Suppose p is inversely proportional to the cube of q. if p=14 when q=9, what is p if q is 4

If the sum of the measures of two angles is 180 degrees, then the angles are called supplementary angles.

Supplementary angles are a pair of angles that, when added together, result in a sum of 180 degrees. This means that if you have two angles, and their measures add up to 180 degrees, then those angles are considered supplementary to each other. For example, let's say we have Angle A and Angle B. If the measure of Angle A is 60 degrees, and the measure of Angle B is 120 degrees, we can check if they are supplementary by adding their measures: 60 + 120 = 180 degrees.

Since the sum is 180 degrees, we can conclude that Angle A and Angle B are supplementary angles. Supplementary angles can be found in various scenarios. For instance, consider a straight line. A straight line forms an angle of 180 degrees. So, if we divide this line into two angles, each angle will be 90 degrees. Since 90 + 90 equals 180, these angles are supplementary.In such cases, we can refer to the angles as non-supplementary. In summary, if the sum of the measures of two angles is 180 degrees, those angles are called supplementary angles. They are commonly found in situations where a straight line is divided into two angles, each measuring 90 degrees.

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Maka should plan to spend $13.10 + $7.10 = $20.20.

Based on the given menu, the price of a combo meal is the same as purchasing each item separately.

Maka wants 2 burritos and 1 enchilada, so let's calculate the cost.

From combo meal 1, the price of one burrito is $6.55.
From combo meal 2, the price of one enchilada is $7.10.

Since Maka wants 2 burritos, she will spend $6.55 x 2 = $13.10 on burritos.
She also wants 1 enchilada, so she will spend $7.10 on the enchilada.

Adding the two amounts together, Maka should plan to spend $13.10 + $7.10 = $20.20.

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from the foci, it is clear that the center is at (6,1) and

c = 2

Since the major axis has length 10, a=5

b^2 = 25-4 = 21

so, the equation is

(x-6)^2/25 + (y-1)^2/21 = 1

The given function can be written in vertex form as y = (x + 1)² + 4. The vertex of the parabola is (-1, 4).

The vertex form of a quadratic function is y=a(x−h)2+k. To write the given function in vertex form, complete the square and transform it accordingly. Solution:

Given function is y = x² + 2x + 5

To write in vertex form, complete the square and transform it accordingly.Square half of coefficient of x and add and subtract it in the function. Let's do that now.We have to add (-1)² in order to complete the square. The given function becomes:(x² + 2x + 1) + 5 - 1⇒ (x + 1)² + 4This is the vertex form of a quadratic function, where the vertex is (-1, 4).

Explanation:We know that vertex form of a quadratic function is given byy = a(x - h)² + k where (h, k) is the vertex of the parabola.In the given function, y = x² + 2x + 5. The coefficient of x² is 1. Hence we can write the function asy = 1(x² + 2x) + 5.

Now, let's complete the square in x² + 2x.The square of half of the coefficient of x is (2/2)² = 1.So, we can add and subtract 1 inside the parenthesis of x² + 2x as follows.y = 1(x² + 2x + 1 - 1) + 5y = 1[(x + 1)² - 1] + 5y = (x + 1)² - 1 + 5y = (x + 1)² + 4

Therefore, the vertex form of the given function is y = (x + 1)² + 4. The vertex of the parabola is (-1, 4).

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Hello!

1/4 + 2/3

= 1*3/4*3 + 2*4/3*4

= 3/12 + 8/12

= 11/12

the theoretical probability of getting a sum of 7 when rolling two standard number cubes is 6/36, which can be simplified to 1/6 or approximately 0.167.

The theoretical probability of getting a sum of 7 when rolling two standard number cubes can be calculated by determining the number of favorable outcomes and dividing it by the total number of possible outcomes.

To calculate the number of favorable outcomes, we need to find the combinations of numbers on the two cubes that sum up to 7. These combinations are: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). So, there are 6 favorable outcomes.

To calculate the total number of possible outcomes, we need to consider that each cube has 6 sides, and therefore, 6 possible outcomes for each cube. Since we are rolling two cubes, we multiply the number of outcomes for each cube, resulting in a total of 6 x 6 = 36 possible outcomes.

