Yesterday, a movie theater sold 266 bags of popcorn. A large bag of popcorn costs $4. A small bag


of popcorn costs $1. In all, the movie theater made $437 from popcorn sales. Write and solve a system


of equations to find how many bags of each size popcorn were sold. WRITE VARIABLE STATEMENT (LET)


AND EQUATION ONLY.

Janice is incorrect in claiming that there will be over 200,000 bacteria 20 hours after their population reached 9,500 in an observational study.

In an exponential growth model, the population of bacteria follows a pattern where it increases by a fixed proportion over a constant time interval. To determine if Janice's claim is correct, we need to calculate the growth rate of the bacteria and project it forward 20 hours from the time the population reached 9,500.

To find the growth rate, we can use the formula for exponential growth: N = N0 * e^(rt), where N is the final population, N0 is the initial population, r is the growth rate, and t is the time in hours. We know that N0 = 6,000 and N = 9,500 after 3 hours.

Rearranging the formula to solve for r, we have r = ln(N/N0) / t. Plugging in the values, we get r = ln(9,500/6,000) / 3 ≈ 0.193. Therefore, the growth rate is approximately 0.193 per hour.

Now, let's project the population 20 hours after reaching 9,500. Using the formula N = N0 * e^(rt) and plugging in the values, we have N = 9,500 * e^(0.193 * 20) ≈ 144,168. This indicates that there will be around 144,168 bacteria after 20 hours, which is significantly lower than Janice's claim of over 200,000 bacteria. Therefore, Janice's statement is incorrect based on the given information and calculations.

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The inequality that best describes w, the weight in pounds of Erin’s backpack after she packs it is w + 5 < 25. Explanation:

Let's assume the weight of Erin’s backpack as w and the weight of the cooking kit as 5 pounds.

The backpack must weigh less than 25 pounds when packed.

So, the inequality can be written as:w + 5 < 25 (as the cooking kit weighs 5 pounds)Subtracting 5 from both sides,w < 20So, the weight of Erin's backpack should be less than 20 pounds,

otherwise it will exceed the limit of 25 pounds.

Therefore, the inequality that best describes w, the weight in pounds of Erin’s backpack after she packs it is w + 5 < 25.

Therefore, the weight of Erin's backpack should be less than 20 pounds to ensure it does not exceed the limit of 25 pounds.

Hence, the inequality that best describes w, the weight in pounds of Erin's backpack after she packs it, is w + 5 < 25.

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Aiden's money would take about 16.28 years to double, while Olivia's money would take about 16.15 years. So, it would take Aiden around 0.13 years longer.



To determine the time it takes for the investments to double, we can use the compound interest formula:

Future Value = Principal * (1 + Interest Rate)^Time

For Aiden's investment, the interest is compounded annually, so the formula becomes:

2 * 600 = 600 * (1 + 0.0425)^Time

Simplifying the equation:

2 = (1.0425)^Time

For Olivia's investment, the interest is compounded daily, so the formula becomes:

2 * 600 = 600 * (1 + 0.0475/365)^(365*Time)

Simplifying the equation:

2 = (1 + 0.0475/365)^(365*Time)

Let's start with Aiden's investment:

2 = (1.0425)^Time

Taking the natural logarithm of both sides:

ln(2) = ln(1.0425)^Time

Time * ln(1.0425) = ln(2)

Time = ln(2) / ln(1.0425)

Using a calculator:

Time ≈ 16.28 years

Now let's calculate the time for Olivia's investment:

2 = (1 + 0.0475/365)^(365*Time)

Taking the natural logarithm of both sides:

ln(2) = ln[(1 + 0.0475/365)^(365*Time)]

Time * ln[(1 + 0.0475/365)] = ln(2) / (365 * ln[(1 + 0.0475/365)])

Using a calculator:

Time ≈ 16.15 years

Therefore, it would take approximately 0.13 years (or around 1.56 months) longer for Aiden's money to double compared to Olivia's money.

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After 5 years, the population is expected to double if the growth pattern continues.

To determine after how many years the population will double, we need to consider the growth rate and the initial population.

