Which number sequence follows the rule subtract 15 starting from 105? 15, 30, 45, 60, 75 15, 10, 25, 20, 35 105, 100, 95, 90, 85 105, 90, 75, 60, 45

The recursive sequence representing how many papers Khloe has remaining to grade after working for n hours is given by a_n = a_{n-1} - 10, where a_0 = 90.

Let a_n denote the number of papers Khloe has remaining to grade after n hours of work. After the first hour of work, she will have 90 - 10 = 80 papers remaining. Therefore, we have a_1 = 90 - 10 = 80.

After the second hour of work, she will have a_2 = a_1 - 10 = 80 - 10 = 70 papers remaining. Similarly, after the third hour of work, she will have a_3 = a_2 - 10 = 70 - 10 = 60 papers remaining.

In general, after n hours of work, Khloe will have a_n = a_{n-1} - 10 papers remaining to grade. This is a recursive sequence, where the value of a_n depends on the value of a_{n-1}. The initial value of a_0 is given as 90, since she starts with 90 papers to grade. Therefore, the recursive sequence is given by a_n = a_{n-1} - 10, where a_0 = 90.

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Clarissa needs a total of 251.25 square inches of construction paper for her project.

To find the total area of construction paper needed, we need to find the area of each rectangle and add them together.

The first rectangle has an area of 10 inches × 14.5 inches = 145 square inches.

The second rectangle has an area of 10.25 inches × 10.5 inches = 107.625 square inches.

Adding these two areas together, we get a total of 145 + 107.625 = 252.625 square inches.

Therefore, Clarissa needs a total of 251.25 square inches (rounded to two decimal places) of construction paper for her project.

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Answer: To determine the percent of birds in Forest A that are robins, we need to know the number of robins and the total number of birds in Forest A. Let's assume that we have conducted a bird survey and found the following information:

Number of robins in Forest A = 50

Number of other birds in Forest A = 150

Total number of birds in Forest A = 50 + 150 = 200

To calculate the percentage of birds in Forest A that are robins, we can use the following formula:

Percentage = (Number of robins / Total number of birds) x 100%

Plugging in the values, we get:

Percentage = (50 / 200) x 100% = 25%

Therefore, 25% of the birds in Forest A are robins.

Step-by-step explanation:

The workers would have traveled a total of 32 km while erecting the fence over 40 days.

To determine the total distance the workers traveled while erecting the fence, we can use the following terms: daily distance, number of days, and total distance.

Step 1: Determine the daily distance traveled.
The workers erect 0.8 km of new fence each day.

Step 2: Determine the number of days it takes to erect the fence.
It takes 40 days to erect the fence.

Step 3: Calculate the total distance traveled.
To find the total distance, multiply the daily distance (0.8 km) by the number of days (40).

Total distance = Daily distance × Number of days
Total distance = 0.8 km × 40

Total distance = 32 km

So, the workers would have traveled a total of 32 km while erecting the fence over 40 days.

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The equation of the line passing through the points (0, 1) and (2, -3) is y = -2x + 1.

The formula for equation of line is expressed as;

y = mx + b

Where m is slope and b is y-intercept.

First, we determine the slope of the line.

Given the two points are (0, 1) and (2, -3).

We can find the slope of the line by using the slope formula:

m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Substituting the values, we get:

m = (-3 - 1) / (2 - 0)

m = -4 / 2

m = -2

Using the point-slope form, plug in one of the given points and slope m = -2 to find the equation of the line.

Let's use the point (0, 1):

y - y₁ = m(x - x₁)

y - 1 = -2(x - 0)

y - 1 = -2x

y = -2x + 1

Therefore, the equation of the line is y = -2x + 1.

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The area of the rhombus is 1140 square inches.

Let the side length of the rhombus be "a" and let the length of the other diagonal be "d".