To find the theoretical probability, we divide the number of favorable outcomes (6) by the total number of possible outcomes (36).

Therefore, the theoretical probability of getting a sum of 7 when rolling two standard number cubes is 6/36, which can be simplified to 1/6 or approximately 0.167.

Regarding the second part of your question, there are 36 total outcomes when rolling two standard number cubes because each cube has 6 sides and there are 6 possible outcomes for each cube.

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The slopes of opposite sides of a quadrilateral can be observed to be equal if the quadrilateral is a parallelogram. There are several types of quadrilaterals, such as squares, rectangles, rhombuses, and trapezoids etc.

The slopes of opposite sides of a quadrilateral can be observed to be equal if the quadrilateral is a parallelogram. This is a property of parallelograms, where opposite sides are parallel and have the same slope.

However, if the slopes of opposite sides are different, then the quadrilateral is not a parallelogram.
As for the type of quadrilateral A B D C, I would need more information or a diagram to accurately determine its classification.

There are several types of quadrilaterals, such as squares, rectangles, rhombuses, and trapezoids, each with their own unique properties. Without additional information, it is not possible to determine the specific type of quadrilateral A B D C.

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To calculate the value of the error in the expression z = x/y, where x = 9.4 ± 0.1 and y = 3.7 ± 0, we can use the formula for propagating uncertainties.

The formula for the fractional uncertainty in a quotient is given by:

δz/z =[tex]\sqrt((\sigma x/x)^2 + (\sigma y/y)^2),[/tex]

where δz is the uncertainty in z, δx is the uncertainty in x, δy is the uncertainty in y, and z is the calculated value of the expression.

Substituting the given values:

x = 9.4 ± 0.1

y = 3.7 ± 0

We can calculate the fractional uncertainty as:

δz/z = [tex]\sqrt((0.1/9.4)^2 + (0/3.7)^2)[/tex]

     = sqrt(0.00001117 + 0)

     ≈ sqrt(0.00001117)

     ≈ 0.0033

To obtain the value of the error with one decimal place, we round the fractional uncertainty to one significant figure:

δz/z ≈ 0.003

Therefore, the value of the error with one decimal place for z = x/y is 0.003.

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To simplify the expression 3(x + 5) - 2x, let's break it down step by step:

First, apply the distributive property by multiplying 3 with each term inside the parentheses:

3(x + 5) - 2x = 3x + 15 - 2x

Next, combine like terms by grouping the x terms together:

3x - 2x + 15 = (3x - 2x) + 15

Simplifying the x terms, we get:

(3x - 2x) + 15 = x + 15

Therefore, the simplified expression is x + 15.

This means that the original expression, 3(x + 5) - 2x, is equivalent to x + 15.

To further explain, the expression 3(x + 5) - 2x represents three times the quantity of x plus 5, subtracted by two times x. By distributing the 3, we get 3x + 15, and then combining the x terms yields x + 15.

So, the expression x + 15 is equivalent to 3(x + 5) - 2x. It represents the same mathematical relationship and simplifies the original expression by grouping like terms.

It's important to note that this simplification assumes x is a variable and not a specific value. If x has a specific value, then the simplified expression x + 15 will represent a numerical result based on that value.

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To solve the equation 2x² + 6x = -4 by factoring, we first rearrange the equation to bring all terms to one side: 2x² + 6x + 4 = 0

Now, we look for factors of the quadratic expression that sum up to 6x and multiply to 2x² * 4 = 8x².

The factors that satisfy these conditions are 2x and 2x + 2:

2x² + 2x + 4x + 4 = 0

Now, we group the terms and factor by grouping:

(2x² + 2x) + (4x + 4) = 0

Factor out the common factors:

2x(x + 1) + 4(x + 1) = 0

Now, we have a common binomial factor of (x + 1):

(2x + 4)(x + 1) = 0

Now, we set each factor equal to zero and solve for x:

2x + 4 = 0 or x + 1 = 0

From the first equation, we have:

2x = -4

x = -2

From the second equation, we have:

x = -1

Therefore, the solutions to the equation 2x² + 6x = -4 are x = -2 and x = -1.

