Write the equation of the line that passes through the points (-6, 4) and (-3,-2).
Put your answer in fully reduced point-slope form, unless it is a vertical or horizontal
line.

\(( ~~ \csc(\theta )-\cot(\theta ) ~~ )^2=\cfrac{1-\cos(\theta )}{1+\cos(\theta )} \\\\[-0.35em] ~\dotfill\\\\ ( ~~ \csc(\theta )-\cot(\theta ) ~~ )^2\implies \csc^2(\theta )-2\csc(\theta )\cot(\theta )+\cot^2(\theta ) \\\\\\ \cfrac{1^2}{\sin^2(\theta )}-2\cdot \cfrac{1}{\sin(\theta )}\cdot \cfrac{\cos(\theta )}{\sin(\theta )}+\cfrac{\cos^2(\theta )}{\sin^2(\theta )}\implies \cfrac{1}{\sin^2(\theta )}-\cfrac{2\cos(\theta )}{\sin^2(\theta )}+\cfrac{\cos^2(\theta )}{\sin^2(\theta )}\)

\(\cfrac{\cos^2(\theta )-2\cos(\theta )+1}{\sin^2(\theta )}\implies \cfrac{[\cos(\theta )-1][\cos(\theta )-1]}{\sin^2(\theta )} \\\\\\ \cfrac{[\cos(\theta )-1][\cos(\theta )-1]}{1-\cos^2(\theta )}\implies \cfrac{[\cos(\theta )-1][\cos(\theta )-1]}{-[\cos^2(\theta )-1]}\)

\(\cfrac{[\cos(\theta )-1][\cos(\theta )-1]}{-[\cos^2(\theta )-1^2]}\implies \cfrac{[\cos(\theta )-1][\cos(\theta )-1]}{-[\cos(\theta )-1][\cos(\theta )+1]} \\\\\\ \cfrac{\cos(\theta )-1}{-[\cos(\theta )+1]}\implies \cfrac{-[\cos(\theta )-1]}{\cos(\theta )+1}\implies \cfrac{1-\cos(\theta )}{1+\cos(\theta )}\)

Answer:

Martin family: 20 hours

Lewis family: 35 hours

Step-by-step explanation:

Let's say that the Lewis family sprinklers were out for L hours and the Martin family's sprinklers were out for M hours. We know that for each hour that the Lewis family sprinkler was on, 30L of water was put out. We can thus write the Lewis family sprinkler water output as 30L per each hour of L = 30 * L. Similarly, the Martin family sprinkler water output = 15 * M .

We know that the total hours for the sprinklers is 55, so L + M = 55. The total water output for the sprinklers is the sum of the sprinkler outputs, so 30 * L + 15 * M = 1350

L + M = 55

30 * L + 15 * M = 1350

One way to solve this would be to solve for L in the first equation and substitute that into the second

subtract M from both sides in the second equation

55 - M = L

30 * (55-M) + 15 * M = 1350

30 * 55 - 30 * M + 15 * M = 1350

1650 - 15M = 1350

subtract 1650 from both sides to isolate the M and its coefficient

-15M = -300

divide both sides by -15 to isolate M

M = 20

L = 55-20 = 35

Answer:

3 is 15 degrees and 1 is 67 degrees

Answer: 12:32

Step-by-step explanation: 3:8 x 4:4 = 12:32

3:8 6:12 9:24 12:32

Symbolically, they correspond to the relationships between the coefficients of the equations.

A system of 2 linear equations in 2 variables can have one of the following three possible solutions:

One unique solution: In this case, the two lines intersect at exactly one point, and this point is the solution to the system. Visually, the two lines are not parallel, but they are not the same either. Symbolically, the system is represented as:

a1x + b1y = c1

a2x + b2y = c2

where a1, b1, c1, a2, b2, and c2 are constants, and x and y are variables.

Infinitely many solutions: In this case, the two lines coincide and are on top of each other, meaning they have the same slope and the same y-intercept. Visually, the two lines are identical. Symbolically, the system is represented as:

a1x + b1y = c1

ka1x + kb1y = kc1

where a1, b1, and c1 are constants, x and y are variables, and k is any non-zero constant.

No solution: In this case, the two lines are parallel and never intersect. Visually, the two lines are distinct and never meet. Symbolically, the system is represented as:

a1x + b1y = c1

a2x + b2y = c2

where a1, b1, c1, a2, b2, and c2 are constants, and x and y are variables, and the slope of one line is not equal to the slope of the other.