If the population is increasing every year, it indicates exponential growth. In this case, we can use the formula for exponential growth:

P(t) = P(0) * (1 + r)^t

Where:

P(t) is the population at time t,

P(0) is the initial population,

r is the growth rate per year, and

t is the number of years.

In this scenario, we know that the initial population (P(0)) is 1200. We want to find the value of t when the population (P(t)) doubles, which means it will be 2 * P(0).

2 * P(0) = P(0) * (1 + r)^t

Dividing both sides by P(0):

2 = (1 + r)^t

Now, let's check the given options to determine which one satisfies the equation:

01: (1 + r)^1 = 2 (Not satisfied)

02: (1 + r)^2 = 2 (Not satisfied)

03: (1 + r)^3 = 2 (Not satisfied)

05: (1 + r)^5 = 2 (Satisfied)

So, after 5 years, the population is expected to double if the growth pattern continues.

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IIf the two linear functions are given, we can find their slopes and determine which one has the greater slope (m-value). That function will have the greater range of change. Without the functions, it is not possible to determine which function has the greater range of change.

The two linear functions are not shown below, please provide them or a detailed description of them to determine the function that has the greater range of change.

Let's recall the definition of linear functions. A linear function is a function whose graph is a straight line. Linear functions are known to have a constant rate of change. When written in the slope-intercept form, the equation of a linear function appears as

y = mx + b,

where m represents the slope of the line and b represents the y-intercept. This equation can also be written as

f(x) = mx + b.

Linear functions have a set of specific features that make them easy to analyze and graph, such as domain and range. The domain is the set of all possible x-values, while the range is the set of all possible y-values for a function. Because linear functions are straight lines, they have an infinite domain and range unless there is some other restriction present in the problem statement.The function that has the greater range of change is the one with the greater slope (m-value). This is because the slope determines how fast or slow the line is rising or falling, and thus how much the y-value changes for each unit increase in the x-value. The greater the slope, the greater the change in y for a given change in x.

Therefore, if the two linear functions are given, we can find their slopes and determine which one has the greater slope (m-value). That function will have the greater range of change. Without the functions, it is not possible to determine which function has the greater range of change.

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In an isosceles triangle, the angles opposite the equal sides are congruent, thus, angle JML≈= angle LKJ

Since JM is perpendicular to ML whereas JK is perpendicular to KL, the congruence of triangles JMK and LMK can be deduced using the Side-Angle-Side (SAS) congruence principle.

This suggests that angle JKM and angle LKM are equivalent in measure. In other words, JK is almost the same length as ML, suggesting that the LKJ triangle is nearly an isosceles triangle.

If a triangle has two equal sides, then the angles facing those sides are also identical. Consequently, angle KJL is nearly identical to angle LJK. Given the nature of transitivity.

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Using a line of best fit with the equation Y equals negative 3X +21 to predict the value of Y when X equals 10 is an example of a linear regression.

A linear regression is a statistical technique used to model the relationship between two variables by fitting a linear equation to the observed data. It is used to determine the strength and direction of the relationship between the two variables .Linear regression is widely used in business, finance, economics, and other fields to predict future trends and make informed decisions.

The equation Y equals negative 3X +21 represents the slope and intercept of the line of best fit for the given data set with integer X-values One through six. By substituting X equals 10 in the equation, we can predict the value of Y when X equals 10 as follows :Y = -3(10) + 21 = -30 + 21 = -9Therefore, the predicted value of Y when X equals 10 is -9.

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The perimeter of rectangle is given by 2 * (length * breadth). The other side of the rectangle labeled x is 120 - 3x.

Let's assume that the rectangle has sides of length L and W. We know that the perimeter of the fenced area is 260 meters, which means that:

2L + 2W = 260

Simplifying this equation, we can divide both sides by 2 to get:

L + W = 130

We also know that one of the sides of the rectangle is labeled 3x + 10. Let's assume that this side is equal to L, so we have:

L = 3x + 10

We can use this equation to substitute L in the previous equation, so we get:

(3x + 10) + W = 130

Simplifying this equation, we can subtract 10 from both sides:

3x + W = 120

Now we have two equations:

L + W = 130

3x + W = 120

We can use the second equation to solve for W in terms of x:

W = 120 - 3x

So the other side labeled x is equal to W, which is:

W = 120 - 3x

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The answer is Option C) The annual interest rate is 3.3%. given that you deposited ​$200 in a savings account 2 years ago and the simple interest rate is 3.3%.