To check our answers, we substitute each solution back into the original equation:

For x = -2:

2(-2)² + 6(-2) = -4

8 - 12 = -4

-4 = -4 (satisfied)

For x = -1:

2(-1)² + 6(-1) = -4

2 - 6 = -4

-4 = -4 (satisfied)

Hence, both solutions satisfy the original equation 2x² + 6x = -4, confirming our answers.

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The value of the expression ₉C₂ is 36. This means that there are 36 different ways to select 2 items from a set of 9 items.

The expression ₉C₂ represents the combination of selecting 2 items from a set of 9 items. To find the value of this expression, we can use the formula for combinations, which is nCr

= n! / (r!(n-r)!),

where n is the total number of items and r is the number of items being selected.
In this case, n is 9 and r is 2. So, we can plug these values into the formula:
₉C₂ = 9! / (2!(9-2)!)

= (9 * 8 * 7!) / (2! * 7!)

= (9 * 8) / (2 * 1)

= 36.
Therefore, the value of the expression ₉C₂ is 36. This means that there are 36 different ways to select 2 items from a set of 9 items.

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The value of the expression ₉C₂ is 36.

The expression ₉C₂ represents the combination of selecting 2 items from a set of 9 items.

To find the value of this expression, we can use the formula for combinations:

nCr = n! / (r!(n-r)!)

In this case, n = 9 and r = 2. Plugging these values into the formula, we have:

₉C₂ = 9! / (2!(9-2)!)

To simplify the expression, we need to calculate the factorial values.

The factorial of a number is the product of all positive integers up to that number.

For example, 4! = 4 x 3 x 2 x 1 = 24.

Calculating the factorials:

9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880
2! = 2 x 1 = 2
(9-2)! = 7!

Now, substituting these values back into the expression:

₉C₂ = 362,880 / (2 x 5,040)

Simplifying further:

₉C₂ = 362,880 / 10,080

Dividing these two values:

₉C₂ = 36

Therefore, the value of the expression ₉C₂ is 36.

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The calculated standard deviation value of 14.5 is much higher than the provided value of 11.1. The computed result differs from the given number by a percentage of 30.6%, which is greater than the threshold of 20% required to determine significance. So, option B is correct.

Percentage = (14.5 - 11.1) / 11.1 × 100

= 30.6%

Which is greater than 20%. Hence,

The computed value is greater than the given value.

Option B is correct.

The calculated percentage difference is bigger than the problem's 20% cutoff point at 30.6%. A discrepancy of 20% or more is deemed substantial by the provided standards. We can therefore conclude that the computed value of 14.5 is much higher than the provided value of 11.1, as it surpasses this threshold.

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The complete question is-

Consider a difference of 20% between two values of a standard deviation to be significant. How does the computed value, 14.5, compare with the given standard deviation, 11.1?

A. The computed value is significantly less than the given value.

B. The computed value is significantly greater than the given value.

C. The computed value is not significantly different from the given value.

The surface area of the glass sculpture in the shape of a right square prism can be represented by the equation 10s^2, where s represents the side length of the base square.

A glass sculpture in the shape of a right square prism is shown. The base of the sculpture's outer shape is a square. To find the surface area of the sculpture, we need to calculate the area of each face and then add them together.

To calculate the surface area, we can use the formula: Surface Area = 2lw + 2lh + 2wh, where l, w, and h represent the length, width, and height of the prism.

Since the base of the sculpture is a square, we know that the length (l) and width (w) are equal. Let's call this side length s.

To find the surface area, we can substitute the values into the formula:
Surface Area = 2s^2 + 2s*h + 2s*h.

Since the sculpture is a right square prism, we can assume that the height (h) is also equal to the side length (s).

Substituting the values:
Surface Area = 2s^2 + 2s*s + 2s*s.

Simplifying the equation:
Surface Area = 2s^2 + 4s^2 + 4s^2.

Combining like terms:
Surface Area = 10s^2.

So, the surface area of the glass sculpture in the shape of a right square prism can be represented by the equation 10s^2, where s represents the side length of the base square.

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Answer:

base = 35 cm

Step-by-step explanation:

the area (A) of a triangle is calculated as

A = [tex]\frac{1}{2}[/tex] bh ( b is the base and h the height )

given A = 840 and h = 48 , then

[tex]\frac{1}{2}[/tex] × b × 48 = 840

24b = 840 ( divide both sides by 24 )

b = 35

then base is 35 cm

The per-tree yield is initially 1100 peaches per tree when 65 or fewer trees are planted per acre.

