Help solve for C Law of Cosines

The equation of a line that passes through points (2,5) and (4,3) is

y = -x+7.

Finding the equation of a line:

First, we need to find out the slope for the given points.

(X1,Y1) = (2,5)

(X2,Y2) = (4,3)

formula for slope(m) = [tex]\frac{Y2 - Y1}{X2 - X1}[/tex]

substitute the points in the above formula

[tex]\frac{3 - 5}{4 - 2}[/tex] = [tex]\frac{-2}{2}[/tex]

[tex]\frac{-2}{2}[/tex] = -1

slope for the given points(m) = -1.

m = -1

The equation of a line is y-y1 = m(x-x1), where x and y are variables.

substituting the values in the above equation then :

y-5 = -1(x-2)

y-5 = -x+2

y+x = 2+5

x+y = 7

y = -x+7

Therefore, the equation of the line passing through the points (2,5) and (4,3) is y = -x+7

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A) Two equations are,

S = 18 + 4t

L = 30 + 3t

Where, "t" represents the number of weeks that have passed.

B)  It will take 12 weeks for Samuel and Lewis to have the same number of rocks in their collections.

C) when the amount of rocks in their collection is equal, Samuel and Lewis will have 66 rocks each.

Part A:

For a system of equations to represent the number of rocks in each person's collection.

Let "S" be the number of rocks in Samuel's collection and "L" be the number of rocks in Lewis's collection.

Hence, We can represent the situation as follows:

Equation 1:

S = 18 + 4t

Equation 2:

L = 30 + 3t

Where, "t" represents the number of weeks that have passed.

Part B:

To find out how many weeks it will take for Samuel and Lewis to have the same number of rocks in their collections, we need to set Equation 1 equal to Equation 2 and solve for "t".

This gives us the following:

S = L

18 + 4t = 30 + 3t

1t = 12

t = 12 weeks

Hence, It will take 12 weeks for Samuel and Lewis to have the same number of rocks in their collections.

Part C:

For number of rocks Samuel and Lewis will have when the amount of rocks in their collection is equal, we can substitute the value of "t" that we found in Part B into either Equation 1 or Equation 2.

Hence, substituting "t = 12" into Equation 1 gives,

S = 18 + 4(12)

S = 18 + 48

S = 66 rocks

Therefore, when the amount of rocks in their collection is equal, Samuel and Lewis will have 66 rocks each.

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This problem is relatively simple. Before we begin, it might be beneficial to imagine the graph as a vertical number line. Imagining it this way makes it easy to count the units between them and gives you the answer: [tex]$\boxed{4}$[/tex].

Here's a slightly more advanced way to think about it:

The example question tells us to subtract the distance between the two points. The formula for this is [tex]\sqrt{\big{(}(x_{2}-x_{1})+(y_{2}-y_{1})\big{)}^2[/tex], but for this question, let's say that it is [tex]\sqrt{(y_{2}-y_{1})^2[/tex], or, to simplify it even more, [tex]$|y_{2}-y_{1}|$[/tex]. Now that we know the formula, we can substitute our y-values into the last formula and solve. Let's say that Point C is our first point and Point D is our second.

[tex]\big{|}(-8)-(-4)\big{|} = \big{|}-8+4\big{|} = \big{|}-4\big{|} = \boxed{4}[/tex] , so [tex]$4\text{ units}$[/tex] is our answer.

Disclaimer: Neither of the last two formulas I provided is the actual formula, just a version of the Distance Formula that might be easier to understand. The first formula is the actual thing, but you will encounter this in math later in life, probably around 8th or 9th grade.

As per the given triangle, the base of the triangle, b, is 360 millimeters.

We may use the formula for the area of a triangle to determine the triangle's base, or b, given its 360 square millimetres in area, 41 mm for each side, and 40 millimetres in height.

The formula for calculating a triangle's area is:

Area = (base * height) / 2

Given that,

Area = 360 square millimetres

Height = 40 mm

Sides = 41 mm

So, 360 = (b * 40) / 2

720 = b * 2

b = 720 / 2

b = 360

Thus, the base of the triangle is 360 mm.

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Apologies, but your questions seem to have errors and incomplete information.

For the first question about Jessica and Melissa sharing 12 pieces of dried pears, it is unclear how much Jessica ate as the fraction is missing. Similarly, information about how many pears Melissa ate is also missing. To answer the question accurately, I need the missing information.