Since a rhombus has all sides congruent, the perimeter is given by:

4a = 136

Simplifying, we get:

a = 34

We can use the formula for the area of a rhombus:

Area = (diagonal 1 x diagonal 2)/2

Substituting the given values:

Area = (60 x d)/2

Area = 30d

Now we can substitute the value of "a" in terms of "d" into the formula for the length of the diagonal:

d = √(a² + b²)

d = √(34² + b²)

d = √(1156 + b²)

We also know that the perimeter of the rhombus is given by:

4a = 136

Substituting the value of "a" we found earlier:

4(34) = 136

So the length of the other diagonal can be found by subtracting the length of the given diagonal from the perimeter, and dividing by 2:

d = (136 - 60)/2 = 38

Therefore, the length of the other diagonal is 38 inches.

To find the area, we can substitute the value we found for "d" into the formula we derived earlier:

Area = 30d = 30(38) = 1140

So the area of the rhombus is 1140 square inches.

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The measure of the unknown arc is 76°

An arc is a smooth curve joining two endpoints. It can also be defined as a portion of a circle.

Arc angle relationship states that If two chords intersect inside a circle, then the measure of each angle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Therefore angle 30° = 1/2 ( x-61)

30 = 1/2( x -61)

15 = x -61

x = 15 + 61

x = 76°

therefore the measure of the unknown arc is 76°

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1. The coordinates are: START: (0,0) END: (6,-4), 2. The perimeter of rectangle YARD is 20 units,3. The lengths of YX and YZ are 4 units and 6 units, respectively, 4. The area of Mary's room is 24 square units,

1-To get from one corner of his yard to another, Jason travels 4 units down and then 6 units right. Starting from the origin, his starting point is (0,0). From there, he moves 4 units down to the point (0,-4), and then 6 units right to reach his endpoint, which is at (6,-4).

2-The rectangle has two sides of length 4 and two sides of length 6. The perimeter is the sum of the lengths of all sides, so it is equal to 2(4) + 2(6) = 8 + 12 = 20 units.

3-The coordinates of points Y, X, and Z are not given, so we cannot calculate the lengths directly. However, we know that the sides of a rectangle are perpendicular, so we can use the Pythagorean theorem to find the lengths. Let Y be the origin (0,0), and let X be the point (0, -4). Then YX has length 4 units. Similarly, let Z be the point (6, 0), so YZ has length 6 units.

4.To find the area of a rectangle, we can multiply the lengths of its sides. From problem 3, we know that the lengths of the sides are 4 and 6 units, so the area is 4 x 6 = 24 square units.

5. The sides AB and AC form a right angle.

To determine which sides of the triangle form a right angle, we need to find the slope of each side. The slope of AB is (3-8)/(6-3) = -5/3, and the slope of AC is (3-3)/(6-3) = 0. Since the product of the slopes of two perpendicular lines is -1, we can see that AB is perpendicular to AC. Therefore, sides AB and AC form a right angle.

6. Tyler walks 9 blocks in all.

To find the distance Tyler walks, we need to calculate the length of sides AB and AC. Using the distance formula, we can find that the length of AB is sqrt[(6-3)² + (3-8)²] =√[(34) units, and the length of AC is 3 units. Therefore, Tyler walks 3 + √[34 units along the two sides that form the right angle. This is approximately 9.4 blocks, so he walks 9 blocks in all.

7. The distance between two points in a coordinate plane can be found using the distance formula:

d = √[(x₂-x₁)² + (y₂-y₁)²]

where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points, and d is the distance between them. The formula is derived from the Pythagorean theorem, which relates the sides of a right triangle.

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steal it

Step-by-step explanation:

Answer:

A = 28 units²

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 perpendicular height )

here b = FG = 8 and h = FE = 7 , then

A = [tex]\frac{1}{2}[/tex] × 8 × 7 = 4 × 7 = 28 units²

The amount for the fencing cost  is $903. 36

From the information given, we have that the shape of the garden is  semi -circle.

