Traffic Jam
There are 8 cans of strawberry jam, 7 raspberry jam,
and 5 cherry jam in the cellar. You're trying to sneak
some out, but don't want to attract attention or take
too many. It's dark, so you can't tell what kind of jam
you're taking.
How many cans can you sneak out of the
cellar in the dark with the certainty that there
will still be at least 4 cans of one kind of jam
and 3 cans of another left over?

Critical numbers: 0, -3, 3
Increasing intervals: (-∞, -3), (3, ∞)
Decreasing intervals: (-3, 0), (0, 3)

Let's analyze the given function and find the critical numbers, as well as the intervals where it is increasing or decreasing.

Function: f(x) = x^2 / (x^2 - 9)

1. Find the critical numbers:
To find the critical numbers, we need to compute the first derivative of the function and set it equal to zero or identify where it's undefined.

f'(x) = (2x(x^2 - 9) - x^2(2x)) / (x^2 - 9)^2 = (2x^3 - 18x - 2x^3) / (x^2 - 9)^2 = -18x / (x^2 - 9)^2

Setting f'(x) equal to zero:
-18x = 0
x = 0

Now, let's check where f'(x) is undefined:
x^2 - 9 = 0
x = ±3

Thus, the critical numbers are 0, -3, and 3.

2. Find the open intervals where the function is increasing or decreasing:
To determine the intervals, we will examine the sign of f'(x) in the intervals determined by the critical numbers: (-∞, -3), (-3, 0), (0, 3), (3, ∞).

Interval (-∞, -3): Choose x = -4, then f'(-4) > 0. So, f(x) is increasing on this interval.

Interval (-3, 0): Choose x = -1, then f'(-1) < 0. So, f(x) is decreasing on this interval.

Interval (0, 3): Choose x = 1, then f'(1) < 0. So, f(x) is decreasing on this interval.

Interval (3, ∞): Choose x = 4, then f'(4) > 0. So, f(x) is increasing on this interval.

Your answer:
Critical numbers: 0, -3, 3
Increasing intervals: (-∞, -3), (3, ∞)
Decreasing intervals: (-3, 0), (0, 3)

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Answer:

72 + 7x + 24 = 180

7x + 96 = 180

7x = 84

x = 12

The intercept of 5.3 suggests that even if a mother had no income, the expected minimum fathers' income would be $5.30.

In mathematical terms, the equation y=1.23x+5.3 is in slope-intercept form, where y represents the fathers' incomes and x represents the mothers' incomes. The slope of the line, 1.23, represents the change in fathers' incomes for every one unit increase in mothers' incomes. The y-intercept of the line, 5.3, represents the minimum fathers' income when the mothers' income is zero.

With this equation, we can make several conclusions about this sample. Firstly, the positive slope suggests a positive correlation between the incomes of mothers and fathers. As the mothers' incomes increase, the fathers' incomes also tend to increase.

Secondly, the slope value of 1.23 suggests that fathers' incomes increase by $1.23 for every $1 increase in mothers' incomes.

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If the scatter plot shows no discernible relationship between the variables and the points appear randomly scattered, then the r-value of 0.01 is accurate. However, if there is a visible relationship, the value of r may need to be recalculated or checked for errors in data input or analysis.

To find whether a correlation coefficient of r = 0.01 is an accurate value for the data based on the scatter plot, it's essential to consider the following factors:

1. Visual inspection of the scatter plot: Observe the overall pattern of data points in the scatter plot. If the points seem randomly scattered with no discernible pattern, then a correlation coefficient close to 0, such as r = 0.01, would be accurate. However, if there is a clear linear or non-linear relationship between the variables, the value of r = 0.01 may not be accurate.

2. Strength of the relationship: The correlation coefficient r ranges from -1 to 1, where -1 represents a strong negative relationship, 0 represents no relationship, and 1 represents a strong positive relationship. An r-value of 0.01 indicates a very weak or no relationship between the variables. Confirm that this is consistent with the scatter plot pattern.

