Steve determines that sides DK and BC are congruent. He also measures

The 11.43 minutes is less than 12 minutes, Jillian has a good chance of making the team. Therefore, Jillian might make the cross country team.

Jillian is trying for the cross country team. To make it she must run 3 1/2 miles in less than 40 minutes.

To find out if Jillian will make the cross country team, we must check if she can run 3 1/2 miles in less than 40 minutes. The time required for Jillian to run one mile is found by dividing 40 minutes by 3.5:40 / 3.5 = 11.43Jillian must complete one mile in 11.43 minutes to be eligible for the cross country team.

Since ,11.43 minutes is less than 12 minutes, Jillian has a good chance of making the team. Therefore, Jillian might make the cross country team.

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Given data set: Mean: 65.8Median: 63.5Mode: 65Range: 11 Standard Deviation: 3.9To find the mean, median, mode, range, and standard deviation when each value of the data set is increased by 8, we need to add 8 to each data value.

Mean: 65.8 + 8 = 73.8Median: 63.5 + 8 =  there are no changes in the frequency of numbers, the mode will remain the same.Mode: 65Range: 11 Standard Deviation: 3.9 The standard deviation of a data set is not affected by adding or subtracting a constant from every value in the data set.

Therefore, the standard deviation remains the same.Standard Deviation: 3.9Answer:Mean: 73.8Median: 71.5Mode: 65Range: 11Standard Deviation: 3.9.Mean: 65.8 + 8 = 73.8Median: 63.5 + 8 = 71.5Since there are no changes in the frequency of numbers, the mode will remain the same.Mode: 65Range: 11 Standard Deviation: 3.9

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The range of the function g(x) = x^2 - 2, when the domain is {-2, -1, 1, 3}, is C. {-7, -2, -1, 1}.

To find the range of the function g(x) = x^2 - 2, we need to substitute each value from the given domain into the function and observe the corresponding outputs.

For x = -2, g(-2) = (-2)^2 - 2 = 4 - 2 = 2.

For x = -1, g(-1) = (-1)^2 - 2 = 1 - 2 = -1.

For x = 1, g(1) = (1)^2 - 2 = 1 - 2 = -1.

For x = 3, g(3) = (3)^2 - 2 = 9 - 2 = 7.

Thus, when the domain is {-2, -1, 1, 3}, the corresponding range values are {-7, -2, -1, 1}. Therefore, the correct option is C. {-7, -2, -1, 1}.

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The length of the midsegment LM in Trapezoid ABCD is 85.5 units. The length of the midsegment is equal to the average of the lengths of the two bases.

In a trapezoid, the midsegment is a line segment that connects the midpoints of the two non-parallel sides. The length of the midsegment is equal to the average of the lengths of the two bases.

Given that AB = 46 and DC = 125, we can find the length of the midsegment (LM) by calculating the average of these two values.

LM = (AB + DC) / 2

LM = (46 + 125) / 2

LM = 171 / 2

LM = 85.5

Therefore, the length of the midsegment LM in Trapezoid ABCD is 85.5 units.

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the weight of the crackers originally in the box was 120/23 pounds.

Let the weight of the entire box be x pounds. Now, Roger served 5/8 pound of crackers, which was 2/3 of the entire box.

Therefore, the weight of the crackers left in the box = (1 - 2/3) x = 1/3 xSince the crackers served by Roger was 5/8 pound, the weight of the crackers left in the box = x/3, then we can set up the following equation to find the value of x:5/8x + 1/3x = x

Multiplying the equation by 24 (the least common multiple of 8 and 3) on both sides gives us:

15x + 8x = 24x

Therefore, 23/24 x = 5/8 pound of crackers served by Roger.So, x = (5/8) x (24/23) pounds = 15/23 pounds

To solve the given question, let us suppose that the weight of the entire box of crackers is x pounds. Now, the given information is that Roger served 5/8 pound of crackers which was 2/3 of the entire box.