The interest earned in those 2 years is ​$13.20.

Then, we have to choose the true statements from the options below.

Option A) The current balance of the account is $226.40.

Option B) The annual interest rate is 6.6%.Option C) The annual interest rate is 3.3%.Option D) The interest rate for the second year is 3.3%.

Solution:We know that the formula for simple interest is:

Simple Interest (I) = (P × R × T) / 100

Where P is the Principal (amount), R is the rate of interest and T is the time period given.

Let’s first calculate the rate of interest earned per year.

r = (I * 100) / P * t

Rate = (13.2 * 100) / 200 * 2Rate = 6.6%

Therefore, the annual interest rate is 6.6%

Now, we can find the current balance of the account using the formula:

Amount = P + I

Amount = 200 + 13.20

Amount = 213.20

Hence, the statement A is false.

So, the option A is not true.The statement B is false because the annual interest rate is 3.3% only.The statement C is true as it is already found that the annual interest rate is 3.3%.The statement D is false as we have already calculated that the annual interest rate is 6.6%.

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Duplicating an angle involves constructing an angle that has the same measure as the original angle. It can be done using a compass and a straightedge, following specific constructions. On the other hand, a line segment duplication refers to creating an identical copy of a given line segment, typically by using a ruler or other measuring tools.

Duplicating an angle is based on the concept of angle congruence, where two angles are considered congruent if they have the same measure. To duplicate an angle, one can use a compass to transfer the angle's measure onto another location or construct a new angle with the same measure using geometric constructions.

In contrast, duplicating a line segment refers to creating an exact copy of a given line segment. This can be achieved by using a ruler or other measuring tools to measure the length of the segment and then transferring that length to a different location. The resulting line segment will have the same length as the original.

In summary, duplicating an angle involves reproducing an angle with the same measure, while duplicating a line segment entails creating an identical copy of a given line segment in terms of its length. The methods and tools used for these operations differ, as duplicating an angle relies on geometric constructions, while duplicating a line segment involves measurement and replication techniques.


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The quotient obtained when dividing the least common multiple (LCM) of 8, 12, and 3 by their highest common factor (HCF) is 6.

To find the LCM of 8, 12, and 3, we first determine their prime factorizations:

- 8 can be expressed as [tex]2^3[/tex].

- 12 can be expressed as [tex]2^2 * 3^1[/tex].

- 3 is already a prime number.

To calculate the LCM, we consider the highest power of each prime factor that appears in any of the numbers. In this case, we have:

- [tex]2^3 * 3^1[/tex], as the highest powers of 2 and 3.

Multiplying these prime factors together, we get [tex]2^3 * 3^1 = 8 * 3 = 24[/tex].

Next, we find the HCF of the three numbers. The HCF is the highest power of each prime factor that is common to all the numbers. In this case, we have:

- The highest power of 2 is [tex]2^2[/tex].

- The highest power of 3 is [tex]3^0[/tex] (as it is not a factor of 8).

Multiplying these prime factors together, we get [tex]2^2 * 3^0 = 4 * 1 = 4[/tex].

Finally, we divide the LCM by the HCF to find the quotient: 24 / 4 = 6.

Therefore, the quotient obtained when dividing the LCM of 8, 12, and 3 by their HCF is 6.

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To represent how much more Company X charges than Company Y for n minutes, we can subtract the total cost of Company Y from the total cost of Company X. The expression that represents this is (35 + 5n) - (20 + 4n) cents. Simplifying this expression yields 15 + n cents.

To find how much more Company X charges than Company Y for n minutes, we need to calculate the difference in their total costs. For Company X, it costs 35 cents to connect and an additional 5 cents for each minute, resulting in a total cost of (35 + 5n) cents for n minutes. For Company Y, it costs 20 cents to connect and an additional 4 cents for each minute, giving a total cost of (20 + 4n) cents for n minutes.