To determine if the r-value of 0.01 is accurate for the data, carefully examine the scatter plot and consider these factors. If the scatter plot shows no discernible relationship between the variables and the points appear randomly scattered, then the r-value of 0.01 is accurate. However, if there is a visible relationship, the value of r may need to be recalculated or checked for errors in data input or analysis.

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The difference between the costs of a hovercraft and a boat in the year 2010 was approximately $1,558.

(a) The difference in costs between a hovercraft and a boat can be expressed as follows:

Cost difference = Hovercraft cost - Boat cost

Cost difference = (-1012 + 350t + 400) - (30 + 140t)

Cost difference = -1012 + 350t + 400 - 30 - 140t

Cost difference = 220t - 642

Therefore, the difference in costs between a hovercraft and a boat can be expressed as 220t - 642.

(b) To find the difference in the year 2010, we need to substitute t = 10 in the expression we derived in part (a):

Cost difference = 220t - 642

Cost difference = 220(10) - 642

Cost difference = 2200 - 642

Cost difference = 1558

Therefore, the difference between the costs of a hovercraft and a boat in the year 2010 was approximately $1,558.

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Since we need to round our answer to the nearest whole number, the insect population after 5 days will be approximately 367 insects.

Given the rate at which the insect population increases, we can determine the population after 5 days using the given formula: 280 + 64 + 0.9t^2 insects per day.
First, we need to substitute t with the number of days, which is 5:

280 + 64 + 0.9(5)^2

Now, calculate the value of the equation:

280 + 64 + 0.9(25) = 280 + 64 + 22.5 = 366.5

OR,

The population of insects after 5 days can be found by plugging in t = 5 into the given equation:

Population = 280 + 64 + 0.9(5)^2
Population = 280 + 64 + 22.5
Population = 366.5

Rounding to the nearest insect, the population after 5 days is 367 insects.

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If a scale model of the plane is 40.6 cm long, the scale factor used to create the model is approximately 1:142.857 or 0.007 rounded to the nearest thousandth.

To determine the scale factor used to create the model, we need to divide the length of the actual airplane by the length of the model airplane:

58 m / 40.6 cm

To perform this calculation, we need to convert the units so that they match. We can convert 58 m to cm by multiplying by 100:

58 m = 58 × 100 cm = 5800 cm

Now we can divide:

5800 cm / 40.6 cm = 142.857...

Rounding to the nearest thousandth, we get:

142.857... ≈ 142.857

Therefore, the scale factor used to create the model is approximately 1:142.857 or 0.007 rounded to the nearest thousandth.

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Point estimate of μ  [tex]26.15[/tex]

99% confidence interval: [tex](24.213, 28.087)[/tex]

Margin of error:[tex]1.437[/tex]

The point estimate of  (population mean) is calculated by finding the sample mean , which is the average of the given sample data. Adding up all the observations and dividing by the sample size (18), we get:

To make a 99% confidence interval for we can use the t-distribution and the sample mean. Since the sample size is 18, we have (18 - 1) = 17 degrees of freedom. Using a t-distribution table or a calculator, we find the critical value for a 99% confidence level and 17 degrees of freedom to be approximately 2.898.

Next, we calculate the standard error (SE) using the sample standard deviation (s) divided by the square root of the sample size .Assuming the population standard deviation is unknown, we estimate it using the sample standard deviation, which is approximately 3.858.

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u = 28.8

The 99% confidence interval for μ is (27.01, 30.59).

28.4 + 27.3 + 25.5 + 25.5 + 31.1 + 23.0 + 26.3 + 24.6 + 28.4 + 37.2 + 23.9 + 28.7 + 27.9 + 25.1 + 27.2 + 25.3 + 22.6 + 22.7 = 518.3

The sample mean (or the point estimate of μ) is:

518.3 / 18 = 28.8

Calculate each data point's deviation from the mean, square it, and then sum up these values.

(28.4-28.8)² + (27.3-28.8)² + (25.5-28.8)² + (25.5-28.8)² + (31.1-28.8)² + (23.0-28.8)² + (26.3-28.8)² + (24.6-28.8)² + (28.4-28.8)² + (37.2-28.8)² + (23.9-28.8)² + (28.7-28.8)² + (27.9-28.8)² + (25.1-28.8)² + (27.2-28.8)² + (25.3-28.8)² + (22.6-28.8)² + (22.7-28.8)²

= 116.4

Divide by (n - 1), where n is the number of observations. In this case, n = 18, so n - 1 = 17.