Now, the formula that is used for calculating the circumference of a semicircle is expressed as;

C = πr + 2r

Given that the parameters of the equation are;

From the information given,

Substitute the values, we have;

Circumference = 3.14(19) + 2(19)

expand the bracket

Circumference = 59. 66 + 38

Add the values

Circumference = 97. 66 feet

Then,

if 1 feet = $9.25

Then, 97. 66 feet = x

x = $903. 36

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The amount of time Carl represent is 25-m on the problems.

The statement "Davis spent 25 minutes working on math problems. Carl worked on math problems for m fewer minutes" means that Carl spent some amount of time working on math problems, but that amount is m minutes less than what Davis spent.

To represent the amount of time Carl worked on math problems, we can use the variable X. We know that X is equal to the amount of time Carl worked on math problems, and that X is equal to 25 minus m.

This is because Davis spent 25 minutes on math problems, and Carl worked on them for m fewer minutes. So if we subtract m from 25, we get the amount of time Carl worked on math problems.

Therefore, the equation X = 25 - m represents the amount of time Carl worked on math problems, where X is the amount of time in minutes and m is the number of minutes Carl worked less than Davis.

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The number of zeros are in the product of the number 50 and 6000 is 50 x 6000 = 300,000 are five.

Integers, natural numbers, fractions, real numbers, complex numbers, and quaternions are examples of typical special instances where it is possible to define the product of two numbers or the multiplication of two numbers.

A product is the outcome of multiplication in mathematics, or an expression that specifies the elements (numbers or variables) to be multiplied.

The commutative law of multiplication states that the result is independent of the order in which real or complex numbers are multiplied. The result of a multiplication of matrices or the elements of other associative algebras typically depends on the order of the components. For instance, matrix multiplication and multiplication in general in other algebras are non-commutative operations.

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The average price of milk in 2018 was 6.45 dollars per gallon.

The average price of milk in 2021 was 189.95 dollars per gallon.

The given function is: Price = 3.55 + 2.90(1 + x)³

In order to find the average price of milk in 2018, we need to set x = 0:

Price in 2018 = 3.55 + 2.90(1 + 0)³ = 3.55 + 2.90(1) = 6.45 dollars per gallon

Price in 2021 = 3.55 + 2.90(1 + 3)³ = 3.55 + 2.90(64) = 189.95 dollars per gallon

Hence, the average price of milk in 2021 was 189.95 dollars per gallon.

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To solve this problem, we can use substitution. Let's set u = x-2. Then, du/dx = 1 and dx = du. Using this substitution, we can rewrite the integral as:

∫ (u+2)² - 2(u+2) 3 (u+2) - Su du
Expanding the terms inside the integral, we get:
∫ (u² + 4u + 4) - 2(u+2)³ (u+2) - Su du
Simplifying, we get:

∫ u⁴ - 4u³ + 4u² - u³ + 6u² - 12u - u² + 6u - 9 du
Combining like terms, we get:
∫ u⁴ - 5u³ + 9u² - 6u - 9 du
Now, we can integrate each term separately using the power rule of integration:
∫ u⁴ - 5u³ + 9u² - 6u - 9 du = (1/5)u⁵ - (5/4)u⁴ + (9/3)u³ - 3u² - 9u + C
Substituting back u = x-2, we get:
(1/5)(x-2)⁵ - (5/4)(x-2)⁴ + (3)x³ - 3(x-2)² - 9(x-2) + C
Therefore, the indefinite integral of x² - 2x 3 X - S dx x⁴ - 4x + 4 - X x - 2x⁴ - 4x + 4 dx is (1/5)(x-2)⁵ - (5/4)(x-2)⁴ + (3)x³ - 3(x-2)² - 9(x-2) + C.
Hi! I'd be happy to help you with your integration problem. To find the indefinite integral using substitution, let's first rewrite the given integral:
∫(x² - 2x) / (x⁴ - 4x² + 4) dx

