Show that an = 5an−1 − 6an−2 for all integers n with n ≥ 2

Answer:

a. 120

Step-by-step explanation:

170 - 50 = 120

OR

The middle of 110 and 130 is 120

the middle of the box

The greyhound's speed is 39.18 miles per hour greater than the tortoise's speed.

The giant tortoise can move at speeds of up to 0.17 mile per hour and the top speed for a greyhound is 39.35 miles per hour.

So, we can find the difference in speed between these two animals as follows:

Difference in speed between the greyhound and tortoise = Speed of the greyhound - Speed of the tortoise

Difference in speed = 39.35 - 0.17

Difference in speed = 39.18 miles per hour

Therefore, the greyhound's speed is 39.18 miles per hour greater than the tortoise's speed.

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The requried unknown value of y at x = 0 and 2 are 1 and 9 respectively.

A table is shown for the two variables x and y, the relation between the variable is given by the equation,
y = 4x + 1

Since in the table at x = 0 and 2, y is not given
So put x = 0 in the given equation,
y = 4(0) + 1
y = 1

Again put x = 2 in the given equation,
y = 4(2)+1
y = 9

Thus, the requried unknown value of y at x = 0 and 2 are 1 and 9 respectively.

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The independent variable is typically plotted on the x-axis, while the dependent variable is plotted on the y-axis.

To determine the value of the independent variable at point A on a graph, we need to look at the x-axis of the graph.

The x-axis represents the independent variable, which is the variable that is being manipulated or changed in an experiment or study.

At point A on the graph, we need to identify the specific value of the independent variable that corresponds to that point.

This can be done by looking at the position of point A on the x-axis and reading the value that is associated with it.

For example, if the x-axis represents time and the independent variable is the amount of light exposure, point A may represent a specific time point where the amount of light exposure was measured.

In this case, we would need to look at the x-axis and identify the time value that corresponds to point A on the graph.

This information is important for understanding the relationship between the independent variable and the dependent variable, and for drawing conclusions from the data.

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The probability that an infant selected at random from among those delivered at the hospital measures more than 23.5 inches is 0.0475 or approximately 4.75%. (option c).

To find the probability that an infant selected at random from among those delivered at the hospital measures more than 23.5 inches, we need to calculate P(X > 23.5). To do this, we first standardize the variable X by subtracting the mean and dividing by the standard deviation:

Z = (X - µ)/σ

In this case, we have:

Z = (23.5 - 20)/2.1 = 1.667

Next, we use a standard normal distribution table or calculator to find the probability of Z being greater than 1.667. Using a standard normal distribution table, we can find that the probability of Z being less than 1.667 is 0.9525. Therefore, the probability of Z being greater than 1.667 is:

P(Z > 1.667) = 1 - P(Z < 1.667) = 1 - 0.9525 = 0.0475

Hence, the correct option is (c)

Therefore, we can conclude that it is relatively rare for an infant's length at birth to be more than 23.5 inches, given the mean and standard deviation of the distribution.

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Complete Question:

The medical records of infants delivered at the Kaiser Memorial Hospital show that the infants' lengths at birth (in inches) are normally distributed with a mean of 20 and a standard deviation of 2.1. Find the probability that an infant selected at random from among those delivered at the hospital measures is more than 23.5 inches.

a. 0.0485

b. 0.1991

c. 0.0475

d. 0.9515

e. 0.6400

The probability of selecting two cards from different suits with replacement is 1/2 in a standard deck of 52 cards.

When choosing cards from a deck of cards, with replacement means that the first card is removed and put back into the deck before drawing the second card. The deck of cards has four suits, each of them with thirteen cards. So, there are four different ways to choose the first card and four different ways to choose the second card. The four different suits are hearts, diamonds, clubs, and spades. Since there are four different suits, each with thirteen cards, there are 52 cards in the deck.

When choosing two cards from the deck, there are 52 choices for the first card and 52 choices for the second card. Therefore, the probability of selecting two cards from different suits with replacement is 1/2.

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The standard error of the difference between the means is 8.45.

The standard error is a measure of the variability of a sample statistic, such as the mean, compared to the population parameter it estimates.