So, the variance (s²) is 116.4 / 17 = 6.85 (approximately)

s = √6.85 = 2.62

CI = x ± (t * (s/√n))

CI = 28.8 ± (2.898 * (2.62/√18))

CI = 28.8 ± 1.79

So, the 99% confidence interval for μ is (27.01, 30.59).

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Answer:

Step-by-step explanation:

The Pythagorean theorem has the formula a squared(leg) + b squared(leg) = c squared(longest leg). This means [tex]12^{2} +16^{2} =20^{2} -- > 144+256=400[/tex] which is true meaning number 1 is a right triangle. [tex]10^{2} +49.5^{2} = 50.5^{2} -- > 100+2450.25=2550.25[/tex] is true meaning number 2 is also a right triangle because the sum of the shortest legs squared are equal to the longest leg (hypotenuse) squared.

The height of the house by the given data is 4000ft.

We are given that;

Number of BTUs for heating= 52000

Now,

The time from minutes to hours by dividing by 60:

t=606.24​ hr

t≈0.104 hr

Then, we can plug in the values into the heat loss formula and solve for A, which is the surface area of the house:

Q=UAΔTt

52,000=0.25A×50×0.104

A=0.25×50×0.10452,000​ ft2

A≈4000 ft2

Therefore, by the algebra the answer will be 4000 ft.

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Answer:3.75 BTUs/ft

height=18 ft

Step-by-step explanation:

Using basic mathematical procedures, we can determine that the total cost of the 8 bags of popcorn and 5 drinks is $53.

An operation, in mathematics, is a mathematical function that transforms zero or more input values into a precisely defined output value.

The quantity of operands affects the operation's arity.

The rules that specify the order in which we should carry out the operations required to solve an equation are referred to as the order of operations.

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. (from left to right).

The whole price is thus:

Popcorn costs $3.50 for one bag.

One beverage costs $5.

Total cost for 5 beverages and 8 bags of popcorn:

(8 × 3.50) + (5 × 5) 28 + 25 $53

Therefore ,Using basic mathematical procedures, we can determine that the total cost of the 8 bags of popcorn and 5 drinks is $53.

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The appropriate response is provided below:

Zahra is going to the cinema with a few of her friends. A bag of popcorn costs $3.50 and a drink costs $5 at the theatre. How much would it cost if they purchased 5 beverages and 8 bags of popcorn?

Answer:

Solution: (x-3)(x+1)

Step-by-step explanation:

1) Using Mid term break:

x²-2x-3

x²+x-3x-3

2) Take Common Factors from the two pairs

(x²+x) + (-3x-3)

x(x+1) -3(x+1)

3) Rewrite in Factored Form:

(x-3)(x+1)

The measure of angle zxy is 125 degrees.

To solve the problem, we can use the fact that the sum of the measures of two adjacent angles is 180 degrees. Let's call the measure of angle zxy "x".

We know that the ratio of m angle wxz to m angle zxy is 11:25, which means that:

m angle wxz : m angle zxy = 11 : 25

We can write this as an equation:

m angle wxz / m angle zxy = 11/25

We also know that the two angles are adjacent, so their measures add up to 180 degrees:

m angle wxz + m angle zxy = 180

Now we can use these two equations to solve for x:

m angle wxz / x = 11/25

m angle wxz = (11/25)x

Substituting this into the second equation:

(11/25)x + x = 180

(36/25)x = 180

x = (25/36) * 180

x = 125

Therefore, the measure of angle zxy is 125 degrees.

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The complete relative frequency table include the following missing values:

Early bird Night Owl Total

Coffee. 0.21 0.44 0.65

Tea. 0.19. 0.16. 0.35

Total. 0.4. 0.6. 1

The total of each column is being determined from the grand total which can then be use to fill in the various missing parts of the relative frequency table.