Now, let's perform substitution:
Let u = x² - 2x
Then, du/dx = 2x - 2
And also let v = x⁴ - 4x² + 4
Then, dv/dx = 4x³ - 8x
We need to find du in terms of dx, so:
du = (2x - 2) dx
Now, we can rewrite the integral in terms of u and v:
∫(u) / (v) (du / (2x - 2))
Now we can integrate:
(1/2) ∫(u) / (v) du

Unfortunately, this integral does not have a straightforward elementary antiderivative. However, you can use numerical integration methods or special functions to approximate the indefinite integral if necessary.

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The function that could be described is f(x) = 10cos(2πx/3), where the amplitude is 10, the period is 3, and the maximum value is 20.

In a cosine function, the amplitude represents the vertical distance from the midline to the maximum or minimum value. Here, the maximum value is 20, which means the amplitude is half of that, i.e., 10. The period of the function is the distance it takes for one complete cycle, and in this case, it is 3 units.

By using the formula f(x) = A*cos(2πx/P), where A is the amplitude and P is the period, we can determine that the given function matches the described characteristics.

The function f(x) = 10cos(2πx/3) has a maximum value of 20 and a minimum value of 0, and it completes one cycle over the interval of the period, which is 3 units.

In conclusion, the function f(x) = 10cos(2πx/3) satisfies all the given conditions and represents the described function.

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The value of x that maximizes the concentration is 119 weeks, and the maximum concentration is approximately 7.19 ml.

Based on the provided equation, the concentration C(x) is a quadratic function of x with a negative coefficient for the quadratic term, which means that it has a maximum point.

(a) To find the value of x that maximizes the concentration, we can take the derivative of the concentration function with respect to x, set it equal to zero, and solve for x. The derivative of C(x) is:

C'(x) = 56x + 7

Setting C'(x) equal to zero, we get:

56x + 7 = 0

Solving for x, we get:

x = -7/56 = -0.125

However, x represents the number of weeks from now, which cannot be negative. Therefore, the maximum concentration occurs at the endpoint of the interval we are considering, which is x = 119 weeks.

To find the maximum concentration, we can substitute x = 119 into the concentration function:

C(119) = 28²(2119)+7 - 119²-2119+2 ≈ 7.19 ml

So, the value of x that maximizes the concentration is 119 weeks, and the maximum concentration is approximately 7.19 ml.

(b) For very large values of x, the quadratic term (-x²) dominates the concentration function, and the concentration appears to decrease without bound.

This is because the negative quadratic term becomes much larger than the linear term (2x) and the constant term (2), causing the concentration to become more and more negative as x increases.

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If the donation to charity b is 5/6 of the donation to charity d, charity d's donation is twice the donation to charity c and the ratio of donations for charity c to charity a is 3:4 then the donation to charity b is £1250.

Let's denote the donation to charity a as x. Then the donation to charity c is (3/4)x, and the donation to charity d is 2(3/4)x = (3/2)x.

We know that the donation to charity b is 5/6 of the donation to charity d, so:

donation to charity b = (5/6)(3/2)x = (5/4)x

We also know that the total donation is £4500, so we can set up an equation:

x + (3/4)x + (3/2)x + (5/4)x = £4500

Multiplying through by 4 to get rid of the fractions, we have:

4x + 3x + 6x + 5x = £18,000

18x = £18,000

x = £1000

So the donation to charity b is: (5/4)x = (5/4)(£1000) = £1250

Therefore, the donation to charity b is £1250.

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Using the chain rule, we have: f'(x) = 2ln(x^4 + 5) * 2(x^4 + 5)^1 * 4x^3
f'(x) = 16x^3 * ln(x^4 + 5) * (x^4 + 5)

To find f'(-2), we plug in -2 for x:

f'(-2) = 16(-2)^3 * ln((-2)^4 + 5) * ((-2)^4 + 5)
f'(-2) = -128 * ln(21) * 21
f'(-2) ≈ -599.92 (rounded to 2 decimal places)

Therefore, f'(-2) is approximately -599.92.
To find f'(-2) for the function f(x) = ln((x^4 + 5)^2), we will first find the derivative of the function, and then evaluate it at x = -2.