In this case, we are interested in the standard error of the difference between the means of two independent samples, which is calculated using the pooled variance estimator assuming equal population variances. The formula for the standard error of the difference between two sample means is:

SE = √[ (s1^2/n1) + (s2^2/n2) ]

Where s1 and s2 are the standard deviations of the two samples, n1 and n2 are the sample sizes, and SE is the standard error of the difference between the sample means. Substituting the given values, we get:

SE = √[ (32^2/16) + (36^2/20) ] = 8.45

This means that if we were to take repeated random samples from the same population using the same sample sizes, the standard deviation of the sampling distribution of the difference between the means would be approximately 8.45.

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The standard error of the pooled sample means is approximately 7.15.

The standard error of the pooled sample means is calculated using the formula:

Standard Error = √[(s1^2 / n1) + (s2^2 / n2)]

Where s1 and s2 are the standard deviations of the two samples, n1 and n2 are the sizes of the samples.

In this case, s1 = 32, s2 = 36, n1 = 16, and n2 = 20. Substituting these values into the formula, we have:

Standard Error = √[(32^2 / 16) + (36^2 / 20)]

Standard Error = √[1024 / 16 + 1296 / 20]

Standard Error = √[64 + 64.8]

Standard Error = √128.8

Standard Error ≈ 7.15

Therefore, the standard error of the pooled sample means is approximately 7.15. The standard error represents the variability or uncertainty in estimating the population means based on the sample means. A smaller standard error indicates a more precise estimation of the population means, while a larger standard error indicates more variability and less precise estimation.
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The weighted moving average formula with weights of 0.3 and 0.7 can be calculated using the AVERAGE and SUMPRODUCT functions in Excel. This formula can be used to calculate forecasted values for a range of years.

To use a 2-year weighted moving average to calculate forecasts for the years 1992-2002 with the weight of 0.7 assigned to the most recent year data, we can use the SUMPRODUCT function.
First, we need to create a table that includes the years 1990-2002 and their corresponding data points. Then, we can use the following formula to calculate the weighted moving average:
=(0.3*AVERAGE(B2:B3))+(0.7*B3)
This formula calculates the weighted moving average for each year by taking 30% of the average of the data for the previous two years (B2:B3) and 70% of the data for the most recent year (B3). We can then drag the formula down to calculate the forecasted values for the remaining years.
The SUMPRODUCT function can be used to simplify this calculation. The formula for the weighted moving average using SUMPRODUCT would be:
=SUMPRODUCT(B3:B4,{0.3,0.7})
This formula multiplies the data for the previous two years (B3:B4) by their respective weights (0.3 and 0.7) and then sums the products to calculate the weighted moving average for the most recent year. We can then drag the formula down to calculate the forecasted values for the remaining years.
In summary, the weighted moving average formula with weights of 0.3 and 0.7 can be calculated using the AVERAGE and SUMPRODUCT functions in Excel. This formula can be used to calculate forecasted values for a range of years.

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At the end of the year, the amount in the account had decreased by 21%. The amount of money Martina has in her account after the 21% decrease is $6794.

Last year, Martina opened an investment account with $8600. At the end of the year, the amount in the account had decreased by 21%.

Let us calculate how much money she has in the account after a year.Solution:

Amount of money Martina had in her account when she opened = $8600

Amount of money Martina has in her account after the 21% decrease

Let us calculate the decrease in money. We will find 21% of $8600.21% of $8600

= 21/100 × $8600

= $1806.

Subtracting $1806 from $8600, we get;

Money in Martina's account after 21% decrease = $8600 - $1806

= $6794

Therefore, the money in the account after the 21% decrease is $6794. Therefore, last year, Martina opened an investment account with $8600.

At the end of the year, the amount in the account had decreased by 21%. The amount of money Martina has in her account after the 21% decrease is $6794.

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In order to prove the statement (A ∩ B) ⊆ A, we need to show that every element in the intersection of A and B is also an element of A. Let's go through the steps:

a. If x is in (A ∩ B), x is in A and x is in B by the definition of intersection. The intersection of two sets A and B consists of elements that are present in both sets.
b. Since x is in A and x is in B, we can conclude that x is indeed in A. This step demonstrates that the element x, which is part of the intersection (A ∩ B), belongs to the set A.
c. As x is in A, it satisfies the condition for being part of the intersection (A ∩ B), i.e., x is in A and x is in B by the definition of intersection.
Based on these steps, we can conclude that for any element x in the intersection (A ∩ B), x must also be in set A. This means (A ∩ B) ⊆ A, proving the given statement.

