+x), we get:

f''(x) = -1/(4(1+x)^(3/2))

Substituting x = 0.02 (since we're approximating √1.02), we have:

f''(c) = -1/(4(1+c)^(3/2))

To find the error E, we evaluate the remainder term using the maximum value of f''(c) in the interval [0, 0.02]. To approximate this, we use a calculator:

E = |R1(0.02)| = |0.02 - 0|²/2! * |-1/(4(1+c)^(3/2))|

Calculating this expression, we find E ≈ 0.00002083.

Using a calculator, we can evaluate the actual error by subtracting the approximation 1 + 0.5(0.02) from the actual value of √1.02:

Actual error = √1.02 - (1 + 0.5(0.02))

Calculating this, we find the actual error ≈ 0.00002082.

Comparing the actual error to the error bound E, we can see that they are very close in magnitude.

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The value of x in the equation is 19.

We have,

To find the value of x, we need to set the expressions bd and ac equal to each other and solve for x.

Given:

bd = 8x - 7

ac = 6x + 31

Setting bd = ac:

8x - 7 = 6x + 31

Now, solve this equation for x:

8x - 6x = 31 + 7

2x = 38

x = 38/2

x = 19

Therefore,

The value of x in the equation is 19.

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The decision would depend on the specific cost of the cousin's ticket and the cost of the closer seats relative to the available budget.

Part A: In order to determine if Jonah's claim is correct, we need to know the cost of the closer seats. If the cost of the closer seats is less than or equal to Jonah's savings of $103, then it would be possible for Jonah's dad to move the family to those seats by using Jonah's savings. However, if the cost of the closer seats exceeds $103, then Jonah's claim would not be correct, as his savings alone would not be sufficient to cover the cost. Without information about the cost of the closer seats, we cannot definitively determine if Jonah is correct.

Part B: If Jonah's cousin wants to come to the game and they have a total of $350, the family's seating options would be influenced by this additional expense. The cost of the cousin's ticket would need to be deducted from the $350 budget. If there is enough remaining after purchasing the cousin's ticket, Jonah's dad could consider using the combined savings of $103 from Jonah and the remaining budget to move the family to closer seats, provided the cost of those seats is within the remaining budget. However, if the cost of the closer seats, including the cousin's ticket, exceeds the remaining budget, then it would not be possible to move the family to closer seats. The decision would depend on the specific cost of the cousin's ticket and the cost of the closer seats relative to the available budget.

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Answer:

(a) Please refer to explanation (in part 1)

(b) [tex]V=\frac{2}{3}x(22-x^{2})[/tex]

(c) [tex]\frac{176}{9}\sqrt{\frac{22}{3}} \text{cm}^{3}[/tex]

Step-by-step explanation:

The explanation is attached below.

The system of equations are solved and the numbers are 15 and 7

Given data ,

The difference of the two numbers is 8, which can be expressed as:

x - y = 8

It is also given that twice the first number (2x) added to three times the second number (3y) equals 51:

2x + 3y = 51

We now have a system of two equations with two variables. We can solve this system using various methods, such as substitution or elimination.

Let's solve the system using the substitution method:

From equation (1), we can express x in terms of y:

x = y + 8

Substituting this expression for x into equation (2), we get:

2(y + 8) + 3y = 51

2y + 16 + 3y = 51

5y + 16 = 51

5y = 51 - 16

5y = 35

y = 35/5

y = 7

Substituting the value of y back into equation (1):

x - 7 = 8

x = 8 + 7

x = 15

Hence , the two numbers are x = 15 and y = 7

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At the hot wings restaurant, a fraction of 5/9 of the patrons ordered the inferno hot wings, and 1/8 of those patrons passed out from the intensity of the sauce. The fraction of patrons who passed out are 5/72.

Given that 5/9 of the patrons ordered the inferno hot wings, this fraction represents the portion of patrons who were exposed to the intense sauce. Out of this group, 1/8 passed out due to the sauce's intensity.

To find the fraction of patrons who passed out, we multiply the fractions 5/9 and 1/8:

(5/9) * (1/8) = 5/72.

Therefore, the fraction 5/72 represents the proportion of patrons who passed out from the intensity of the inferno hot wing sauce.

This information is important for understanding the effects of the spicy sauce and can be used by the restaurant to gauge the intensity of the dish and potentially make adjustments to cater to different preferences. Additionally, it provides insights into the customer experience and can influence future menu decisions or considerations regarding the heat levels of their offerings.

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The correct answer is d) p = 20 and q = 18. For these values of p and q

[tex]x^{36} + px^q + 100[/tex] is a perfect square for all integer values of x

To explain why p = 20 and q = 18 are the correct values, let's analyze the expression [tex]x^{36} + px^q + 100[/tex]. For this expression to be a perfect square for all integer values of x, it must be in the form (ax^18 + b)^2, where a and b are integers.