For coffee;

To determine the total = 1-0.35 = 0.65

Coffee early Bird = 0.65-0.44 =0.21

For tea ;

Tea night owl = 0.35-0.19 = 0.16

For grand total = 1-0.6 = 0.4

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The value of the speed of the satellite in meters per second is,

⇒ Speed = 7.5 x 10³ m /sec

We have to given that;

A satellite orbiting Earth travels 4.95×10⁷ meters for each orbit.

And, It takes 6,600 seconds to make one orbit.

Hence, the speed of the satellite in meters per second is,

⇒ Speed = Distance / Time

⇒ Speed = 4.95×10⁷/ 6,600

⇒ Speed = 0.00075 x 10⁷

⇒ Speed = 7.5 x 10⁻⁴ x 10⁷

⇒ Speed = 7.5 x 10³ m /sec

Thus, The value of the speed of the satellite in meters per second is,

⇒ Speed = 7.5 x 10³ m /sec

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A rectangular prism is a three-dimensional shape with six faces, all of which are rectangles. The volume of a rectangular prism can be found by multiplying the length, width, and height of the prism.

In this case, Sam's box has a height of 1/6 inches, a width of 1/3 inches, and a length of 1/2 inches. To find the volume, we need to multiply these three dimensions:

(1/6) x (1/3) x (1/2) = 1/36 cubic inches.

Therefore, the volume of Sam's box is 1/36 cubic inches, which is an improper fraction because the numerator is greater than the denominator.

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Answer:

39.27 square units

Step-by-step explanation:

In circle E with m∠DEF = 70∘ and DE=8, we want to find the area of sector DEF. To do this, we can use the formula for the area of a sector:

A_sector=1/2 r^2⋅

α

where r is the radius of the circle and α is the central angle in radians. In our case, r=8 and α=70 ∘ ⋅ π/180∘ = 7π/18 radians

Now, we can plug these values into the formula:

A_sector = 1/2(8^2) ⋅ 7π/18 = 32 ⋅ 7π/18 ≈ 39.27

So, the area of sector DEF is approximately 39.27 square units, rounded to the nearest hundredth.

The area of sector DEF in circle E is approximately 6.10 square units, rounded to the nearest hundredth.

To find the area of sector DEF in circle E, we need to use the formula for the area of a sector:

Area of sector = (Central angle / 360°) * π * r^2

where "Central angle" is the measure of angle DEF in degrees, "r" is the radius of the circle (DE in this case).

Given that m/DEF = 70° and DE = 8, we can calculate the area of sector DEF as follows:

Area of sector = (70° / 360°) * π * 8^2

Now, perform the calculations step by step:

Area of sector = (70/360) * π * 64

Area of sector = (7/36) * π * 64

Area of sector ≈ 3.14 * (7/36) * 64

Area of sector ≈ 3.14 * 1.9444

Area of sector ≈ 6.10 (rounded to two decimal places)

Therefore, the area of sector DEF in circle E is approximately 6.10 square units, rounded to the nearest hundredth.

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Normal Distribution: Finding Values

Press the "2nd" key, then press the "vars" key to access the DISTR menu.

Scroll down to "Normalcdf" and press enter.

Enter the lower bound, upper bound, mean, and standard deviation. For example, if you want to find the probability that X is between 60 and 70, given that the mean is 65 and the standard deviation is 5, you would enter "60, 70, 65, 5".

Press enter to get the result. The calculator will give you the probability that X falls between the two values you entered.

Press the "2nd" key, then press the "vars" key to access the DISTR menu.

Scroll down to "InvNorm" and press enter.

Enter the probability you want to find. For example, if you want to find the z-score that corresponds to the 90th percentile, you would enter "0.9".

Enter the mean and standard deviation. For example, if the mean is 65 and the standard deviation is 5, you would enter "65, 5".

Press enter to get the result. The calculator will give you the z-score that corresponds to the probability you entered.

Sampling Distributions and the Central Limit Theorem: Press the "2nd" key, then press the "vars" key to access the DISTR menu.

Scroll down to "1-PropZTest" or "2-PropZTest" and press enter, depending on the type of test you want to perform.