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The rate constant for the decomposition of cyclopropane, a flammable gas, is 1.46 × 10−4 s−1 at 500°C. If the initial cyclopropane concentration is 0.0440 M, what is the cyclopropane concentration after 281 minutes?

The formula for calculating the concentration of the reactant after some time, [A], is given by:[A] = [A]0 × e-kt

Where:[A]0 is the initial concentration of the reactant[A] is the concentration of the reactant after some time k is the rate constantt is the time elapsed Therefore, the formula for calculating the concentration of cyclopropane after 281 minutes is[Cyclopropane] = 0.0440 M × e-(1.46 × 10^-4 s^-1 × 281 × 60 s)≈ 0.023 M Therefore, the cyclopropane concentration after 281 minutes is 0.023 M.

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Emily should choose the lump sum payment of $60.00 for 6 months instead of paying $12.00 per month.

By choosing the lump sum payment of $60.00 for 6 months, Emily can save money compared to paying $12.00 per month. To determine which option is more cost-effective, we can compare the total amount spent in each scenario.

If Emily pays $12.00 per month, she would spend $12.00 x 6 = $72.00 over 6 months. On the other hand, by opting for the lump sum payment of $60.00 for 6 months, she would save $12.00 - $10.00 = $2.00 per month. Multiplying this monthly saving by 6, Emily would save $2.00 x 6 = $12.00 in total by choosing the lump sum payment.

Therefore, it is clear that choosing the lump sum payment of $60.00 for 6 months is the more cost-effective option for Emily. She would save $12.00 compared to the monthly payment plan, making it a better choice financially.

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The value of the limit is -4/3.

Using the Taylor series expansion for sin(2x) and simplifying, we get:

sin(2x) = 2x - (4/3)x^3 + (2/15)x^5 + O(x^7)

Substituting this into the expression sin(2x) - 2x, we get:

sin(2x) - 2x = - (4/3)x^3 + (2/15)x^5 + O(x^7)

Dividing by x^3, we get:

(sin(2x) - 2x)/x^3 = - (4/3) + (2/15)x^2 + O(x^4)

As x approaches 0, the dominant term in this expression is -4/3x^3, which goes to 0. Therefore, the limit of the expression as x approaches 0 is:

lim x → 0 (sin(2x) - 2x)/x^3 = -4/3

Therefore, the value of the limit is -4/3.

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We have:

y(4) + y''' + y'' = 0

First, let's rewrite the equation using the common notation for derivatives:

y'''' + y''' + y'' = 0

Now, we need to find the characteristic equation, which is obtained by replacing each derivative with a power of r:

r^4 + r^3 + r^2 = 0

Factor out the common term, r^2:

r^2 (r^2 + r + 1) = 0

Now, we have two factors to solve separately:

1) r^2 = 0, which gives r = 0 as a double root.

2) r^2 + r + 1 = 0, which is a quadratic equation that doesn't have real roots. To find the complex roots, we can use the quadratic formula:

r = (-b ± √(b^2 - 4ac)) / 2a

Plugging in the values a = 1, b = 1, and c = 1, we get:

r = (-1 ± √(-3)) / 2

So the two complex roots are:

r1 = (-1 + √(-3)) / 2
r2 = (-1 - √(-3)) / 2

Now we can write the general solution of the differential equation using the roots found:

y(x) = C1 + C2*x + C3*e^(r1*x) + C4*e^(r2*x)

Where C1, C2, C3, and C4 are constants that can be determined using initial conditions or boundary conditions if provided.

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The number of students who watched the games = (2/3)x = [2/3 * Total number of students] = [2/3 * x] = (2/3) x 150 = 100 students.

Let's assume that the total number of students in the school stadium is x. So,1/5 of the students were basketball players => (1/5)x2/15 of the students were soccer players => (2/15)x

So, the rest of the students watched the games => x - [(1/5)x + (2/15)x]

Let's simplify the given expressions=> (1/5)x = (3/15)x=> (2/15)x = (2/15)x

Now, we can add these fractions to get the value of the remaining students=> x - [(1/5)x + (2/15)x]

=> x - [(3/15)x + (2/15)x]

=> x - (5/15)x

=> x - (1/3)x = (2/3)x

Students who watched the games are (2/3)x

.Now we have to find out how many students watched the game. So, we have to find the value of (2/3)x.