1. Differentiate the function using the chain rule:
f'(x) = (d/dx) ln((x^4 + 5)^2) = (1/((x^4 + 5)^2)) * (d/dx) ((x^4 + 5)^2)

2. Differentiate the inner function:
(d/dx) ((x^4 + 5)^2) = 2(x^4 + 5) * (d/dx) (x^4 + 5) = 2(x^4 + 5) * (4x^3)

3. Combine the derivatives:
f'(x) = (1/((x^4 + 5)^2)) * (2(x^4 + 5) * (4x^3)) = (8x^3(x^4 + 5))/((x^4 + 5)^2)

4. Evaluate the derivative at x = -2:
f'(-2) = (8(-2)^3((-2)^4 + 5))/((-2)^4 + 5)^2 = (-128(21))/(21^2) = -128/21

So, f'(-2) is -128/21 as an exact fraction.

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Triangle DBG is isosceles and  BD = BG.

A triangle is a closed two-dimensional geometric shape with three straight sides and three angles. It is one of the basic shapes in geometry and has a wide range of applications in mathematics, science, and engineering.

To prove that triangle DBG is isosceles, we need to show that BD = BG.

First, we can use the given similarity to find the length of DF in terms of EB and EC. Since triangle ABC is similar to triangle EDF, we have:

AB:BC = ED:DF

Substituting the given values, we get:

2:3 = ED:DF

Multiplying both sides by DF, we get:

DF = (3÷2)ED

Next, we can use the fact that triangles EDF and EBG are similar (since they share angle E) to find the length of BG in terms of EB and DF:

ED/EB = BG/DF

Substituting the value we found for DF, we get:

ED/EB = BG/(3/2)ED

Multiplying both sides by (3/2)ED, we get:

BG = (3/2)ED²/ EB

Now we can use the Pythagorean theorem to find the lengths of BD and BG in terms of EB and EC:

BD² = BE² + ED²

BG² = BE² + EG²

Since EG = EC - BD, we can substitute BD = EC - EG in the first equation to get:

BD² = BE² + ED² = BE² + (3/2)ED²

Substituting the expression we found for BG in terms of ED and EB in the second equation, we get:

BG² = BE² + (3/2)ED²/EB² * BE²

Simplifying this expression, we get:

BG² = BE²(1 + 3ED²/2EB²)

Since we know that ED/EB = 2/3, we can substitute this value to get:

BG² = BE²(1 + (3/2)(4/9)) = BE²(25/18)

Therefore, we have:

BD² = BE² + (3/2)ED² = BE² + (3/2)(9/4)BE² = (15/8)BE²

BG² = BE²(25/18)

To show that BD = BG, we can compare the squares of these lengths:

BD² = (15/8)BE²

BG² = BE²(25/18)

Multiplying both sides of the first equation by 18/25, we get:

(18/25)BD² = (27/40)BE²

Substituting the expression for BG² in the second equation, we get:

(18/25)BD² = (27/40)BG²

Therefore, we have:

BD² = (27/40)BG²

Taking the square root of both sides, we get:

BD = (3/4)√(10) * BG

Substituting the expression we found for BG in terms of ED and EB, we get:

BD = (3/4)√(10) * (3/2)ED²/EB

Substituting the value of ED/EB = 2/3, we get:

BD = (3/4)√(10) * (3/2)(4/9)ED²

Simplifying this expression, we get:

BD = (2/3)√(10)ED²

Next, we can substitute the value we found for DF in terms of ED to get:

DF = (3/2)

Therefore, triangle DBG is isosceles and  BD = BG.