Therefore, the weight of the crackers left in the box = (1 - 2/3) x = 1/3 x.Now, we need to find out the original weight of the crackers in the box, which is the value of x.

To do that, we can set up an equation as follows:5/8x + 1/3x = xMultiplying both sides by the least common multiple of 8 and 3, which is 24, we get:15x + 8x = 24x

Simplifying further, we get:23x = 120x = 120/23 poundsThis is the weight of the entire box of crackers.

Therefore, the weight of the crackers originally in the box was 120/23 pounds.

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Emma is using x fence panels along the width of her garden. We need to find expressions for f(x), the width of her garden in feet, and g(x), the length of her garden in feet.

To find an expression for f(x), the width of Emma's garden, we need to determine how the number of fence panels (x) relates to the width. Assuming each fence panel has a fixed width, we can express f(x) as:

f(x) = x * width of each fence panel

The width of each fence panel may vary depending on the specific measurements provided. For example, if each fence panel has a width of 4 feet, then the expression for f(x) becomes:

f(x) = 4x

To find an expression for g(x), the length of Emma's garden, we need additional information or assumptions. The given information does not specify how the number of fence panels along the width relates to the length of the garden. Without this information, we cannot determine a specific expression for g(x).

In summary, we can express the width of Emma's garden, f(x), by multiplying the number of fence panels (x) by the width of each fence panel. However, we cannot determine a specific expression for the length of her garden, g(x), without additional information or assumptions.

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Complete question:

Emma wants to enclose her rectangular garden with fence panels. If she uses x fence panels along the width of her garden, find an expression for f(x), the width of her garden in feet.

f(x) = ?

"Next, find an expression for g(x), the length of her garden, in feet.

g(x) = ?

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The linear function in slope-intercept form (y=mx+c) is: y = 2x + fixed amount where fixed amount is the amount that the band makes even if no tickets are sold.

Given that a band has been playing weekly at local venue.

They have been making a fixed amount, plus they receive additional money for every ticket sold.

Last week they sold 40 tickets and were paid $200.

The week before that they sold 70 tickets and were paid $260.

Let x represent the number of tickets sold and y represent the amount the band will be paid.

The linear function that represents the relationship between the number of tickets sold and the amount the band will be paid can be represented as follows:

For the week where 40 tickets were sold, the band received $200.

For the week where 70 tickets were sold, the band received $260.

The slope of the line is given as follows: The slope of the line =Change in y-value/change in x-value

                                                                                                        = 260-200/70-40

                                                                                                         = 60/30

                                                                                                         = 2.

The y-intercept of the line is given by the point (0, fixed amount).

Since the band receives a fixed amount irrespective of the number of tickets sold, the y-intercept of the line is equal to the fixed amount.

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A catalytic ozone destruction cycle requires no atomic oxygen and it incorporates the photochemical decomposition of ClONO₂ to Cl and NO₃, and photochemical decomposition of the later to NO and O₂. The overall reaction is NO + O₃ → NO₂ + O₂

In the lower stratosphere, a proposed mechanism for ozone destruction in the late spring over northern latitudes begins with the photochemical decomposition of ClONO₂ to Cl and NO₃. This reaction is catalyzed by sunlight in the lower stratosphere. The photodissociation of NO₃ is the next step in the cycle, and it results in the production of NO and O₂.

The NO then reacts with O₃ in the following reaction: NO + O₃ → NO₂ + O₂The NO₂ that is produced then reacts with atomic oxygen to form NO₃, and the cycle starts again with the photodissociation of ClONO₂. The NO that is produced during the reaction between NO₂ and O₃ can also react with atomic oxygen to form NO₂, which can then go on to form NO₃.However, the catalytic cycle that has been proposed requires no atomic oxygen to be present. The NO that is produced during the reaction between NO₂ and O₃ reacts with more O₃ to form NO₃ and O₂: NO + O₃ → NO₂ + O₂NO₂ + O₃ → NO₃ + O₂The NO₃ that is produced in this reaction can then go on to react with more O₃, starting the cycle over again. Thus, the overall reaction for the catalytic ozone destruction cycle is:NO + O₃ → NO₂ + O₂NO₂ + O₃ → NO₃ + O₂NO₃ + O₃ → NO + 2O₂The cycle continues as long as the necessary reactants are available.