To find the difference, we subtract the total cost of Company Y from the total cost of Company X: (35 + 5n) - (20 + 4n) cents. Simplifying this expression, we combine like terms and get 15 + n cents. Therefore, the expression (15 + n) represents how much more Company X charges than Company Y for n minutes.

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The linear model with a slope of 2 grows at a constant rate over time. This means that it grows at a consistent pace, regardless of the initial value.

On the other hand, the exponential function with a growth rate of 10% grows at an increasing rate over time. Initially, the exponential model grows faster than the linear model because its growth rate is proportional to its current value.

However, as time goes on, the exponential model continues to accelerate its growth due to the compounding effect of the exponential function. Eventually, the exponential growth surpasses the linear growth, leading to a much faster growth rate.

For short-term investment, the linear model may be preferable as it provides a more predictable and stable growth pattern. The linear model's consistent rate of growth makes it easier to estimate and plan for returns in the short term.

For long-term investment, the exponential model would be better suited. The compounding effect of the exponential growth results in exponential returns over time. This can lead to significantly higher gains compared to the linear model. However, it's important to note that the exponential model may also carry more risk and volatility due to its accelerating growth rate.

Overall, the choice between a linear or exponential model depends on the specific investment goals, risk tolerance, and time horizon. The linear model offers stability and predictability in the short term, while the exponential model offers the potential for higher long-term returns, albeit with increased risk.

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The next two numbers in Antonio's pattern would be 96 and 102. To identify the next two numbers in Antonio's pattern, let's analyze the given sequence: 72, 78, 84, and 90.

Determine the difference between consecutive numbers: By subtracting each number from its preceding number, we find a consistent difference of 6.

Apply the difference to the last number: Add 6 to the last number in the sequence, 90. This gives us the next number: 90 + 6 = 96.

Repeat step 2: Add 6 to the newly obtained number, 96. This gives us the second number: 96 + 6 = 102.

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The perimeter of the polygon with side lengths 12, 12, and 8 is 32 inches. Option C

To calculate the perimeter of a polygon, we need to add up the lengths of all its sides. In this case, the polygon has three sides with lengths 12, 12, and 8 inches. To find the perimeter, we add these lengths together: 12 + 12 + 8 = 32 inches. Therefore, the correct answer is option C: 32 in.

The perimeter represents the total distance around the polygon, so it gives us an idea of the length of the boundary of the shape. In this case, the polygon has two sides of length 12 inches and one side of length 8 inches. Adding these lengths together gives us the total distance around the polygon, which is 32 inches.

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The predicted value of y is 53 for the given equation of the trend line.

Given equation of trend line is Yx = 3 + 10x.

We are supposed to determine the predicted value of y if x is 5.

To find the value of y, we have to substitute the value of x in the given equation:

Yx = 3 + 10x

When x = 5,

Then Yx = 3 + 10(5)

           = 3 + 50 = 53

So, the predicted value of y is 53. Therefore, 53 is the correct answer.

Trend lines are drawn at an angle and are used to determine a trend and help making trading decision.

A trend line should be connected by at least three highs or lows to make it valid.

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No, AB is not parallel to CD. No, AB is not parallel to CD. No, AB is not parallel to CD.

To determine if two lines are parallel, we need to compare their slopes. If the slopes are equal, then the lines are parallel; otherwise, they are not.

For the points A (-5, 12) and B (7, 18), the slope of AB can be calculated as (18 - 12) / (7 - (-5)) = 6 / 12 = 0.5.

For the points C (0, -4) and D (-8, 0), the slope of CD can be calculated as (0 - (-4)) / (-8 - 0) = 4 / -8 = -0.5.

Since the slopes of AB and CD are not equal (0.5 ≠ -0.5), AB is not parallel to CD.

For the points A (-6, 2) and B (4, 6), the slope of AB can be calculated as (6 - 2) / (4 - (-6)) = 4 / 10 = 0.4.