For every extra tree planted beyond 65, the per-tree yield decreases by 20 peaches.

Based on the given information, when 65 or fewer trees are planted per acre, the per-tree yield is equal to 1100. However, when more than 65 trees are planted per acre, the per-tree yield decreases by 20 peaches for every extra tree planted.

To calculate the per-tree yield, we can use the following equation:
Per-tree yield = 1100 - (number of extra trees * 20)

For example, if 70 trees are planted per acre, there would be 5 extra trees (70 - 65 = 5).

Therefore, the per-tree yield would be:
Per-tree yield = 1100 - (5 * 20)

= 1000 peaches per tree.

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In statistical hypothesis testing, the P-value is a significant factor. It is the probability of obtaining a test statistic at least as extreme as the one calculated from the data, assuming the null hypothesis to be true. If the null hypothesis is false, the P-value is the probability of a type I error. It is the probability of rejecting the null hypothesis when it is true.

To interpret the P-value correctly, a P-value of 0.054 means that if the null hypothesis is correct, there is a 5.4% probability that the sample will produce a test statistic as extreme as, or more extreme than the one that was observed. If the calculated P-value is higher than the significance level, which is usually 0.05 or 0.01, we cannot reject the null hypothesis.

In the given situation, the sample provides insufficient evidence to reject the owner's claim that the mean weight of Gala apples this year is heavier than usual because the calculated P-value is higher than the significance level. Hence, the correct option is that the P-value suggests that there is not sufficient evidence to reject the null hypothesis.

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The probabilities of events A ∩ B, A ∪ B, and A ∩ B^c, assuming all 36 sample points have equal probability, are 1/2, 5/6, and 1/4, respectively.

Let's analyze the events described:

Event A: The sum of the faces is odd.

Event B: At least one ace (one comes up).

To describe the events A ∩ B, A ∪ B, and A ∩ B^c, we need to understand the outcomes that satisfy each event.

Event A ∩ B: The sum of the faces is odd and at least one ace comes up. This means we want the outcomes where the sum is odd and there is at least one 1 on either die.

Event A ∪ B: The sum of the faces is odd or at least one ace comes up. This includes the outcomes where either the sum is odd, or there is at least one 1.

Event A ∩ B^c: The sum of the faces is odd, but no aces (1) come up. This means we want the outcomes where the sum is odd and neither die shows a 1.

To find the probabilities of these events, we need to count the favorable outcomes and divide by the total number of possible outcomes.

There are 36 possible outcomes when two dice are thrown (6 possible outcomes for each die)

The favorable outcomes for each event can be determined as follows:

Event A ∩ B: There are 18 favorable outcomes. There are 9 outcomes where the sum is odd (1+2, 1+4, 1+6, 2+1, 2+3, 2+5, 3+2, 4+1, 6+1) and another 9 outcomes where there is at least one ace (1+2, 1+3, 1+4, 1+5, 1+6, 2+1, 3+1, 4+1, 5+1).

Event A ∪ B: There are 30 favorable outcomes. There are 18 outcomes where the sum is odd (as mentioned above) and an additional 12 outcomes where there is at least one ace (1+2, 1+3, 1+4, 1+5, 1+6, 2+1, 3+1, 4+1, 5+1, 6+1, 1+6, 2+6).

Event A ∩ B^c: There are 9 favorable outcomes. These are the outcomes where the sum is odd and neither die shows a 1 (1+3, 1+5, 2+3, 2+5, 3+2, 3+4, 4+3, 4+5, 5+3).

Finally, we can calculate the probabilities by dividing the number of favorable outcomes by the total number of outcomes (36):

P(A ∩ B) = 18/36 = 1/2

P(A ∪ B) = 30/36 = 5/6

P(A ∩ B^c) = 9/36 = 1/4

Therefore, the probabilities of events A ∩ B, A ∪ B, and A ∩ B^c, assuming all 36 sample points have equal probability, are 1/2, 5/6, and 1/4, respectively.