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The Intermediate Value Theorem states that if a function f(x) is continuous on a closed interval [a,b], and if M is any number between f(a) and f(b), then there exists at least one number c in the interval [a,b] such that f(c) = M.

In this case, we are given a continuous function g(x) with g(0) = 3, g(1) = 8, and g(2) = 4. Let s(x) be the inverse of g(x), which means that s(g(x)) = x for all x in the domain of g(x).

Suppose s(x) is invertible. Then for any y in the range of g(x), there exists a unique x such that g(x) = y, and therefore s(y) = x. In particular, let y = 5, which is between g(1) = 8 and g(2) = 4. By the Intermediate Value Theorem, there exists a number c in the interval [1,2] such that g(c) = 5.

However, this means that s(5) is not well-defined, since there are two values of x (namely c and s(5)) that satisfy g(x) = 5. Therefore, s(x) is not invertible.

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], and k is any number between f(a) and f(b), then there exists a number c in the interval [a, b] such that f(c) = k.

Let g(x) be continuous with g(0) = 3, g(1) = 8, and g(2) = 4. Since g(x) is continuous, the Intermediate Value Theorem applies. However, to show that s(x) is not invertible, we need to show that g(x) is not one-to-one.

Notice that g(0) = 3 and g(2) = 4, with g(1) = 8 in between. This means that there must exist a point c1 in the interval (0, 1) such that g(c1) = 4, and another point c2 in the interval (1, 2) such that g(c2) = 3, due to the Intermediate Value Theorem.

Since g(c1) = g(c2) = 4 and c1 ≠ c2, g(x) is not one-to-one. Therefore, its inverse function s(x) does not exist, and s(x) is not invertible.

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We need at least 7 terms of the series to be certain that the partial sum Sn is within 1/100 of the sum.

We want to find the smallest value of n such that the absolute value of the difference between the sum of the first n terms and the sum of the entire series is less than 1/100.

The sum of the first n terms of the series is given by:

Sn = ∑_(k=1[tex])^n[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex]

We can write the sum of the entire series as:

S = ∑_(k=[tex]1)^∞[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex]

The absolute value of the difference between the sum of the first n terms and the sum of the entire series is:

|S - Sn| = |∑_(k=n+1[tex])^∞[/tex] [tex](-1)^(k+1)/2^k|[/tex]

We want to find the smallest value of n such that |S - Sn| < 1/100.

Let's start by evaluating the sum of the series:

S = ∑_(k=1) (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex] = 1/2 - 1/4 + 1/8 - 1/16 + ...

This is a geometric series with first term a = 1/2 and common ratio r = -1/2. The sum of the series is:

S = a/(1-r) = (1/2)/(1+1/2) = 1/3

Now we can write:

|S - Sn| = |∑_(k=n+1[tex])^∞[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k|[/tex] <= 1/[tex]2^(n+1)[/tex]

The last inequality is true because the terms of the series are decreasing in absolute value, and we are summing an infinite number of terms.

Therefore, we need to find the smallest value of n such that 1/2^(n+1) < 1/100. This gives:

n+1 > log2(100)

n > log2(100) - 1

n > 6.64

The smallest integer value of n that satisfies this inequality is n = 7.

Therefore, we need at least 7 terms of the series to be certain that the partial sum Sn is within 1/100 of the sum.

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For two parametric equations, x = cos(t)− sin(t) ; y = cos(t) + sin(t) ; 0≤t≤2π

a) Cartesian equation for the parametric curve is represented by x² + y² = 2.

b) The sketch for this parametric curve, with arrows in the direction of curve tracing is present above figure.