For the second question about the children playing in the park, you mentioned that one-third of the children returned home, but you didn't provide the total number of children initially in the park. Without that information, I cannot determine how many children stayed in the park.

Please provide complete and accurate information for a precise answer.

Answer:

approximately 4.2667%

Step-by-step explanation:

To calculate the percentage above the website's estimate that Jerrold set his asking price, we can use the following formula:

Percentage above = ((Asking price - Website estimate) / Website estimate) * 100

Percentage above = (($750,000 - $715,000) / $715,000) * 100

Percentage above ≈ 4.895

Therefore, Jerrold set his asking price approximately 4.895% above the website's estimate.

To calculate the percentage below his asking price that Jerrold sold for, we can use the following formula:

Percentage below = ((Selling price - Asking price) / Asking price) * 100

Percentage below = (($718,000 - $750,000) / $750,000) * 100

Percentage below ≈ -4.2667

Therefore, Jerrold sold his house approximately 4.2667% below his asking price.

The mean of the given set of numbers is 11.

To find the mean (average) of a set of numbers, we sum up all the numbers and divide the sum by the total count of numbers.

Given the set of numbers: 8, 14, 22, 7, 2, 11, 25, 7, 5, 9

To find the mean, we add up all the numbers:

8 + 14 + 22 + 7 + 2 + 11 + 25 + 7 + 5 + 9 = 110

Next, we divide the sum by the total count of numbers, which is 10:

110 / 10 = 11

Therefore, the mean of the given set of numbers is 11.

The mean is a measure of central tendency and represents the average value of the data set.

In this case, it indicates that, on average, the numbers in the set tend to cluster around the value of 11.

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To calculate the amount of trace element collected by Robert, we need to find the volume of the cylinder-shaped container. Rounded to the nearest tenth, Robert has collected approximately 211.4 grams of the trace element in the second sample.

To calculate the amount of trace element collected by Robert, we need to find the volume of the cylinder-shaped container and then multiply it by the concentration of the trace element.

The volume of a cylinder is given by the formula:

V = π * r^2 * h

where V is the volume, π is approximately 3.14159, r is the radius, and h is the height.

Given:

Radius (r) = 8.7 cm

Height (h) = 17.3 cm

Concentration of trace element = 0.052 grams/cm³

Substituting these values into the volume formula:

V = 3.14159 * (8.7 cm)^2 * 17.3 cm

V ≈ 4068.57196 cm³

Now, we can calculate the amount of trace element collected by multiplying the volume by the concentration:

Amount of trace element = V * concentration

Amount of trace element ≈ 4068.57196 cm³ * 0.052 grams/cm³

Amount of trace element ≈ 211.42941792 grams

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Point A will have coordinates (-2, -3), which overlaps with the coordinates of Point B (3, -1).

To determine the new coordinates of Point A after the translation, we can start with the coordinates of Point B and apply the inverse translation.

Given that Point B has the coordinates (3, -1), we know that Point A after translation will have the same coordinates. We need to determine the inverse translation that will bring Point B back to its original position, and then apply that inverse translation to Point B.

The inverse translation of moving 2 units up and 5 units to the right is moving 2 units down and 5 units to the left. Therefore, we need to subtract 2 from the y-coordinate and subtract 5 from the x-coordinate of Point B.

Applying this inverse translation to Point B (3, -1), we have:

New x-coordinate of Point A = 3 - 5 = -2

New y-coordinate of Point A = -1 - 2 = -3

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Step-by-step explanation:

7! = 5040 ways

The first five terms of the following generalized Fibonacci sequence are -19, 14, -5, 9, 4

From the question, we have the following parameters that can be used in our computation:

The generalized Fibonacci sequence

In the sequence, we can see that the last two terms are added to get the new term

Also, we have

a(1) = -19

a(2) = 14

Using the above as a guide, we have the following:

a(3) = -19 + 14 = -5

a(4) = -5 + 14 = 9

a(5) = 9 - 5 = 4

Hence, the first five terms of the following generalized Fibonacci sequence are -19, 14, -5, 9, 4

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Answer: To factor the expression -14 - (-8) completely using the distributive law, we need to simplify it first.

Remember that when we subtract a negative number, it is equivalent to adding the positive number. Therefore, -(-8) is the same as +8.