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The function for the value of the plasma TV, V(t) = 3700 * (0.75)^t, represents decay. Where,t represents the number of years since the TV was bought, and V(t) represents the value of the TV at time t.

The initial value of $3,700 is multiplied by 0.75 each year, representing a 25% decrease. As time (t) increases, the value of the TV decreases exponentially. This is evident from the exponentiation of 0.75 to the power of t.

Decay functions signify a diminishing quantity or value over time, in this case, the decreasing value of the TV. Therefore, the function reflects the depreciation of the TV's value over successive years, indicating decay rather than growth.

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Using the F distribution table or a calculator, we find the critical F value to be approximately 4.75 at a significance level of 0.05.  The f critical value is used in statistical hypothesis testing to determine whether the difference between two sample means or variances is statistically significant.

The critical F value can be determined using a statistical table or calculator. In this case, with four observations in the numerator and seven in the denominator, we need to find the critical F value at a specific significance level (e.g., α = 0.05).

To find the critical F value, we compare the calculated F statistic to the critical F value from the F distribution table. The calculated F statistic is the ratio of the variances of the two groups being compared.
Since we have four observations in the numerator and seven in the denominator, our degrees of freedom are (4-1) = 3 and (7-1) = 6, respectively.

Using the F distribution table or a calculator, we find the critical F value to be approximately 4.75 at a significance level of 0.05. This means that if the calculated F statistic exceeds 4.75, we can reject the null hypothesis and conclude that there is a significant difference between the variances of the two groups.

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Using Pascal's Triangle the expansion of each binomial. (j+3 k)³ is j^3 + 9j^2 + 27j + 27.

To expand the binomial (j + 3)^3 using Pascal's Triangle, we can utilize the binomial expansion theorem. Pascal's Triangle provides the coefficients of the expanded terms.

The binomial expansion theorem states that for any positive integer n, the expansion of (a + b)^n can be expressed as:

(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n

Here, C(n, r) represents the binomial coefficient, which can be obtained from Pascal's Triangle. The binomial coefficient C(n, r) is the value at the nth row and the rth column of Pascal's Triangle.

In this case, we want to expand (j + 3)^3. Let's find the coefficients from Pascal's Triangle and substitute them into the binomial expansion formula.

The fourth row of Pascal's Triangle is:

1 3 3 1

Using this row, we can expand (j + 3)^3 as follows:

(j + 3)^3 = C(3, 0) * j^3 * 3^0 + C(3, 1) * j^2 * 3^1 + C(3, 2) * j^1 * 3^2 + C(3, 3) * j^0 * 3^3

Substituting the binomial coefficients from Pascal's Triangle:

(j + 3)^3 = 1 * j^3 * 1 + 3 * j^2 * 3 + 3 * j^1 * 3^2 + 1 * j^0 * 3^3

Simplifying each term:

(j + 3)^3 = j^3 + 9j^2 + 27j + 27

Therefore, the expansion of (j + 3)^3 using Pascal's Triangle is j^3 + 9j^2 + 27j + 27.

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The equation 2x⁴ - x³ + 2x² + 5x - 26 = 0 can have at most 4 complex roots, 1 or 0 positive real roots, and no negative real roots. The possible rational roots can be determined by considering all possible combinations of factors of -26 and 2.

To analyze the equation 2x⁴ - x³ + 2x² + 5x - 26 = 0, we can follow these steps:

Number of Complex Roots:

The degree of the equation is 4, so it can have at most 4 complex roots.

Possible Number of Real Roots:

By applying Descartes' Rule of Signs, we count the sign changes in the coefficients. In this equation, there is one sign change, so the number of positive real roots is either 1 or 0. There are no sign changes in the reversed order of coefficients, indicating 0 negative real roots.

Possible Rational Roots:

Using the Rational Root Theorem, we consider all possible combinations of factors of the constant term (-26) and the leading coefficient (2) to find the possible rational roots.

The factors of -26 are ±1, ±2, ±13, ±26, and the factors of 2 are ±1, ±2. By trying out the combinations, we can determine if any of them are roots of the equation.

Therefore, the equation 2x⁴ - x³ + 2x² + 5x - 26 = 0 can have at most 4 complex roots. It can have 1 or 0 positive real roots and no negative real roots. The possible rational roots can be found by considering all possible combinations of factors of -26 and 2.