For the points C (7, -4) and D (-3, -8), the slope of CD can be calculated as (-8 - (-4)) / (-3 - 7) = -4 / -10 = 0.4.

Since the slopes of AB and CD are equal (0.4 = 0.4), AB is parallel to CD.

For the points A (-6, 2) and B (4, 6), the slope of AB can be calculated as (6 - 2) / (4 - (-6)) = 4 / 10 = 0.4.

For the points C (7, -4) and D (-4, -8), the slope of CD can be calculated as (-8 - (-4)) / (-4 - 7) = -4 / -11 = 0.364.

Since the slopes of AB and CD are not equal (0.4 ≠ 0.364), AB is not parallel to CD.

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The multiplication and division equations to represent the problem can be written as 3x = 245 and x = 245/3.

We know that Finn has 245 days to complete three parts of a construction project, and he wants to spend the same number of days on each part. Therefore, let x be the number of days he will spend on each part. Now, since he has to spend the same amount of days on each part, the total number of days will be 3x.

Therefore, we can write a multiplication equation as 3x = 245, which represents the total number of days. Finn has for completing the three parts of the project by spending equal amounts of time on each part. The division equation to represent the problem can be found by dividing both sides of the above equation by 3, which gives us x = 245/3. This equation represents the number of days. Finn will spend on each part of the project, provided he wants to complete all three parts in 245 days.

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Public opinion polls use sample sizes of more than 1000 people because it helps to ensure that the results are statistically significant. This means that the results are likely to be accurate and representative of the population as a whole.

The margin of error for a poll is the amount by which the results of the poll could be off from the actual results. The margin of error is calculated using the sample size, the confidence level, and the standard deviation of the population.

The confidence level is the probability that the results of the poll are within the margin of error. The standard deviation of the population is a measure of how spread out the opinions of the population are.

The larger the sample size, the smaller the margin of error will be. This is because the larger the sample size, the more likely it is that the sample will be representative of the population as a whole.

For example, if a poll has a sample size of 1000 people and a confidence level of 95%, the margin of error will be +/- 3%. This means that there is a 95% chance that the results of the poll are within 3% of the actual results.

If a poll has a sample size of 500 people and a confidence level of 95%, the margin of error will be +/- 4.4%. This means that there is a 95% chance that the results of the poll are within 4.4% of the actual results.

As you can see, the larger the sample size, the smaller the margin of error will be. This is why public opinion polls use sample sizes of more than 1000 people. It helps to ensure that the results are statistically significant and that they are likely to be accurate and representative of the population as a whole.

The present age of the pet mouse is 3 years, and the present age of the owner is 10 years.

Age of pet mouse and owner in three years will be the same as the owner's age four years ago. Find their present ages.

Let's assign variables to represent the present ages of the pet mouse and the owner. We'll use "M" for the mouse's age and "O" for the owner's age.

According to the problem, in three years' time, the pet mouse will be as old as the owner was four years ago. This can be expressed as:

M + 3 = O - 4

Their present ages total 13 years, so we can write another equation:

M + O = 13

Now we have a system of two equations with two variables. We can solve it using substitution or elimination.

Let's solve it by substitution:

From the first equation, we can isolate M:

M = O - 7

Substituting this value of M into the second equation:

(O - 7) + O = 13

2O - 7 = 13

Adding 7 to both sides:

2O = 20

Dividing both sides by 2:

O = 10

Now that we know the owner's age is 10, we can substitute this value back into one of the original equations to find the mouse's age:

M + 10 = 13

M = 3

Therefore, the present age of the pet mouse is 3 years, and the present age of the owner is 10 years.

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the train's average speed was approximately 68 miles per hour.

To calculate the average speed of the train, we need to divide the distance traveled by the time taken.

Given:

Distance: 170 miles

Time taken: 4:18 PM - 1:48 PM = 2 hours and 30 minutes

To calculate the average speed, we first need to convert the time taken to hours. Since there are 60 minutes in an hour, we have:

Time taken = 2 hours + 30 minutes / 60 = 2.5 hours

Now we can calculate the average speed:

Average speed = Distance / Time taken

Average speed = 170 miles / 2.5 hours

Average speed = 68 miles per hour (rounded to the nearest whole number)

Therefore, the train's average speed was approximately 68 miles per hour.