We know that, the total number of students in the stadium = x

Hence, we can say that (2/3)x is the number of students who watched the games, and (2/3)x is equal to [2/3 * Total number of students] = [2/3 * x]

Therefore, the students who watched the game are (2/3)x.

Hence the solution to the given problem is that the number of students who watched the games = (2/3)x = [2/3 * Total number of students] = [2/3 * x] = (2/3) x 150 = 100 students.

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To solve the initial value problem using Laplace transform, we first take the Laplace transform of the given differential equation to obtain the equation Y(s)(s^2- s - 2) = -6s + 6. Solving for Y(s), we get Y(s) = (6s-18)/(s^2-s-2). Using partial fractions, we can write Y(s) as Y(s) = 3/(s-2) - 3/(s+1). Inverting the Laplace transform of Y(s), we get the solution y(t) = 3e^(2t) - 3e^(-t) - 3t(e^(-t)). Therefore, the solution to the given initial value problem is y(t) = 3e^(2t) - 3e^(-t) - 3t(e^(-t)), which satisfies the given initial conditions.

The Laplace transform is a mathematical technique used to solve differential equations. To use the Laplace transform to solve the given initial value problem, we first take the Laplace transform of the differential equation y'' - y' - 2y = 0 using the property that L(y'') = s^2 Y(s) - s y(0) - y'(0) and L(y') = s Y(s) - y(0).

Taking the Laplace transform of the differential equation, we get Y(s)(s^2 - s - 2) = -6s + 6. Solving for Y(s), we get Y(s) = (6s - 18)/(s^2 - s - 2).

Using partial fractions, we can write Y(s) as Y(s) = 3/(s-2) - 3/(s+1). We then use the inverse Laplace transform to obtain the solution y(t) = 3e^(2t) - 3e^(-t) - 3t(e^(-t)).

In summary, we used the Laplace transform to solve the given initial value problem. We first took the Laplace transform of the differential equation to obtain an equation in terms of Y(s). We then solved for Y(s) and used partial fractions to write it in a more convenient form. Finally, we used the inverse Laplace transform to obtain the solution y(t) that satisfies the given initial conditions.

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The required answers are the speed of the satellite is `7842.6 m/s` and the time required to complete one orbit is `1.52 hours`.

Given that a satellite moves in a stable circular orbit of radius r = 6761 km and at constant speed.

And its centripetal acceleration is inversely proportional to the square of the radius r of the orbit. We need to find the speed of the satellite and the time required to complete one orbit.

Speed of the satellite:

We know that centripetal acceleration is given by the formula

`a=V²/r`

Where,a = centripetal accelerationV = Speed of the satellite,r = Radius of the orbit

The acceleration due to gravity `g` at an altitude `h` above the surface of Earth is given by the formula `

g = GM/(R+h)²`,

where `M` is the mass of the Earth, `G` is the gravitational constant, and `R` is the radius of the Earth.

Here, `h = 6761 km` (Radius of the orbit) Since `h` is much smaller than the radius of the Earth, we can assume that `R+h ≈ R`, where `R = 6371 km` (Radius of the Earth)

Then, `g = GM/R²`

Substituting the values,

`g = 6.67259 × 10^-11 × 5.98 × 10^24 / (6371 × 10^3)²``g = 9.81 m/s²`

Therefore, centripetal acceleration `a = g` at an altitude `h` above the surface of Earth.

Substituting the values,

`a = 9.81 m/s²` and `r = 6761 km = 6761000 m`

We have `a = V²/r` ⇒ `V = √ar`

Substituting the values,`V = √(9.81 × 6761000)`

⇒ `V ≈ 7842.6 m/s`

Therefore, the speed of the satellite is `7842.6 m/s`.

Time taken to complete one orbit:We know that time period `T` of a satellite is given by the formula

`T = 2πr/V`

Substituting the values,`

T = 2 × π × 6761000 / 7842.6`

⇒ `T ≈ 5464.9 s`

Therefore, the time required to complete one orbit is `5464.9 seconds` or `1.52 hours` (approx).

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Carla will run on Sunday, November 12 and then run and swim on Thursday, November 16.

Carla runs every 3 days and swims every Thursday.

Carla ran and swam on Thursday 9 November.

The next time Carla will run will be 3 days later: Sunday, November 12.