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3.3% as a decimal is 0.033, and 0.033 as a percent is 3.3%.

To convert a decimal to a percent, we multiply the decimal by 100. Similarly, to convert a percent to a decimal, we divide the percent by 100.

Converting 3.3% to a decimal:

To convert 3.3% to a decimal, we divide 3.3 by 100:

3.3% = 3.3 / 100 = 0.033

Therefore, 3.3% as a decimal is 0.033.

Converting 0.033 to a percent:

To convert 0.033 to a percent, we multiply 0.033 by 100:

0.033 = 0.033 × 100 = 3.3%

Therefore, 0.033 as a percent is 3.3%.

Therefore, 3.3% can be expressed as the decimal 0.033, and 0.033 can be expressed as the percent 3.3%. This means that both forms represent the same value, with one expressed as a decimal and the other as a percentage

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the greatest possible product of a four-digit number and a three-digit number obtained from seven distinct digits is 2,463,534.

To find the greatest possible product of a four-digit number and a three-digit number obtained from seven distinct digits, we can start by considering the largest possible values for each digit.

Since we need to use seven distinct digits, let's assume we have the digits 1, 2, 3, 4, 5, 6, and 7 available.

To maximize the product, we want to use the largest digits in the higher place values and the smallest digits in the lower place values.

For the four-digit number, we can arrange the digits in descending order: 7, 6, 5, 4.

For the three-digit number, we can arrange the digits in descending order: 3, 2, 1.

Now, we multiply these two numbers to find the greatest possible product:

7,654 * 321 = 2,463,534

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The fraction is 3/2. A fraction is a numerical representation that expresses a part of a whole or a ratio between two quantities. It consists of a numerator and a denominator, separated by a slash (/), indicating division.

To find the fraction that satisfies the given conditions, we can set up an equation. Let's call the numerator of the fraction 'x' and the denominator 'y'.

According to the question, the sum of the numerator and denominator is 5. So we can write the equation: x + y = 5.

The fraction is also greater than its square, which means[tex]\frac{x}{y} > \left(\frac{x}{y}\right)^2[/tex].

Simplifying this inequality, we get [tex]\frac{x}{y} > \frac{x^2}{y^2}[/tex].

To find the fraction that satisfies this inequality, we can look for values of x and y that satisfy both the inequality and the equation.

One possible solution is x = 3 and y = 2, because [tex]\frac{3}{2} > \left(\frac{3}{2}\right)^2[/tex] (which simplifies to [tex]\frac{3}{2} >\left\frac{9}{2}[/tex]).

So the fraction is 3/2.

As for how I found this answer, I used algebraic equations to represent the given conditions and then solved for the variables that satisfied both the inequality and the equation.

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To find the distance between points A and B, the surveyor needs to measure the distances AC and BC and apply the Pythagorean theorem to calculate AB. AB = √(x^2 + y^2)

To find the distance between points A and B, a surveyor locates a point C on land such that ∠CAB forms a right angle. This technique is commonly known as using a right triangle to determine the distance.

In this case, we can use the Pythagorean theorem to find the distance between points A and B. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Let's denote the distance between A and C as x, and the distance between C and B as y. Since ∠CAB forms a right angle, we can use the Pythagorean theorem to express the relationship between x, y, and the distance between A and B:

[tex]x^2 + y^2 = AB^2[/tex]

Solving for AB, we have:

AB = √(x^2 + y^2)

So, to find the distance between points A and B, the surveyor needs to measure the distances AC and BC and apply the Pythagorean theorem to calculate AB.

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The required answer is the -

Part a: cos 300° = 0.5.

Part b:  sin 300° = -0.866.

Part a: To determine cos 300° using the cosine sum identity,  write 300° as the sum of two angles: 180° + 120°. The cosine sum identity states that cos(A + B) = cosAcosB - sinAsinB.

Now,  substitute A = 180° and B = 120° into the cosine sum identity equation:
cos(180° + 120°) = cos180°cos120° - sin180°sin120°.

Since cos180° = -1 and sin180° = 0,  simplify the equation to:
cos(180° + 120°) = -1 * cos120° - 0 * sin120°.

Simplifying further:
cos(180° + 120°) = -cos120°.