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The ratio of the number of teams that wear red jerseys is 35% of the teams wear red jerseys.

The percentage of teams that wear red jerseys can be found by dividing the number of teams that wear red jerseys by the total number of teams and then multiplying by 100. In this case, since the ratio of teams wearing red jerseys to teams wearing blue jerseys is 7:13, the percentage of teams wearing red jerseys is (7 / (7 + 13)) * 100 = 35%.

To calculate the percentage, we first add the two parts of the ratio (7 + 13 = 20) to get the total number of teams. Then, we divide the number of teams wearing red jerseys (7) by the total number of teams (20) and multiply by 100 to express it as a percentage. Therefore, 35% of the teams wear red jerseys.

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To find out the original price of the article, we can use the formula:

Original price = (Selling price × 100) / (100 + Profit%)

Let's put the given values in the formula:

Profit% = 10%Selling price

           = Birr 330

Original price = (330 × 100) / (100 + 10)

Original price = (330 × 100) / 110

Original price = Birr 300

So, the original price of the article was Birr 300. The dealer gained a 10% profit by selling it for Birr 330. Profit is the money earned by the dealer above the cost of the article. Here, the profit earned is 10% of the original cost of the article.

The formula to find out the profit is:

Profit = Selling price - Cost price

Let's put the values in the formula:

Profit = 330 - 300

Profit = Birr 30

Therefore, the dealer earned a profit of Birr 30.

To summarize, the original price of the article was Birr 300 and the dealer earned a profit of Birr 30 by selling it for Birr 330.

The original price of the article was Birr 300 and the dealer earned a profit of Birr 30 by selling it for Birr 330. The formula used to find the original price is:

Original price = (Selling price × 100) / (100 + Profit%) and the formula to find the profit is: Profit = Selling price - Cost price.

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In an investigation on accuracy and precision, Myriah measures the mass of a 10. 00 g piece of metal several times. Her measurements are 9. 98 g, 10. 02 g, 10. 01 g, and 9. 99 g. The statement that best describes Myriah's results is that they are precise but not accurate.

Accuracy refers to how close a measured value is to the true or accepted value. Precision refers to the degree of agreement among repeated measurements. The mean of her measurements is: (9.98 g + 10.02 g + 10.01 g + 9.99 g) / 4 = 9.25 g. The results are precise because they are close together, and the mean value was calculated to the nearest hundredth of a gram. The measurements, on the other hand, were not accurate because they were all significantly less than the true value of 10.00 g.

To find the measure of angle A, place the protractor so that the base of the angle is aligned with the bottom line of the protractor. Then, read the measurement where the other side of the angle intersects the protractor. In this case, angle A intersects the protractor at 45 degrees. To find the measure of angle B, place the protractor so that the base of the angle is aligned with the bottom line of the protractor. Then, read the measurement where the other side of the angle intersects the protractor. In this case, angle B intersects the protractor at 120 degrees. Therefore, the measure of angle A is 45 degrees, while the measure of angle B is 120 degrees.

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The two numbers can be represented by the quadratic equation (x + 1)² = x² - 23.

Let's assume the smaller number is x. According to the problem, the second number is 1 more than the first, so the larger number can be represented as x + 1.

The problem states that the square of one number (x²) is 23 less than the square of the second number. Mathematically, this can be represented as:

(x + 1)² = x² - 23

Expanding the left side of the equation:

x² + 2x + 1 = x² - 23

Next, we can simplify the equation by canceling out the common terms:

2x + 1 = -23

Solving for x, we subtract 1 from both sides:

2x = -24

Dividing by 2:

x = -12

Therefore, the smaller number is -12, and the larger number is -12 + 1, which is -11.

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The equation z = xy^2 can be rearranged to find y as y = z / x. This formula allows us to express y in terms of x and z, enabling us to determine y using given x and z values.