Geometrically, these cases correspond to the three possible positions of two lines in the coordinate plane. Symbolically, they correspond to the relationships between the coefficients of the equations.

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Answer:

28 miles

Step-by-step explanation:

subtract the pay to the additional pay

The interpretation of the margin of error at a 90% confidence level is that we can be 90% confident that the true proportion of juniors who plan to attend homecoming lies within a range of plus or minus 0.093 of the sample proportion.

To calculate the margin of error, we need to use the formula:

Margin of Error =\(Z * (sqrt(p * (1-p) / n))\)

Where:
Z is the z-score corresponding to the desired level of confidence
p is the proportion of juniors who said they will attend homecoming
n is the sample size

In this case, the sample size is 75 and the proportion who said they will attend homecoming is 62/75 = 0.827.

To find the z-score for a 90% confidence level, we can use a z-table or a calculator. The z-score for a 90% confidence level is approximately 1.645.

Now we can plug in the values into the formula:

Margin of Error = \(1.645 * (sqrt(0.827 * (1-0.827) / 75))\)

Calculating this, we find that the margin of error is approximately 0.093.

The interpretation of the margin of error at a 90% confidence level is that we can be 90% confident that the true proportion of juniors who plan to attend homecoming lies within a range of plus or minus 0.093 of the sample proportion.

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The correct answer i.e. mean difference is (a) 0.

What is the mean difference?

The mean difference is a statistical measure that represents the average difference between pairs of values in a dataset. It is calculated by taking the sum of all the differences and dividing it by the total number of pairs.

To calculate the mean difference, follow these steps:

Identify the pairs of values in your dataset for which you want to calculate the difference.

Calculate the difference between each pair of values.

Sum up all the differences.

Divide the sum by the total number of pairs.

In a correlated t-test, the null hypothesis assumes that the mean difference between paired observations is zero, indicating no effect of the independent variable. Therefore, if the independent variable has no effect, the sample difference scores are expected to be a random sample from a population where the mean difference score, denoted as μd, equals 0.

Hence, the correct answer is (a) 0.

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Answer:

I think it is C. Number cube

Let 1,2 = pop

Let 3,4 = jazz

Let 5, 6 = rock

Roll cube two times. Repeat.

Step-by-step explanation:

Answer:

Step-by-step explanation:

Part A: Based on the given graph, we can observe that as the temperature of the city increases, the number of people at the swimming pool generally tends to increase as well. This suggests a positive correlation between temperature and the pool's population. In other words, when it gets hotter, more people are likely to visit the swimming pool. The relationship is not strictly linear, but it shows a general trend of increasing pool population with increasing temperature.

Part B: To determine the line of best fit, we can calculate the approximate slope and y-intercept using the given data points. Let's select two points from the data, such as (2.5, 1) and (12, 12):

Slope (m) = (change in y) / (change in x)

= (12 - 1) / (12 - 2.5)

= 11 / 9.5

≈ 1.16

To find the y-intercept (b), we can choose one of the points and substitute the values into the slope-intercept form (y = mx + b). Let's use the point (2.5, 1):

1 = 1.16 * 2.5 + b

1 = 2.9 + b

b ≈ -1.9

Therefore, the approximate slope of the line of best fit is 1.16, and the approximate y-intercept is -1.9.

A function rule that relates the total distance a car can travel d as a function of the number of gallons of gas in its tank g is given by: d = 20g

A proportion can be defined as an equation which is typically used to represent the equality of two (2) ratios. This ultimately implies that, proportions can be used to establish that two (2) ratios are equivalent and solve for all unknown quantities.

Mathematically, a rule that relates the total distance (d) which a car can travel as a function of the number of gallons (g) of gas in its tank:

d = 20g

Where:

In conclusion, the total distance (d) traveled by most cars is directly proportional to the number of gallons (g) of gas in their tank at a constant rate of 20 miles per gallon of gas.

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Explanatory variables in a regression model are typically considered to be random when they are subject to variability or uncertainty. There are several reasons why explanatory variables may be treated as random:

Measurement error: Explanatory variables may be measured with some degree of error or imprecision. This measurement error introduces randomness into the values of the variables. Accounting for this randomness is important to obtain unbiased and accurate estimates of the regression coefficients.

Sampling variability: In many cases, the data used to estimate the regression model are obtained through sampling. The values of the explanatory variables in the sample may differ from the true population values due to random sampling variability. Treating the explanatory variables as random helps capture this uncertainty and provides more robust inference.