So the expression becomes:

-14 + 8

To factor it further using the distributive law, we can rewrite the addition as multiplication by distributing the -14 to both terms:

(-14) + (8) = -14 * 1 + (-14) * 8

This can be simplified as:

-14 + 8 = -14 * 1 + (-14) * 8 = -14 + (-112)

Finally, we can add the two negative numbers to get the result:

-14 + (-112) = -126

Therefore, the expression -14 - (-8) factors completely as -126.

The remaining balance on a CD with a $2,200 initial investment and a 2.95% interest rate after three years would be roughly $2,393.10.

The compound interest formula is as follows:

Balance = Principal * (Rate/100 + Rate * 1)Time

Where

Principal (P) is equal to $2,200.

Rate (R) = 2.95% = 2.95/100 = 0.0295 in decimal form.

Time (T) = 3 years.

The following formula is substituted for the values:

Balance =[tex]$2,200 * (1 + 0.0295)^3.[/tex]

Exponent calculation:

Balance = [tex]$2,200 * (1,0295)^3[/tex]

Calculating the result:

Balance ≈ $2,200 * 1.089135

Balance ≈ $2,393.10

So, the remaining balance on a CD with a $2,200 initial investment and a 2.95% interest rate after three years would be roughly $2,393.10.

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[tex]\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\\ P=\textit{initial amount}\dotfill &5918\\ r=rate\to 6\%\to \frac{6}{100}\dotfill &0.06\\ t=years\dotfill &8\\ \end{cases} \\\\\\ A = 5918(1 - 0.06)^{8} \implies A = 5918( 0.94 )^{8}\implies A \approx 3607.43[/tex]

The possible lengths of the tree lot are 25 + 5√5 yards and 25 - 5√5 yards.

Let's denote the length of the rectangular tree lot as "l" and the width as "w".

We know that the perimeter of a rectangle is given by the formula:

Perimeter = 2(l + w)

Given that the perimeter of the tree lot must be 100 yards, we can write the equation as:

2(l + w) = 100

Next, we know that the area of a rectangle is given by the formula:

Area = l × w

Given that the area of the tree lot must be at least 500 square yards, we can write the inequality as:

l × w ≥ 500

Now, let's solve the equations simultaneously to find the possible lengths of the tree lot.

Perimeter equation:

2(l + w) = 100

l + w = 50

w = 50 - l

Area inequality:

l × w ≥ 500

Substituting the value of w from the perimeter equation into the area inequality, we have:

l × (50 - l) ≥ 500

50l - l^2 ≥ 500

l^2 - 50l + 500 ≥ 0

Now, we need to find the values of l that satisfy the inequality. Since the coefficient of the squared term is positive, the graph of this quadratic opens upward. This means that the values of l that satisfy the inequality will be either the entire range of possible values or a portion of it.

To find the possible lengths, we can either factor the quadratic or use the quadratic formula. Let's use the quadratic formula:

l = (-(-50) ± √((-50)^2 - 4(1)(500))) / (2(1))

l = (50 ± √(2500 - 2000)) / 2

l = (50 ± √500) / 2

l = (50 ± 10√5) / 2

l = 25 ± 5√5

Therefore, the possible lengths of the tree lot are 25 + 5√5 yards and 25 - 5√5 yards.

In summary, the possible lengths of the rectangular tree lot are 25 + 5√5 yards and 25 - 5√5 yards, respectively.

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There are 52 students taking none of the three languages.

There are 14 students taking both Arabic and Bulgarian. Of these, some may also take Chinese. We can find out how many by subtracting the students taking only Arabic or only Bulgarian from the total number of students taking Arabic and Bulgarian (14 + 15 + 19 = 48). So, 48 - 30 - 19 = 15 students are taking both Arabic and Bulgarian, and some of them may also take Chinese.

Next, we can subtract the students taking only Arabic, only Bulgarian, and only Chinese from the total number of students taking each language to find out how many students are taking all three languages.

From the students taking Arabic, we subtract the 15 students taking only Arabic: 30 - 15 = 15 students who may also be taking Bulgarian and Chinese.

From the students taking Bulgarian, we subtract the 19 students taking only Bulgarian: 36 - 19 = 17 students who may be taking Arabic and Chinese.

From the students taking Chinese, we subtract the 19 students taking only Chinese: 35 - 19 = 16 students who may be taking Arabic and Bulgarian.

So, there are 15 + 17 + 16 = 48 students who are taking some combination of the three languages.