Finally, substitute cos120° with its value on the unit circle, which is -0.5:
cos(180° + 120°) = -(-0.5) = 0.5.

Therefore, cos 300° = 0.5.

Part b: To determine sin 300° using the sine difference identity, we can write 300° as the difference of two angles: 330° - 30°. The sine difference identity states that sin(A - B) = sinAcosB - cosAsinB.

Now,  substitute A = 330° and B = 30° into the sine difference identity equation:
sin(330° - 30°) = sin330°cos30° - cos330°sin30°.

Since sin330° = -0.5 and cos330° = 0.866, and sin30° = 0.5 and cos30° = 0.866, simplify the equation to:
sin(330° - 30°) = -0.5 * 0.866 - 0.866 * 0.5.

Simplifying further:
sin(330° - 30°) = -0.433 - 0.433.

Finally, adding the terms:
sin(330° - 30°) = -0.866.

Therefore, sin 300° = -0.866.

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Answer:

0.97

Step-by-step explanation:

[tex]\frac{1}{2^2}[/tex] - 0.54 + 1.26

= [tex]\frac{1}{4}[/tex] - 0.54 + 1.26

= 0.25 - 0.54 + 1.26 ← evaluate from left to right

= - 0.29 + 1.26

= 0.97

We can say that the system has exactly one solution for all values of m and n except the case where mn = 1.

To find the values of m and n for which the given system of equations has exactly one solution, we can use the determinant method. The system of equations is not given, so we cannot use the coefficients of the variables to form the matrix of coefficients and calculate the determinant directly. However, we can use the general form of a system of linear equations to derive the matrix of coefficients and calculate its determinant. The general form of a system of two linear equations in two variables x and y is given by:

ax + by = c

dx + ey = f

The matrix of coefficients is then:

A = [a b d e]

The determinant of this matrix is:

|A| = ae - bdIf

|A| ≠ 0, the system has exactly one solution, which can be found by using Cramer's rule.

If |A| = 0, the system has either no solution or infinitely many solutions, depending on whether the equations are consistent or not.

Now, let's apply this method to the given system of equations, which is not given. We only know that the variables are x and y, and the constants are m and n.

Therefore, the general form of the system is:

x + my = n

x + y = m + n

The matrix of coefficients is:

A = [1 m n 1]

The determinant of this matrix is:

|A| = 1(1) - m(n) = 1 - mn

To have exactly one solution, we need |A| ≠ 0. Therefore, we need:

1 - mn ≠ 0m

n ≠ 1

Thus, the system of equations has exactly one solution for all values of m and n except when mn = 1.

Therefore, we can say that the system has exactly one solution for all values of m and n except the case where mn = 1.

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The value of a is approximately 54.7.

Given, b = 100 and c = 114.

We need to find a.

We can use the Pythagorean theorem to solve this problem as it relates to right-angled triangles according to which,a² + b² = c²

Substituting the values in the above expression, we get:

a² + 100² = 114²

⇒ a² + 10000 = 12996

⇒ a² = 2996

⇒ a = √2996=54.7

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(g⁰f)(6) = 958, starting with the input 6, we apply the transformation f(x) to obtain -31, and then we apply the transformation g(x) to get the final result of 958.

To find (g⁰f)(6), we need to compute the composition of functions g and f evaluated at x = 6.

Let's start by calculating f(6):

f(x) = 5 - x²

Substituting x = 6:

f(6) = 5 - (6)²

= 5 - 36

= -31

Now, let's calculate g(-31):

g(x) = x² - 3

Substituting x = -31:

g(-31) = (-31)² - 3

= 961 - 3

= 958

Therefore, (g⁰f)(6) = 958.

The composition of functions involves applying one function to the output of another function. In this case, we first evaluate f(6) to get -31, and then substitute -31 into g(x) to get 958 as the final result.

To visualize this process, think of f(x) as a transformation that takes an input x and produces an output by subtracting the square of x from 5. Then, g(x) is another transformation that takes its input, squares it, and subtracts 3. By performing (g⁰f)(6), we are applying both transformations successively.

In summary, starting with the input 6, we apply the transformation f(x) to obtain -31, and then we apply the transformation g(x) to get the final result of 958.

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