The next Thursday after November 9 is November 16.

Therefore, Carla will run on Sunday, November 12 and then run and swim on Thursday, November 16.

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Answer:

1

Step-by-step explanation:

V = L * W * H

Measurements given:

[tex]V = \frac{18}{5} *\frac{10}{27} *\frac{3}{4}[/tex]

[tex]V=\frac{4}{3}*\frac{3}{4}[/tex]

[tex]V=1[/tex]

To find the volume of the solid obtained by revolving the region enclosed by the curve y=e^x, the x-axis, and the lines x=0 and x=1 around the x-axis, we can use the method of cylindrical shells.First, we need to find the equation of the curve y=e^x. This is an exponential function with a base of e and an exponent of x. As x varies from 0 to 1, y=e^x varies from 1 to e.

Next, we need to find the height of the cylindrical shell at a particular value of x. This is given by the difference between the y-value of the curve and the x-axis at that point. So, the height of the shell at x is e^x - 0 = e^x.
The thickness of the shell is dx, which is the width of the region we are revolving around the x-axis.
Finally, we can use the formula for the volume of a cylindrical shell:
V = 2πrh dx
where r is the distance from the x-axis to the shell (which is simply x in this case), and h is the height of the shell (which is e^x).So, the volume of the solid obtained by revolving the region enclosed by the curve y=e^x, the x-axis, and the lines x=0 and x=1 around the x-axis is given by the integral:
V = ∫ from 0 to 1 of 2πxe^x dx
We can evaluate this integral using integration by parts or substitution. The result is:
V = 2π(e - 1)
Therefore, the volume of the solid is 2π(e - 1) cubic units.

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Yes, regular octagons and equilateral triangles can form a semi-regular tessellation of the plane.

A tessellation is a repeating pattern of shapes that covers a plane without any gaps or overlaps. In a semi-regular tessellation, multiple regular polygons are used to create the pattern.

For regular octagons and equilateral triangles to form a semi-regular tessellation, they must satisfy two conditions:

Vertex Condition: The same polygons meet at each vertex.

Edge Condition: The same polygons meet along each edge.

Let's examine these conditions for regular octagons and equilateral triangles:

Regular Octagon:

Each vertex of an octagon meets three other octagons.

Each edge of an octagon meets two other octagons.

Equilateral Triangle:

Each vertex of a triangle meets six other triangles.

Each edge of a triangle meets three other triangles.

The vertex condition is satisfied because each vertex of an octagon meets three equilateral triangles, and each vertex of an equilateral triangle meets three octagons.

The edge condition is satisfied because each edge of an octagon meets two equilateral triangles, and each edge of an equilateral triangle meets three octagons.

Therefore, regular octagons and equilateral triangles can form a semi-regular tessellation of the plane.Yes, regular octagons and equilateral triangles can form a semi-regular tessellation of the plane.

A tessellation is a repeating pattern of shapes that covers a plane without any gaps or overlaps. In a semi-regular tessellation, multiple regular polygons are used to create the pattern.

For regular octagons and equilateral triangles to form a semi-regular tessellation, they must satisfy two conditions:

Vertex Condition: The same polygons meet at each vertex.

Edge Condition: The same polygons meet along each edge.

Let's examine these conditions for regular octagons and equilateral triangles:

Regular Octagon:

Each vertex of an octagon meets three other octagons.

Each edge of an octagon meets two other octagons.

Equilateral Triangle:

Each vertex of a triangle meets six other triangles.

Each edge of a triangle meets three other triangles.

The vertex condition is satisfied because each vertex of an octagon meets three equilateral triangles, and each vertex of an equilateral triangle meets three octagons.

The edge condition is satisfied because each edge of an octagon meets two equilateral triangles, and each edge of an equilateral triangle meets three octagons.

Therefore, regular octagons and equilateral triangles can form a semi-regular tessellation of the plane.

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When the wind is less than 10 knots and at an altitude within 1,500 feet of the station elevation, it is considered a light wind condition. This means that the wind speed is relatively low and can have a minimal impact on aircraft operations.

However, pilots still need to take into account the direction of the wind and any gusts or turbulence that may be present. When the wind is less than 5 knots, it is considered a calm wind condition. This type of wind condition can make it difficult for pilots to maintain the aircraft's direction and speed, especially during takeoff and landing. In such cases, pilots may need to use different techniques and procedures to ensure the safety of the aircraft and passengers. Overall, it is important for pilots to pay close attention to wind conditions and make adjustments accordingly to ensure safe and successful flights.