Finally, to find out how many students are taking all three languages, we subtract the students taking only two of the languages from the total number of students taking some combination of the three languages (48 - (15 + 19 + 17) = 48 - 51 = -3). This means there are no students taking all three languages.

To find out how many students are taking none of the three languages, we subtract the total number of students taking some combination of the three languages from the total number of students at the school (100 - 48 = 52).

Hence, there are 52 students taking none of the three languages.

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The length of each edge of the cube =  6 inches, Thus option B is correct.

The volume of a cube = 216 cubic inches,

length of each edge of a cube is an inch

To find the value of a:

The formula for the volume of a cube is [tex]a^{3}[/tex]

The volume of cubic = [tex]a^{3}[/tex]

V = [tex]a^{3}[/tex]

substitute the volume value given in the equation in the above formula

216 = [tex]a^{3}[/tex]

[tex]a = \sqrt[3]{216}[/tex]

[tex]a[/tex] = 6

a = 6inches

Therefore, the length of each edge of the cube is 6 inches.

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The balance after 4 years on $2000 at 4% is equal to $2320.

In Mathematics, simple interest can be calculated by using this formula:

S.I = PRT or S.I = A - P

Where:

Substituting the given parameters into the simple interest formula, we have;

S.I = 2000 × 4/100 × 4

S.I = 2000 × 0.04 × 4

S.I = $320.

Next, we would calculate the future value as follows;

Future value, A = S.I + P

Future value, A = $320 + $2000

Future value, A = $2320.

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The solution for x, is x= (3m - b)/2. (Option-d)

To solve for x in the given equation, m=(2x+b)/3, we can start by isolating x on one side of the equation.

First, we can multiply both sides of the equation by 3 to get rid of the denominator:

3m = 2x + b

Next, we can subtract b from both sides of the equation:

3m - b = 2x

Finally, we can divide both sides of the equation by 2:

x = (3m - b)/2

It's important to note that this is a linear equation with one unknown variable, and we can use algebraic methods to solve for the unknown variable. This equation may arise in many fields of study, including physics, engineering, and economics, among others.(Option-d)

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Answer:

         m∠RQS = 102°

         m∠TQS = 78°

Step-by-step explanation:

    A straight angle is equal to 180 degrees. We will create an equation to solve for x.

         9x° + 3° + 7x° + 1° = 180°

         16x° + 4° = 180°

         16x° = 176°

         x = 11

    Next, we will substitute this value into the expressions representing the angles.

         m∠RQS = 9x° + 3° = 9(11)° + 3° = 102°

         m∠TQS = 7x° + 1° = 7(11)° + 1° = 78°

The complete table for the function are

- f(1) - 3 = -5

- f(0) - 3 = -5.5

- f(2) - 3  = -7

- f(6) - 3 = undefined

- f(14) - 3 = -1

- f3(0) - 3 = 2

From the question, we have the following parameters that can be used in our computation:

The function equation and the incomplete table of values

This is given as

g(x) = - f(x) - 3

From the table, the missing values are at all values of x

So, we have

- f(1) - 3 = -2 - 3 = -5

- f(0) - 3 = -2.5 - 3 = -5.5

- f(2) - 3 = -4 - 3 = -7

- f(6) - 3 = undefined

- f(14) - 3 = 2 - 3 = -1

- f3(0) - 3 = 5 - 3 = 2

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The product of the expression 7 x 63 is 441.

We have,

To find the product of 7 x 63 using the distributive property, we can break down 63 as the sum of its factors, such as 60 and 3:

7 x 63 = 7 x (60 + 3)

Now, we can apply the distributive property by multiplying 7 to each term inside the parentheses:

7 x (60 + 3) = 7 x 60 + 7 x 3

Simplifying further:

7 x 60 + 7 x 3 = 420 + 21

Therefore,

The product of 7 x 63 is 441.

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We can integrate this S = 2π ∫(1.1 to 4.4) (5√(4x + 25))/(2√x) dx over the given interval (1.1 to 4.4) to find the surface area.

We can evaluate the integral using numerical methods or a calculator to find the final answer.

We have,

To find the surface area of the revolution about the x-axis of the function f(x) = 5√x over the interval (1.1 to 4.4), we can use the formula for the surface area of revolution:

S = ∫(a to b) 2πy√(1 + (f'(x))²) dx

In this case,

f(x) = 5√x, so f'(x) = (d/dx)(5√x) = 5/(2√x).