Enter the sample statistics and the population parameters, if known. The calculator will use these values to calculate the test statistic and p-value.

Press enter to get the result. The calculator will give you the test statistic and the p-value, which you can use to make inferences about the population.

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The total area of the flag is the sum of the areas of the three stripes and the percentage of the flag that is gold is approximately 0.459%

Let's call this height "h".

We also know that the side lengths of the flag are in the ratio 2:3. This means that if the shortest side is 2x, then the longest side is 3x. Since the three stripes are of equal height, each one must be h/3 in height.

Now we can use this information to find the area of the gold-colored disk and the total area of the flag. The area of a circle is given by the formula A = πr^2, where r is the radius. Since the diameter is equal to the height of the center stripe (which is h/3), the radius of the disk is h/6. So the area of the disk is:

A_disk = π(h/6)^2 = πh^2/36

The total area of the flag is the sum of the areas of the three stripes. Since each stripe is h/3 in height and the shortest side is 2x, the area of each stripe is:

A_stripe = (2x)(h/3) = 2hx/3

So the total area of the flag is:

A_flag = 3(A_stripe) = 6hx

To find the percentage of the flag that is gold, we need to divide the area of the gold-colored disk by the total area of the flag and multiply by 100.

percentage = (A_disk / A_flag) x 100

Substituting our expressions for A_disk and A_flag, we get:

percentage = (πh^2/36) / (6hx) x 100

Simplifying, we can cancel out the factor of h and get:

percentage = (π/216)x100

So the percentage of the flag that is gold is approximately 0.459% (rounded to three decimal places).

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The maximum value represents (b) the point where most profits are made

Given that the graph is the graph of a company's profit

The maximum value represents (b) the point where most profits are made

Given that the graph is the graph of a company's profit

The x-intercept is when the profit = 0

So

The x-intercept represents (b) the price per pen where no profit is made

This is calculated as

Rate = [f(6) - f(3)]/[6 - 3]

So, we have

Rate = [0 - 120]/[6 - 3]

Rate = -40

The domain of this graph given the situation is (0, 6] because the values do not makes sense beyond those points

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The volume of granola a cylindrical container can hold 5451.6 cubic centimeters if containers have a diameter of 20 cm and a height of 17. 4 cm.

The number of unit cubes (cubes of unit length) that can fit inside a cylinder determines its volume.

Identifying the radius (r) and height (h)
r = 10 cm
h = 17.4 cm

Calculating the volume (V) using the formula V = πr²h
V = π × (10 cm)² × 17.4 cm
V ≈ 3.14 × 100 cm² × 17.4 cm
V ≈ 5451.6 cm³

Approximately, a container will hold 5451.6 cubic centimeters of granola.

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Answer:

m = -2

Step-by-step explanation:

We Know

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points (-3,2) (-2,0)

We see the y decrease by 2, and the x increase by 1, so the slope is

m = -2

Answer:

The correct answer is ** x(y+12) or f.

We can simplify the expression 1/5x(5y+60) by multiplying the factors in the parentheses and then dividing by 5. This gives us:

```

1/5x(5y+60) = 1/5 * 5xy + 1/5 * 60x = x(y+12)

```

The volume of the cones are;

1. 84. 78 in³

2.  564. 15 ft³

3.  4710 yd³

The formula for calculating the volume of a cone is expressed as;

V = 1/3 πr²h

Given that;

From the information given, we have;

1. Volume = 1/3 × 3.14 × 3² × 9

Multiply the values and find the square

Volume = 254. 34/3

divide the values

Volume = 84. 78 in³

2. Volume = 1/3 × 3.14 × 7² × 11

Multiply the values

Volume = 1692. 46/3

Volume = 564. 15 ft³

3. Volume = 1/3 × 3.14 × 15² × 20

Multiply the values

Volume = 4710 yd³

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A definite integral is defined as the signed area under a function between certain limits (bounds) of integration.

An indefinite integral represents the family of antiderivatives of a function and is also known as its general integral or antiderivative.