Random assignment in experiments: In experimental studies, researchers often manipulate or assign values to the explanatory variables randomly. This random assignment ensures that the variables are not influenced by any underlying factors or confounders. Treating the explanatory variables as random reflects the randomization process used in the experiment.

By considering the explanatory variables as random, we acknowledge and account for the inherent variability and uncertainty associated with them. This allows for a more comprehensive and accurate modeling of the relationships between the explanatory variables and the response variable in regression analysis.

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The approximate measure of angle JGH is 70.5°.

According to this question, we have a right triangle in which lengths of its hypotenuse (GH), in centimeters, and one leg (JH) are known and we must determine the measure of an angle (∠JGH), in degrees. A representation of this triangle is included in the image attached below.

By trigonometry we have the following expression for the required angle:

cos∠GJH = HJ/GH               .......(1)

Given, HJ = 2, GH = 6

cos∠GJH = 2/6

     ∠GJH = \(cos^{-1}\frac{2}{6}\)

                ≈ 70.529

The approximate measure of angle JGH is 70.5°.

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P(73 < X < 83) = [probability value] (calculated using z-scores and the standard normal distribution)

The probability P(73 < X < 83) for a normally distributed random variable with a mean μ = 86 and standard deviation σ = 4, we can standardize the values using the z-score formula.

First, calculate the z-score for the lower value (73):

z1 = (73 - 86) / 4

Next, we calculate the z-score for the upper value (83):

z2 = (83 - 86) / 4

Using the standard normal distribution table or a calculator, we can find the probabilities associated with these z-scores. Then, we calculate the difference between the two probabilities to find the desired probability: P(73 < X < 83) = P(z1 < Z < z2).

The final probability value can be determined by subtracting the cumulative probability associated with the lower z-score from the cumulative probability associated with the higher z-score.

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Answer:

5.371 × 10 to 0 power

Answer:

\(\sf 5.371 * 10^0\)

Answer:

8 because if you take the radical of raditian 64 after multiplying 5 and 12.8, you then do the numerator which is radical 0.016. You can invert it and multiply it by 64 to cancel it out and you get 8

Answer:

  11 mi/h

Step-by-step explanation:

You want the speed of the boat when it travels 12 miles upstream against a current of 5 mph in the same time it travels 32 miles downstream with the current.

The relation between time, speed, and distance is ...

  time = distance/speed

If b represents the speed of the boat, the times upstream and downstream are the same, so we have ...

  (12 mi)/(b -5) = (32 mi)/(b+5)

Multiplying by the product of the denominators gives ...

  12(b+5) = 32(b-5) . . . . . . also divided by "mi"

  12b +60 = 32b -160 . . . . multiply by (b+5)(b-5)

  220 = 20b . . . . . . . . . . . . add 160 -12b

  11 = b . . . . . . . . . . . . . . . . . divide by 20

The speed of the boat in still water is 11 miles per hour.

__

Additional comment

Each trip took 2 hours.

Answer:

6.27 x 10 ^ 8

Step-by-step explanation:

Answer:

Part A: there are 264,000 feet in 50 miles.

Part B: there are 50 miles in 88,000 yards.

Step-by-step explanation:

Part A:

1 mile = 5280 feet

50 miles= x

Hence,

\(\displaystyle\\x=\frac{50*5280}{1} \\\\x=50*5280\\\\x=264,000\ feet\)

Part B:

1 mile = 1760 yards

x miles=88,000 yards

Hence,

\(\displaystyle\\x=\frac{88,000*1}{1760} \\\\x=\frac{88,000}{1760} \\\\x=\frac{1760*50}{1760} \\\\x=50 \ miles\)

The number of pores that are on the top of your foot is approximately 13,000 pores on the top of your foot.

To find the number of pores on the top of your foot, we can multiply the area of the top of your foot in square inches by the number of pores per square inch.

The area of the top of your foot is 2.0 x 10^1 in^2 and each square inch of your body has about 6.5 x 10^2 pores. So, the number of pores on the top of your foot can be calculated as:

2.0 x 10^1 in^2 * 6.5 x 10^2 pores/in^2 = 130 x 10^2 pores = 13000 pores

So, there are approximately 13,000 pores on the top of your foot.

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To use Lagrange multipliers, we first need to set up our objective function and constraint equation. Constructing a box with dimensions 12x12x6 feet will minimize the cost of construction.