Let's calculate the surface area:

S = ∫(1.1 to 4.4) 2π(5√x)√(1 + (5/(2√x)²) dx

Simplifying the expression inside the integral:

S = ∫(1.1 to 4.4) x 2π(5√x)√(1 + 25/(4x)) dx

Next, we can integrate this expression over the given interval (1.1 to 4.4) to find the surface area.

To find the surface area of revolution about the x-axis of the function

f(x) = 5√x over the interval (1.1 to 4.4), we need to evaluate the integral:

S = ∫(1.1 to 4.4) 2π(5√x)√(1 + 25/(4x)) dx

Let's calculate the integral:

S = 2π ∫(1.1 to 4.4) (5√x)√(1 + 25/(4x)) dx

To simplify the calculation, let's simplify the expression inside the integral first:

S = 2π ∫(1.1 to 4.4) (5√x)√((4x + 25)/(4x)) dx

Next, we can distribute the square root and simplify further:

S = 2π ∫(1.1 to 4.4) (5√(4x + 25))/(2√x) dx

Thus,

We can integrate this expression over the given interval (1.1 to 4.4) to find the surface area.

We can evaluate the integral using numerical methods or a calculator to find the final answer.

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Answer: 4 small bags, 3 large bags (equations below in explanation)

Step-by-step explanation:

x=number of small bags

y=number of large bags

x+y=7

4x+10y=52

Subtract them after multiplying the first equation by 4.

4x+4y=28

4x+10y=52

-6x=-24

x=4

x+y=7

4+y=7

y=3

The  expression represents the total surface area of the prism is

2 (5 * 7) + 2 (4 * 7) +  2  (4 * 5)

The term "TSA" stands for "Total Surface Area" of a prism. The Total Surface Area represents the  sum of the areas of all the faces (including the bases) of the prism.

for the rectangular prism, the Total Surface Area can be calculated using the formula:

TSA = 2lw + 2lh + 2wh

where

l = 5

w = 4

h = 7

plugging in the values gives

TSA = 2 (5 * 7) + 2 (4 * 7) +  2  (4 * 5)

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She sold 27 appliances.

Given,

Lucy earns $400 a month in salary and she receives a commission of $18.

Last Month income = $886

Now,

Equation:

Monthly income = Fixed income + income from commission

$886 = $400 +  income from commission

Income from commission = $486

Commission for one appliance = $18

So,

Total items sold for the commission of  $486,

$486/$18

= 27

Hence total 27 appliances lucy sold.

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Answer: 18.78 cubic feet

Step-by-step explanation:

The detailed analysis is attached below.

Based on the given data, the relationship appears to be exponential.

To determine if the relationship between the given values is linear, exponential, or neither, we can examine the pattern in the data.

If the relationship is linear, we would expect the values of y to increase or decrease at a constant rate as the values of x increase. In other words, there would be a constant slope between the points.

If the relationship is exponential, we would expect the values of y to change by a constant factor as the values of x increase. In this case, the relationship would exhibit exponential growth or decay.

Looking at the given values of x and y, we can observe the following differences between consecutive x-values: 7, 9, 9, 9. These differences are not constant, indicating that the relationship is not linear.

To determine if the relationship is exponential, we can examine the ratios between consecutive y-values: 5/10 = 0.5, 10/20 = 0.5, 20/40 = 0.5. These ratios are constant, suggesting that the relationship may be exponential with a common ratio of 0.5.

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Answer:

y = -1/2x

Step-by-step explanation:

Currently, y = 2x - 5 is in slope-intercept form, whose general equation is y = mx + b, where

The slopes of perpendicular lines are negative reciprocals of each other.  We can show this with the following formula;

m2 = -1 / m1, where

Finding the slope of the other line:

Since 2 is the slope of y = 2x - 5, we can plug in 2 for m1 in perpendicular slope formula to find m2, the slope of the other line:

m2 = -1 / 2

m2 = -1/2

Thus, the slope of the other line is m = -1/2.

Finding the y-intercept of the other line:

We can find the y-intercept of the other line (i.e., b in the slope-intercept equation) by plugging in (-2, 1) for x and y and -1/2 for m in the slope-intercept form:

1 = -1/2(-2) + b

1 = 1 + b

0 = b

Thus, y = -1/2x is the line that passes through (-2, 1) and is perpendicular to y = 2x - 5.