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To identify the beverage that consistently yielded the highest profit according to the Smith's BBQ Report, we need to compare the profit margins of liquor, beer, and wine. By analyzing the profit margins over time, we can determine which beverage consistently had the highest margin, indicating the highest profit.

To determine which beverage consistently yielded the highest profit, we need to analyze the data provided in the Smith's BBQ Report. The report likely includes information on the sales and profits generated from liquor, beer, and wine. By comparing the profit margins of each beverage over a period of time, we can identify the one that consistently yielded the highest profit.

1. Analyzing profit margins: To determine the beverage with the highest profit, we examine the profit margins for liquor, beer, and wine. Profit margin is calculated by subtracting the cost of goods sold (COGS) from the revenue and dividing the result by the revenue. By comparing the profit margins of each beverage, we can identify which one consistently had the highest margin.

For example, if the profit margin for beer is consistently higher than that of liquor and wine across different time periods, it suggests that beer consistently yielded the highest profit. The profit margin analysis would provide insights into the beverage that generated the most profit for Smith's BBQ consistently.

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The length of AD, denoted as x, is less than 3/2.

Let's denote the length of AD as x.

According to the given information:

AB is 4 times as large as AD, so AB = 4x.

AC is 3 more than AD, so AC = x + 3.

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Applying this rule to triangle ABC, we can set up the following inequalities:

AD + AC > AB

x + (x + 3) > 4x

Simplifying the inequality:

2x + 3 > 4x

Subtracting 2x from both sides:

3 > 2x

Dividing both sides by 2:

3/2 > x

Therefore, the length of AD, denoted as x, is less than 3/2.

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The correct inequality for the given condition is,

⇒ x + 75 ≥ 200

We have,

Minimum amount to be saved = $200

And, She has $75 saved.

Let x is the amount needed to reach her good.

Hence, The correct inequality for the given condition is,

⇒ x + 75 ≥ 200

Therefore, Option A is correct.

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Complete question is shown in attached image.

The carnival is in town for 21 days, and to determine how many weeks it is in town, we use the equation 21 days ÷ 7 days/week = 3 weeks.

To find the number of weeks the carnival is in town, we need to divide the total number of days (21) by the number of days in a week (7). This can be represented by the equation of division operation:

Number of weeks = Total number of days ÷ Number of days in a week

Plugging in the values, we have:

Number of weeks = 21 days ÷ 7 days/week

Dividing 21 days by 7 days/week, we get:

Number of weeks = 3 weeks

Therefore, the carnival is in town for 3 week

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One and one/third times four and two/fifths` is equal to `88/15`.

To find the value of `one and one/third times four and two/fifths`, lets convert these mixed numbers to improper fractions, then multiply them and simplify the result :

Step 1: Converting mixed numbers to improper fractions`one and one/third` can be written as:

$$1\frac13 = \frac{3}{3}+\frac{1}{3}=\frac{4}{3}$$`

four and two/fifths` can be written as:

$$4\frac{2}{5}=4+\frac{2}{5}=\frac{20}{5}+\frac{2}{5}=\frac{22}{5}$$

Step 2: Multiplying the improper fractions$\frac43\times\frac{22}{5}=\frac{4\times 22}{3\times 5}=\frac{88}{15}$

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The answer to the expression "One and one-third times four and two-fifths" is 6/5.

To multiply fractions, follow these steps:

Step 1: Multiply the numerators together.

Step 2: Multiply the denominators together.