In the equation z = xy2, we can rearrange the equation to find y and then express y in terms of x and z. Let's go over each step:

1. Begin by solving the equation z = xy2. 2. To find the y2 term, divide both sides by x: z/x = y2.

3. To find y, take the square root of both sides: (z/x) = y.

4. Condense the formula to y = z / x.

As a result, y may be represented as y = z / x in terms of x and z. Using the x and z values as inputs, this equation enables us to determine the value of y. Remember that depending on the values of z and x, taking the square root may result in both positive and negative results.

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The algebraic expression for "one third of Danielle's height" is (1/3)h, where h represents Danielle's height.

In algebraic expressions, variables are used to represent unknown quantities. In this case, we are given the phrase "one third of Danielle's height," which means we need to find an expression that represents a third of Danielle's height. Let's use the variable h to represent Danielle's height.

To express "one third of Danielle's height" algebraically, we can multiply the fraction 1/3 by the variable h. This results in the expression (1/3)h, where (1/3) represents one third and h represents Danielle's height. The expression (1/3)h indicates that we are taking a fraction of h, specifically one third of h.

By using this algebraic expression, we can easily calculate one third of Danielle's height by substituting the value of h into the expression. For example, if Danielle's height is 150 centimeters, we can substitute h = 150 into the expression (1/3)h and evaluate it as (1/3) * 150 = 50 centimeters. Therefore, the expression (1/3)h provides a concise and flexible way to represent one third of Danielle's height in algebraic terms.

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RS = SP - 7, PQ = RS = QR = SP Perimeter of ▱PQRS = 124We are supposed to find the length of each side of ▱PQRS under the given conditions. Therefore, let us use the formula of the perimeter of a polygon (quadrilateral) to find the length of each side of the polygon.

The formula of the perimeter of a polygon is given by: P = a + b + c + d Where P is the perimeter and a, b, c, and d are the length of each side of the polygon. Let's substitute the given values in the above formula and solve it: P = PQ + QR + RS + SP124 = PQ + QR + RS + SP... (Equation 1)Now, we know that: PQ = RS = QR = SP We can rewrite the equation 1 as follows:124 = PQ + PQ + PQ - 7 + PQ + PQ - 7

Simplifying, we get;124 = 5PQ - 14Adding 14 on both sides;124 + 14 = 5PQ138 = 5PQDividing by 5 on both sides;138/5 = PQPQ = 27.6Now, we know that; PQ = RS = QR = SP We can substitute the value of PQ in any of the above equations. Let's use the equation PQ = RS. Then; RS = 27.6Using the equation; PQ = SPSP = 27.6QR = 27.6Using the equation; RS = SP - 7SP = RS + 7SP = 27.6 + 7 = 34.6Therefore, the length of each side of ▱PQRS is: PQ = 27.6, QR = 27.6, RS = 27.6, SP = 34.6

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The volume of the balloon before the addition of the extra gas was 7.01 liters.

We will use ideal gas law equation, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant and T is the temperature.

Given data:

Initial number of moles of helium in the balloon (before addition) = 4.51 moles

Additional number of moles of helium added = 1.25 moles

The total number of moles of helium in the balloon (after addition) is:

= 4.51 moles + 1.25 moles

= 5.76 moles

Volume of the balloon after addition = 8.97 liters

To find the volume of the balloon before the addition, we can rearrange the ideal gas law equation to solve for V:

V = (nRT)/P

The volume before the addition will be found using: V_initial = (n_initial * V_final) / n_final

V_initial = (4.51 * 8.97) / 5.76

V_initial = 7.02 liters.

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The area of a towelette with double the base and double the height is 12 square inches.

A triangular towelette with a base of 2 inches and a height of 3 inches has an area of 3 square inches. If the base and height of the triangular towelette double, the area will quadruple (increase by a factor of 4) since the area of a triangle is proportional to the product of its base and height.Area of the original towelette is given by:(1/2)bh= (1/2)×2×3= 3 square inchesLet the new base and height be b and h respectively.

The new area of the towelette is given by:(1/2)bh= (1/2)×2(2)×3(2)= 12 square inchesTherefore, the area of a towelette with double the base and double the height is 12 square inches.

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