Expanding (ax^18 + b)^2 gives us [tex]ax^{36} + 2abx^{18} + b^2[/tex]. Comparing this with the given expression [tex]x^{36} + px^q + 100[/tex], we can deduce the following:

1. The constant term in both expressions must be the same, which gives us b^2 = 100. The only possible integer value for b is 10, as it is the only square root of 100.

2. The coefficient of x^36 in both expressions must also be the same, which gives us a^2 = 1. The only possible integer value for a is 1.

3. The coefficient of x^18 in the expanded form is 2ab, which should be equal to px^q in the given expression. Therefore, we have 2ab = px^q. Since a = 1, this simplifies to 2b = px^q.

We know that b = 10, so we can substitute it into the equation: 2 * 10 = px^q. Simplifying further, we get 20 = px^q.

Now, we need to find a value for p and q that satisfies the equation for all integer values of x. If we set q = 18, then x^q = x^18, and the equation becomes 20 = px^18. This is satisfied for any value of x.

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The equality of partial and Pearson correlations provides some insights into the relationship between X and Y, however, it is not sufficient to determine statistical significance, causality, range of scores.

The fact that the partial correlation is equal to the Pearson correlation does not automatically imply statistical significance. Statistical significance is determined by conducting hypothesis tests or calculating p-values, which require additional information such as sample size and significance level.

The statement (b) does not provide evidence of a causal relationship between X and Y. Correlation alone does not establish causality, as there may be other confounding factors or alternative explanations for the observed relationship.

The range of scores on X and Y cannot be inferred solely from the equality of partial and Pearson correlations. The range of scores depends on the actual data and variability within X and Y, which is not addressed in the statement.

The statement (d) suggests that the variable that was partialed out does not fully account for the correlation between X and Y.

However, it does not specify the nature of the variable or the method used for partial correlation. Further analysis and context are needed to draw conclusions about the role of the partialed-out variable.

In summary, while the equality of partial and Pearson correlations provides some insights into the relationship between X and Y, it is not sufficient to determine statistical significance, causality, range of scores, or the full explanation for the correlation observed. Additional analysis and considerations are necessary to make conclusions in these areas.

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The measure of the amount of random sampling error in a survey's result is known as margin of error.

The margin of error is a statistical concept that quantifies the degree of uncertainty or sampling error associated with survey results. It provides an estimate of the range within which the true population parameter is likely to fall. The margin of error is typically expressed as a percentage and is based on the sample size and the level of confidence desired.

In survey research, random sampling error refers to the natural variability that occurs when a subset of individuals, known as the sample, is selected to represent a larger population. It arises because the sample is not an exact replica of the entire population. The margin of error takes into account this inherent variability and provides a measure of how much the survey results might deviate from the true population values.

A larger sample size generally leads to a smaller margin of error, as it reduces the random variability associated with sampling. Similarly, a higher level of confidence, such as 95% confidence level, results in a larger margin of error to account for a wider range of potential values.

By considering the margin of error, survey researchers can assess the reliability and precision of their findings, providing a range of values within which the true population parameter is likely to reside.

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The general solution to the given differential equation is

y = ± √(1 / (2ln|t| + 4/t - C2))

To solve the given differential equation, we can use the method of separable variables. Let's go through the steps:

Rearrange the equation to separate the variables:

t^2y' + 2ty - y^3 = 0

Divide both sides of the equation by t^2:

y' + (2y/t) - (y^3/t^2) = 0

Now, we can rewrite the equation as:

y' + (2y/t) = (y^3/t^2)

Separate the variables by moving the y-related terms to one side and the t-related terms to the other side:

(1/y^3)dy = (1/t - 2/t^2)dt

Integrate both sides of the equation:

∫(1/y^3)dy = ∫(1/t - 2/t^2)dt

To integrate the left side, let's use a substitution. Let u = y^(-2), then du = -2y^(-3)dy.

-1/2 ∫du = ∫(1/t - 2/t^2)dt

-1/2 u = ln|t| + 2/t + C1

-1/2 (y^(-2)) = ln|t| + 2/t + C1

Multiply through by -2:

y^(-2) = -2ln|t| - 4/t + C2

Now, take the reciprocal of both sides to solve for y:

y^2 = (-1) / (-2ln|t| - 4/t + C2)

y^2 = 1 / (2ln|t| + 4/t - C2)

Finally, taking the square root:

y = ± √(1 / (2ln|t| + 4/t - C2))

Therefore, the general solution to the given differential equation is:

y = ± √(1 / (2ln|t| + 4/t - C2))

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Answer:

B

Step-by-step explanation:

x² - 9x + 20 = 0 ← in standard form

consider the factors of the constant term (+ 20) which sum to give the coefficient of the x- term (- 9)

the factors are - 4 and - 5 , since

- 4 × - 5 = + 20 and - 4 - 5 = - 9 , then

(x - 4)(x - 5) = 0 ← in factored form

equate each factor to zero and solve for x

x - 4 = 0 ( add 4 to both sides )

x = 4

x - 5 = 0 ( add 5 to both sides )

x = 5

solutions are x = 4 , x = 5

To prove the given trigonometric identity:

cot A · cos (30° - A) - sin (30° - A) = √3/2 · cosec A

We'll start by simplifying each side of the equation separately using trigonometric identities:

Left-hand side (LHS):

cot A · cos (30° - A) - sin (30° - A)

Using the identity cot A = cos A / sin A, we can rewrite cot A as cos A / sin A:

(cos A / sin A) · cos (30° - A) - sin (30° - A)

Expanding the cos (30° - A) using the cosine difference formula cos (x - y) = cos x · cos y + sin x · sin y:

(cos A / sin A) · (cos 30° · cos A + sin 30° · sin A) - sin (30° - A)

cos 30° = √3/2 and sin 30° = 1/2:

(cos A / sin A) · (√3/2 · cos A + 1/2 · sin A) - sin (30° - A)

Multiply through by (2 / 2) to simplify:

(2cos A / 2sin A) · (√3/2 · cos A + 1/2 · sin A) - sin (30° - A)

Cancel out the 2's:

(cos A / sin A) · (√3/2 · cos A + 1/2 · sin A) - sin (30° - A)

Apply the quotient identity for sine and cosine: cos x / sin x = cot x:

cot A · (√3/2 · cos A + 1/2 · sin A) - sin (30° - A)

Right-hand side (RHS):

√3/2 · cosec A

Since cosec A = 1 / sin A, we can rewrite the RHS:

√3/2 · (1 / sin A)

Multiply the √3/2 into the parentheses:

(√3/2) · (1 / sin A)

Multiply √3/2 by 1:

(√3/2) / (sin A)

Now, we need to simplify the expression further to match the LHS:

To combine the terms on the LHS, we'll multiply (√3/2) by (cos A / cos A) to get a common denominator with sin A:

(√3/2) · (cos A / cos A) / (sin A)

Simplifying the numerator:

(√3cos A) / (2cos A) / (sin A)

Cancel out the common factor of cos A in the numerator and denominator:

(√3) / 2 / (sin A)

Since (√3) / 2 = sin 60°:

sin 60° / (sin A)

Using the identity sin 60° = √3/2, we have:

(√3/2) / (sin A)

which matches the RHS.

Therefore, the left-hand side (LHS) is equal to the right-hand side (RHS), proving the given trigonometric identity:

cot A · cos (30° - A) - sin (30° - A) = √3/2 · cosec A

To prove the given trigonometric identity:

cot A · cos (30° - A) - sin (30° - A) = √3/2 · cosec A

We'll start by simplifying each side of the equation separately using trigonometric identities:

Left-hand side (LHS):

cot A · cos (30° - A) - sin (30° - A)

Using the identity cot A = cos A / sin A, we can rewrite cot A as cos A / sin A:

(cos A / sin A) · cos (30° - A) - sin (30° - A)

Expanding the cos (30° - A) using the cosine difference formula cos (x - y) = cos x · cos y + sin x · sin y:

(cos A / sin A) · (cos 30° · cos A + sin 30° · sin A) - sin (30° - A)

cos 30° = √3/2 and sin 30° = 1/2:

(cos A / sin A) · (√3/2 · cos A + 1/2 · sin A) - sin (30° - A)

Multiply through by (2 / 2) to simplify:

(2cos A / 2sin A) · (√3/2 · cos A + 1/2 · sin A) - sin (30° - A)

Cancel out the 2's:

(cos A / sin A) · (√3/2 · cos A + 1/2 · sin A) - sin (30° - A)

Apply the quotient identity for sine and cosine: cos x / sin x = cot x:

cot A · (√3/2 · cos A + 1/2 · sin A) - sin (30° - A)

Right-hand side (RHS):

√3/2 · cosec A

Since cosec A = 1 / sin A, we can rewrite the RHS:

√3/2 · (1 / sin A)

Multiply the √3/2 into the parentheses:

(√3/2) · (1 / sin A)

Multiply √3/2 by 1:

(√3/2) / (sin A)

Now, we need to simplify the expression further to match the LHS:

To combine the terms on the LHS, we'll multiply (√3/2) by (cos A / cos A) to get a common denominator with sin A:

(√3/2) · (cos A / cos A) / (sin A)

Simplifying the numerator:

(√3cos A) / (2cos A) / (sin A)

Cancel out the common factor of cos A in the numerator and denominator:

(√3) / 2 / (sin A)

Since (√3) / 2 = sin 60°:

sin 60° / (sin A)

Using the identity sin 60° = √3/2, we have:

(√3/2) / (sin A)

which matches the RHS.

Therefore, the left-hand side (LHS) is equal to the right-hand side (RHS), proving the given trigonometric identity:

cot A · cos (30° - A) - sin (30° - A) = √3/2 · cosec A

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Answer:

------------------------

As per diagram, point E has coordinates (5, 2).

Reflection across the y-axis results in the x-coordinate flip the sign, while the y-coordinate remains unchanged.

Hence the point E' is (- 5, 2).