Step 3: Simplify the result obtained in step 1 and step 2 by reducing it to the lowest term possible.

Let's calculate the given expression:

One and one-third can be converted to an improper fraction by multiplying the denominator 3 by 1 and adding the numerator 1 to the product, which gives 4/3.

The same can be done with four and two-fifths. 5 is multiplied by 4, resulting in 20. Then, 2 is added to 20, resulting in 22/5.

Now we have:

One and one-third times four and two-fifths = 4(4) + 2 / 5(3) = 16 + 2 / 15 = 18/15 = 6/5

Therefore, the answer to the expression "One and one-third times four and two-fifths" is 6/5.

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The correct answer is:p = 3 × 4

Let's write the equation for the given statement:

Randy baked 4 pies

Let the number of pies that Liza baked be p

Liza baked 3 times as many pies as Randy.

Thus, the equation for the above statement can be written as:

p = 3 × 4Simplifying the above equation we get:p = 12Thus, Liza baked 12 pies.

So, the equation that can be used to find how many pies Liza made is:

p = 3 × 4The equation can be simplified to p = 12.

Therefore, the correct answer is:p = 3 × 4

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Ahmad's whole journey lasted for 13 hours and 40 minutes.

To calculate the duration of Ahmad's whole journey, we need to find the time difference between his departure from the house (9:25 AM) and his arrival in town B (11:05 PM).

First, let's convert the time to a 24-hour format for easier calculation.

9:25 AM in 24-hour format is 09:25.

11:05 PM in 24-hour format is 23:05.

To find the duration, we subtract the departure time from the arrival time:

23:05 - 09:25 = 13:40

The duration is 13 hours and 40 minutes.

Therefore, Ahmad's whole journey lasted for 13 hours and 40 minutes.

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the correct answer is 2.39 ft, which corresponds to option 2.

To determine the height of the cylindrical rain barrel, which has a radius of 2 feet and holds 30 cubic feet of water, we need to solve for the height using the given information and the formula for the volume of a cylinder. The answer choices provided are: 1. 58 ft, 2. 39 ft, 3. 57 ft, and 4. 78 ft.

The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height. In this case, we are given the radius as 2 feet and the volume as 30 cubic feet.

Substituting the given values into the formula, we have:

30 = 3.14 * 2² * h

Simplifying the equation:

30 = 12.56 * h

h = 30 / 12.56

h ≈ 2.39 ft

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The specific solutions to the equation cos θ = sin θ on the interval [0, 2π) are θ = 0, π, 2π, 3π.

To find the specific solutions to the equation cos θ = sin θ on the interval [0, 2π), we can use trigonometric identities and properties.

Let's rewrite the equation cos θ = sin θ as sin θ - cos θ = 0.

We know that sin θ = cos (π/2 - θ) from the complementary angle identity.

So, we can rewrite the equation as sin θ - sin (π/2 - θ) = 0.

Using the identity sin A - sin B = 2 sin((A - B)/2) cos((A + B)/2), we get:

2 sin((θ - (π/2 - θ))/2) cos((θ + π/2 - θ)/2) = 0.

Simplifying further:

2 sin(θ/2) cos(π/4) = 0.

Since cos(π/4) = 1/√2 is a nonzero constant, the equation reduces to:

sin(θ/2) = 0.

Now, we need to find the values of θ/2 that make sin(θ/2) = 0.

Sin(θ/2) = 0 when θ/2 = 0, π, 2π, 3π, ...

So, θ = 0, π, 2π, 3π are the specific solutions to the equation cos θ = sin θ on the interval [0, 2π).

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The area of the figure is given by 34cm²

The figure is made up of  triangle and a square.