Our objective is to minimize the cost of constructing the box, which is equal to:

C = 5xy + 2(3xy + 3xz + 3yz)

where x, y, and z are the dimensions of the box, and we have broken up the cost into the bottom and sides/top respectively. Our constraint is that the volume of the box must be 480 cubic feet, so:

xyz = 480

To use Lagrange multipliers, we set up the following equation:

∇C = λ∇(xyz)

where ∇C is the gradient of our objective function, ∇(xyz) is the gradient of our constraint equation, and λ is our Lagrange multiplier.

Taking the partial derivatives of our objective and constraint functions, we get:

∇C = (5y, 5x, 6x + 6y)
∇(xyz) = (yz, xz, xy)

Setting these equal and solving for λ, we get:

5y / yz = 5x / xz = (6x + 6y) / xy = λ

Simplifying, we get:

x = y = 2z

Substituting this into our constraint equation, we get:

4z^3 = 480

z = 6

Substituting this back into our dimensions, we get:

x = y = 12

So the minimum cost of constructing the box is:

C = 5(12)(12) + 2(3(12)(12) + 3(12)(6) + 3(12)(6)) = $1,296

Therefore, constructing a box with dimensions 12x12x6 feet will minimize the cost of construction.

To find the minimum cost for constructing a 480 cubic feet box with the bottom costing $5 per square foot and the top and sides costing $3 per square foot, we can use Lagrange multipliers. Let the dimensions of the box be length (x), width (y), and height (z).

First, we need to define the constraint function:
G(x, y, z) = xyz - 480 (the volume constraint)

Next, define the cost function:
C(x, y, z) = 5xy + 3(xz + yz + xy) (cost of bottom + cost of top and sides)

Now, we use the Lagrange multiplier λ and set the gradient of the cost function equal to the gradient of the constraint function multiplied by λ:
∇C(x, y, z) = λ∇G(x, y, z)

This results in a system of equations:

5y + 3z + 3y = λy
5x + 3z + 3x = λx
3x + 3y = λz

We also have the constraint equation:
xyz = 480

Solve this system of equations to find the optimal dimensions x, y, and z. Then, plug these dimensions into the cost function C(x, y, z) to find the minimum cost for constructing the 480 cubic feet box.

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Given: The dimension of the right-trinagle as

To Determine: The second leg of the triangle to the nearest tenth

Solution:

Let us represent the given using a right-triangle

The hypothenuse is the logest side and the side facing the right angle

Applying pythagoras theorem which state that the square of the hypothenuse is equal to the sum of the squares of the other sides

Let the other leg be y

Therefore

Hence, the second leg of the triangle is 20.8 inches rounded to the nearest tenth

Answer:

answer is 15 and second one i dont know yet sorry am doing scoo

Step-by-step explanation:

To get all whole number subscripts in the empirical formula, you would need to multiply the subscripts by a common factor that will give you the smallest possible whole numbers. In this case, you would need to multiply all subscripts by 3 to get C8H9O3.

This is because 2.67 is approximately equal to 8/3, 3 is approximately equal to 9/3, and 1 is equal to 3/3. Multiplying by 3 will simplify the subscripts and give you a whole number ratio of atoms in the empirical formula. This method works for any pseudo-formula with non-whole number subscripts, as long as you find a common factor to multiply by that will give you whole numbers.

In this case, you can divide each subscript by the smallest one (1) to get the ratio: C2.67: H3: O1. Now, find the smallest whole number that can convert 2.67 into a whole number when multiplied. In this case, that number is 3. So, multiply all subscripts by 3:

C(2.67 x 3)H(3 x 3)O(1 x 3) = C8H9O3

This gives you an empirical formula of C8H9O3.

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Answer:

B 7feet

Step-by-step explanation:

Side a = 24

Side b = 25

Side c = 7

Angle ∠A = 73.74° = 73°44'23" = 1.287 rad

Angle ∠B = 90° = 1.5708 rad = π/2

Angle ∠C = 16.26° = 16°15'37" = 0.28379 rad

C=16.26°B=90°A=73.74°b=25a=24c=7

Area = 84

Perimeter p = 56

Semiperimeter s = 28

Height ha = 7

Height hb = 6.72

Height hc = 24

Answer: 7.28

Step-by-step explanation:

not quite sure what this means but if you plug it into a calculator,

27.13 - 19.85 = 7.28

the equation used is

27.13 - 19.85 = n , where n is the value you're looking for