A parametric curve in the x-t plane has the equations x=x(t), y=y(t). The curve associates a point of the plane (x,y) to a value of the parameter t. The rectangular form of the curve can be determined by eliminating the parameter t, i.e. determine the parameter in one equation and Substituting this value in the other equation. We have the following parametric equations,

x = cos⁡t - sin⁡ty = cos⁡(t)+ sin⁡t, 0 ≤ t ≤ 2π

(a) we have to eliminate parameter t to determine a cartesian equation for the parametric curve, use x²+ y² = (cos⁡(t) − sin⁡(t))²+ (cos⁡(t) + sin⁡(t))²

=> x² +y² = cos²t + sin²t - 2cos⁡t sin⁡t + cos²t + sin²t + 2cos⁡t sin⁡t

=> x² + y² = 2 ( sin²t + cos²t) = 2

which represents a circle curve centered at the origin and having radius √2.

(b) A sketch of this parametric curve is shown above figure and arrows are used to indicate the direction of curve trace.

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the length of PM is 98.

What is congruent of the triangle?

The shapes maintain their equality regardless of how they are turned, flipped, or rotated before being cut out and stacked. We'll see that they'll be placed entirely on top of one another and will superimpose one another. Due to their identical radius and ability to be positioned directly on top of one another, the following circles are considered to be congruent.

OM/MN = OP2/P2M

[tex]OM/(r_1 - r_2) = (r_2 + y - 12)/yOM = (r_1 - r_2)*(r_2 + y - 12)/y[/tex]

Similarly, since LQ is tangent to both circles at L and Q respectively, we have OL1 and OQ2 perpendicular to LQ. Therefore, triangle LOM and triangle QOM are similar triangles. Using this similarity, we can find the length of OM in terms of r1 and r2:

OM/ML = OQ2/Q2M

[tex]OM/(r_1 + r_2 - 31) = (r_2 + 62 - 4)/yOM = (r_1 + r_2 - 31)*(r_2 + 62 - 4)/y[/tex]

Since both expressions above represent the same length of OM, we can equate them:

[tex](r_1 - r_2)(r_2 + y - 12)/y = (r_1 + r_2 - 31)(r_2 + 62 - 4)/y[/tex]

Simplifying and solving for y, we get:

y = 22

Therefore, PM = 5y - 12 = 98.

Hence, the length of PM is 98.

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The fully expanded expression using logarithm properties is:

[tex]In(8x^2 - 72x + 112) = 2.079 + In(x - 2) + In(x –-7)[/tex]

To expand the given expression[tex]In(8x^2 - 72x + 112)[/tex], we can use the following logarithmic properties:

Product Rule: [tex]logb (xy) = logb x + logb y[/tex]

Quotient Rule: [tex]logb (x/y) = logb x - logb y[/tex]

Power Rule:[tex]logb (x^a) = a logb x[/tex]

We can first factor out a common factor of 8 from the expression inside the logarithm:

[tex]In(8x^2 - 72x + 112) = In[8(x^2 - 9x + 14)][/tex]

Using the distributive property, we can expand the expression inside the logarithm:

[tex]In[8(x^2 - 9x + 14)] = In(8) + In(x^2 - 9x + 14)[/tex]

Now, we need to expand the second logarithm. We notice that the expression inside the logarithm can be factored as follows:

[tex]x^2 - 9x + 14 = (x - 2)(x - 7)[/tex]

Using the product rule, we can write:

[tex]In(x^2 - 9x + 14) = In[(x - 2)(x - 7)][/tex]

[tex]= In(x - 2) + In(x - 7)[/tex]

Putting all the pieces together, we get:

[tex]In(8x^2 - 72x + 112) = In(8) + In(x^2 - 9x + 14)[/tex]

[tex]= In(8) + In(x -2) + In(x - 7)[/tex]

Finally, we can simplify the numerical expression In(8) by using the fact that ln(e) = 1:

[tex]In(8) = ln(8)/ln(e) = 2.079/1 = 2.079[/tex]

So the fully expanded expression using logarithm properties is:

[tex]In(8x^2 - 72x + 112) = 2.079 + In(x - 2) + In(x -7)[/tex]

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