The area of the figure is given by area of the square + area of the triangle

The area of a triangle is the total space occupied by the three sides of a triangle in a 2-dimensional plane. The basic formula for the area of a triangle is equal to half the product of its base and height, i.e., A = 1/2 b h. This formula is applicable to all types of triangles, whether it is a scalene triangle, an isosceles triangle, or an equilateral triangle

area of triangle = 1/2bh

Area of triangle = 1/2*10*6

Area = 30 com²

But the area of the square is S²

Where s = side

Area of square = 2*2 = 4cm²

therefore area of the shape is( 4+30)cm² = 34cm²

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The function f(x) = 2x2 - x + 2 is continuous at x = 2, the correct option is C. Yes, because the function is defined at x = 2 and approaches y = 8 on the left and right sides of x = 2.

A continuous function is a type of function in mathematics that has no abrupt changes or breaks in its graph. It is a function where the values change smoothly as the input values vary. In other words, a function is continuous if its graph can be drawn without lifting the pen from the paper.

Given the function f(x) = 2x² - x + 2.

Determine whether the function is continuous at x = 2.

Explanation: For a function to be continuous at x = a, it must satisfy the following conditions:

1. The function must be defined at x = a.

2. The limit of the function at x = a must exist.

3. The limit of the function at x = a must be equal to the value of the function at x = a.

Let us verify these conditions for the given function

f(x) = 2x² - x + 2 at x = 2.

1. The function is defined at x = 2.

2. We need to calculate the left-hand limit and the right-hand limit of the function as x approaches 2.

Let us first calculate the left-hand limit:

lim f(x) as x → 2- = lim (2x² - x + 2)

as x → 2- = 2(2)² - 2 + 2

= 6

Now, let us calculate the right-hand limit:

lim f(x) as x → 2+ = lim (2x² - x + 2)

as x → 2+ = 2(2)² - 2 + 2

= 6

Since both the left-hand limit and the right-hand limit of the function exist and are equal to 6, the limit of the function at x = 2 exists and is equal to 6.

3. We need to verify whether the limit of the function at x = 2 is equal to the value of the function at x = 2.

Let us calculate the value of the function at x = 2:

f(2) = 2(2)² - 2 + 2

= 8

Since the limit of the function at x = 2 is equal to the value of the function at x = 2,

we can say that the given function f(x) = 2x² - x + 2 is continuous at x = 2.

Thus, the correct option is C.

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The circle in the equation p<-18 is open. In mathematical notation, the symbol "<" represents "less than." Therefore, the inequality p<-18 means that the value of p is less than -18.

When graphing this inequality on a number line, we use an open circle to represent the endpoint, which in this case is -18. An open circle indicates that the value of p cannot equal -18.

To understand this concept, consider the inequality p<5. In this case, the graph would show an open circle at 5, indicating that p can be any value less than 5 but not equal to 5. Similarly, in p<-18, the open circle at -18 signifies that p can take on any value less than -18 but cannot be equal to -18. This distinction is crucial when interpreting inequalities and their graphs.

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The dividend is the result of multiplying the divisor and quotient and adding the remainder. In this case, the divisor is 6, the quotient is 90, and the remainder is 4.

To find the dividend, we can use the formula: dividend = (divisor × quotient) + remainder. Substituting the given values, we have: dividend = (6 × 90) + 4. Simplifying this expression, we get: dividend = 540 + 4. Adding 540 and 4, we find that the dividend is 544.

The dividend represents the total quantity or value that is being divided. In this context, if we divide the dividend (544) by the divisor (6), we would obtain the quotient of 90 with a remainder of 4. So, when dividing 544 by 6, we can expect the quotient to be 90 and a remainder of 4.

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Given that a bee flies at 12 feet per second directly to a flowerbed from its hive. The bee stays at the flowerbed for 12 minutes, and then flies directly back to the hive at 8 feet per second.