An indefinite integral represents the family of antiderivatives of a function and is also known as its general integral or antiderivative. An indefinite integral does not have specific limits of integration; its result includes a constant of integration (usually denoted +C), which accounts for all possible constant shifts within its antiderivative.

A definite integral is defined as the signed area under a function between certain limits (bounds) of integration. The real number that represents its net area between it and x-axis during an interval.

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Using the inscribed angle theorem, the value of x in the circle shown is calculated as: m<x = 36°.

According to the inscribed angle theorem an inscribed angle has an angle measure that is always half of the measure of the arc it intercepts.

Measure of arc EFG = 2(25 + 47) [inscribed angle theorem]

Measure of arc EFG = 144°

Measure of angle EHG = 1/2(m(EFG)) [inscribed angle theorem]

Substitute:

Measure of angle EHG = 1/2(144) = 72°

m<EHG = m<EGH = 72° [this is because triangle EHG is an isosceles triangle since sides EG = EH].

Therefore, we have:

m<x = 180 - 2(72)

m<x = 36°

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The total area of the first floor of the tiny house is 121 square feet

The total area of the first floor of the tiny house can be expressed as the sum of the areas of each room. The area of a rectangle is calculated by multiplying the length by the width. Therefore, we can write:

Total area = Area of kitchen + Area of bathroom

The area of the kitchen is given by the product of the length and the width of the kitchen, which is 11 feet and 7 feet, respectively. Therefore, the area of the kitchen can be written as:

Area of kitchen = 11 x 7 = 77 square feet

Similarly, the area of the bathroom is given by the product of the length and the width of the bathroom, which is 11 feet and 4 feet, respectively. Therefore, the area of the bathroom can be written as:

Area of bathroom = 11 x 4 = 44 square feet

Substituting these expressions into the equation for the total area, we get:

Total area = Area of kitchen + Area of bathroom

= 77 + 44

= 121 square feet

Therefore, the total area of the first floor of the tiny house is 121 square feet, which can also be expressed as the sum of the areas of the kitchen and bathroom.

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The average speed that your brother was traveling is approximately 45.63 miles per hour

The average speed is the total distance traveled divided by the total time taken. So we have:

Average speed = total distance ÷ total time

We are given the total distance as 115 miles and the total time as 2.52 hours. Therefore, the average speed is:

Average speed = 115 miles ÷ 2.52 hours

Average speed = 45.63 miles per hour

So, the average speed that your brother was traveling is approximately 45.63 miles per hour.

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You should choose Account 1 to obtain the greater amount in 10 years. Account 1 will have a higher balance after 10 years.

Account 1 pays 6% annual interest compounded quarterly, which means the interest is calculated and added to the principal four times a year. You can use the formula A = P(1 + r/n)^(nt) to calculate the future value, where A is the final amount, P is the principal ($2500), r is the annual interest rate (0.06), n is the number of times compounded per year (4), and t is the number of years (10).

Account 2 pays 4% annual interest compounded continuously. You can use the formula A = Pe^(rt) to calculate the future value, where A is the final amount, P is the principal ($2500), e is the base of the natural logarithm (approximately 2.71828), r is the annual interest rate (0.04), and t is the number of years (10).

When you compare the final amounts for both accounts, Account 1 will have a higher balance after 10 years.

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The solution of the trigonometric equation

cos(2x) = cos(x) + 2

is x = 180°

Here we want to solve the equation:

cos(2x) = cos(x) + 2

First, we know that:

cos(2x) = 2cos(x)²  - 1

Then we can rewrite:

2cos(x)²  - 1 = cos(x) + 2

We can define:

cos(x)= y

2y² - 1 = y + 2

Then we need to solve the quadratic:

2y² - 1 - y - 2  =0

2y² - y - 3 = 0

Using the quadratic formula we will get:

[tex]y = \frac{1 \pm \sqrt{(-1)^2 - 4*2*2*-3} }{2*2} \\\\y = \frac{1 \pm 5}{4}[/tex]

so the solutions are:

y = (1 + 5)/4 = 6/4

y = (1-  5)/4 = -1

And remember that y = cos(x), then y =  6/4 can be discarded.

Then the solution comes from:

cos(x) = -1

then x = pi = 180°

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