It is away from the hive for a total of 17 minutes. We are to determine the equation to find the distance of the flowerbed from the hive and the distance of the flowerbed from the hive.(a) We know that distance = speed × time. Let us use the variable d to represent the distance of the flowerbed from the hive. Using the formula distance = speed × time, the distance the bee traveled from the hive to the flowerbed is:d = 12 × 60The bee stays at the flowerbed for 12 minutes, which is equivalent to 12 × 60 seconds,

so the distance the bee traveled from the flowerbed to the hive is: d = 8 × 60To find the total distance traveled, we need to add the distance from the hive to the flowerbed to the distance from the flowerbed to the hive. The total distance is d = (12 × 60) + (8 × 60) Combining like terms gives us: d = 20 × 60Therefore, the equation that can be used to find the distance of the flowerbed from the hive is: d = 1200. (b) The distance of the flowerbed from the hive is 1200 feet since the equation used to find the distance is: d = 1200.

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7.4 inches is the length in inches of the pool on the map.

To find the length of the pool on the map in inches, we can set up a proportion using the given scale:

1/3 inch represents 15 yards

Let's denote the length of the pool on the map as "x" inches.

Using the proportion, we have:

(1/3) / 15 = x / 333

To solve for x, we can cross-multiply:

15 * x = (1/3) * 333

15x = 333/3

15x = 111

Dividing both sides by 15, we find:

x = 111 / 15

x ≈ 7.4

Therefore, the length of the pool on the map is approximately 7.4 inches.

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To determine the missing term in the perfect square trinomial, we need to look at the pattern and properties of perfect square trinomials.

A perfect square trinomial has the form (a ± b)^2 = a^2 ± 2ab + b^2. In this case, we have x^2 + 40x + 100, which fits the form of a perfect square trinomial.

We can identify the missing term by finding the square of half of the coefficient of the linear term, which in this case is 40. Half of 40 is 20, and squaring 20 gives us 400.

So, the missing term is 400. The complete perfect square trinomial is:

x^2 + 40x + 400

Therefore, the missing term in the perfect square trinomial x^2 + 40x + 100 is 400.

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To calculate the earnings per share, we need to divide the company's profit by the number of outstanding shares. The given information includes the company's profit of $117,667,008 and the share price of $56.97.

To determine the earnings per share, we need to know the number of outstanding shares. Since the number of outstanding shares is not provided in the given information, it is not possible to calculate the earnings per share with the given data alone.

The earnings per share (EPS) is calculated by dividing the company's profit by the number of outstanding shares. It represents the portion of the company's profit that is allocated to each outstanding share. By dividing the profit by the number of shares, we can determine how much profit is attributable to each individual share.

However, without the number of outstanding shares, we cannot calculate the exact earnings per share. The market capitalization and current share price do not provide enough information to determine the number of shares outstanding. Additional information, such as the number of shares issued by the company, is needed to calculate the earnings per share accurately.

In summary, the earnings per share cannot be determined with the given information alone. The calculation requires the number of outstanding shares, which is not provided. The earnings per share is a measure of the company's profitability allocated to each share, obtained by dividing the company's profit by the number of outstanding shares. To calculate the earnings per share accurately, the number of shares outstanding must be known.

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Davidson has just arrived at the foothills of the Rockies and is on a cross-country motorcycle trip. He intends to take an hour longer on the east side of the 245 km trip than on the west side of the 225 km trip. On the downhill side, he intends to average 20 km/h more to achieve this. The trip through the Rockies will take approximately 17.375 hours.

Let the speed of the motorcycle on the west side of the trip be x km/h.

So, the time required to complete the 225 km trip will be:

Time for the west side of the trip = 225/x

Let the speed of the motorcycle on the east side of the trip be x + 20 km/h.

So, the time required to complete the 245 km trip will be:

Time for the east side of the trip = 245 / (x + 20)

We know that the time Davidson takes on the east side of the trip will be an hour longer than on the west side. Therefore, we can form the following equation:

245/(x + 20) = 225/x + 1

Multiplying both sides by x(x + 20),

we get:245x = 225(x + 20) + x(x + 20)

Simplifying the equation:20x = 400x = 20 km/h

Time taken on the west side of the trip is 225/20 = 11.25 hours

Time taken on the east side of the trip is 245/40 = 6.125 hours

So the total time for the trip through the Rockies is 11.25 + 6.125 = 17.375 hours, or about 17 hours and 22.5 minutes (rounded to the nearest minute).

Therefore, the trip through the Rockies will take approximately 17.375 hours.

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In 175 graduants, 100 could be expected to seek higher education

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

Rate = 4 out of every 7 graduating seniors

Graduating seniors = 175

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

Higher education seeker = Rate * Graduating seniors

Substitute the known values in the above equation, so, we have the following representation:

Higher education seeker = 4/7 * 175

Evaluate

Higher education seeker = 100

Hence, 100 could be expected to seek higher education

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To find out the total combined area of the living room and deck of the Bain's house, we first need to know the area of the living room and the deck. Once we have found out the areas of both, we can then add them up to get the total combined area.

Area of the living room: The area of a rectangle is calculated by multiplying its length by its width. If the length and width of the living room are 20 feet and 15 feet respectively, then the area of the living room will be: Area of the living room = Length × Width= 20 ft × 15 ft= 300 ft²Area of the deck: The area of a rectangle is calculated by multiplying its length by its width. If the length and width of the deck are 12 feet and 10 feet respectively, then the area of the deck will be: Area of the deck = Length × Width= 12 ft × 10 ft= 120 ft²Total combined area of the living room and deck: Now that we know the area of the living room and the deck, we can add them together to get the total combined area of the living room and deck .Total combined area of the living room and deck= Area of the living room + Area of the deck= 300 ft² + 120 ft²= 420 ft²Therefore, the total combined area of the living room and deck of the Bain's house is 420 square feet.

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By using double grouping, the expression can be factored as (x + 7)(x - 6).

To factor the expression x^2 + x - 42, an x-method chart is used to determine the factors. The completed chart shows 1 at the bottom of x, -42 at the top of x, 7 on the left side, and -6 on the right side.

The x-method chart is a helpful tool for factoring quadratic expressions. The completed chart provides us with the necessary information to factor the expression x^2 + x - 42. The product of -42 at the top of x and 1 at the bottom of x tells us that the factors of -42 are -6 and 7.

To factor the expression, we can use double grouping. We group the terms x and 7 together, as well as the terms x and -6 together. This gives us x(x + 7) - 6(x + 7). Notice that both groups have a common factor of (x + 7). We can factor out this common factor to obtain (x + 7)(x - 6).

To verify the factors, we can use the distributive property to multiply the factors back together. When we multiply (x + 7)(x - 6), we get x^2 + x - 6x - 42. Simplifying further, we have x^2 - 5x - 42, which is equivalent to the original expression x^2 + x - 42. Therefore, (x + 7)(x - 6) is the correct factored form of the given expression.

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The 27th term of the arithmetic sequence with the first term [tex]\(a_1 = -13\)[/tex] and a common difference of 4 is 91.

To find the 27th term of an arithmetic sequence, we can use the formula:

[tex]\[a_n = a_1 + (n - 1)d\][/tex]

where [tex]\(a_n\)[/tex] represents the [tex]\(n\)[/tex]th term, [tex]\(a_1\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] is the term number.

Given that [tex]\(a_1 = -13\)[/tex] and the common difference [tex]\(d = 4\)[/tex], we will simply substitute these values into the given formula:

[tex]\[a_{27} = -13 + (27 - 1) \cdot 4\][/tex]

Simplifying the equation, we have:

[tex]\[a_{27} = -13 + 26 \cdot 4\][/tex]

Calculating the expression, we get:

[tex]\[a_{27} = -13 + 104\][/tex]

Finally, evaluating the sum, we find:

[tex]\[a_{27} = 91\][/tex]

Therefore, the 27th term of the arithmetic sequence with the first term [tex]\(a_1 = -13\)[/tex] and a common difference of 4 is 91.

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