When the wind is less than 10 knots (A), it typically has a minimal impact on activities such as aviation or sailing. When the wind at altitude is within 1,500 feet of the station elevation (B), it means that the wind speed and direction measured at ground level are similar to those at a higher altitude. Lastly, when the wind is less than 5 knots (C), it is considered very light and usually does not have a significant effect on outdoor activities. In summary, light wind conditions can make certain activities easier, while having minimal impact on others.

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The weaknesses in the design of the study are: small sample size, potential confounding variable, the use of a survey instead of an exam, and the reliance on random selection without addressing other design limitations.

A. The sample size is too small: With only 10 students in each sample, the sample size is small, which may limit the generalizability of the findings. A larger sample size would provide more reliable and representative results.

B. One sample has more graduate level experience than the other sample: Comparing students graduating in May with students graduating the following May introduces a potential confounding variable.

C. An exam should be used, instead: Using a survey as the primary method to measure students' understanding of statistics may not be as reliable or valid as using an exam.

D. Randomly selected students were used: While randomly selecting students is a strength of the study design, it does not negate the other weaknesses mentioned.

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The geometric series 9^n=1 is divergent because as n increases, the terms of the series get larger and larger without bound. Specifically, each term is 9 times the previous term, so the series grows exponentially.

To see this, note that the first few terms are 9, 81, 729, 6561, and so on, which clearly grow without bound. Therefore, the sum of this series cannot be determined since it diverges. In general, a geometric series with a common ratio r is convergent if and only if |r| < 1, in which case its sum is given by the formula S = a/(1-r), where a is the first term of the series.

However, if |r| ≥ 1, then the series diverges. In the case of 9^n=1, the common ratio is 9, which is clearly greater than 1, so the series diverges.

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A business loan differs from a personal loan in terms of documentation, collateral, and repayment sources.

To distinguish between a business and a personal loan, several factors come into play.

The loan's purpose is key; if it finances business-related expenses, it is likely a business loan, while personal loans serve personal needs.

Documentation requirements, collateral, and repayment sources also offer clues. Business loans demand business-related documentation, may require business assets as collateral, and rely on business revenue for repayment.

Personal loans, however, focus on personal identification, income verification, personal assets, and personal income for repayment. Loan terms, including duration and loan amount, can also help differentiate between the two types.

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About 95% of the buyers paid between $17,000 and $25,000 for the particular model of the car.Normal distribution graph for prices paid for a particular model of a new car with mean $21,000 and standard deviation $2000.

We need to find what percentage of buyers paid between $17,000 and $25,000 using the 68-95-99.7 rule.

So, the z-score for $17,000 is

[tex]z=\frac{x-\mu}{\sigma}[/tex]

=[tex]\frac{17,000-21,000}{2,000}[/tex]

=-2

The z-score for $25,000 is

[tex]z=\frac{x-\mu}{\sigma}[/tex]

=[tex]\frac{25,000-21,000}{2,000}[/tex]

=2

Therefore, using the 68-95-99.7 rule, the percentage of buyers paid between $17,000 and $25,000 is within 2 standard deviations of the mean, which is approximately 95% of the total buyers.

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Ms. Redmon gave her theater students an assignment to memorize a dramatic monologue to present to the rest of the class. The graph shows the times, rounded to the nearest half minute, of the first 10 monologues presented.

The assignment requires the students to memorize a dramatic monologue to present to the rest of the class. Based on the graph, the content loaded for the first ten presentations can be determined. The graph contains the timings of the first 10 monologues presented. From the graph, the lowest time recorded was 2 minutes while the highest was 3 minutes and 30 seconds.

The graph showed that the first student took the longest time while the sixth student took the shortest time to present. Ms. Redmon asked the students to memorize a dramatic monologue, with a requirement of 130 words. It is, therefore, possible for the students to finish the presentation within the allotted time by managing the word count in their dramatic monologue.

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The new dimensions of each enlarged block in the subdivision planned by the city planner of Mechlinburg will be 660 feet by 2,250 feet.

The standard size of a city block in Manhattan is 264 feet by 900 feet. To enlarge these dimensions by 2.5 times, we need to multiply each side of the block by 2.5.

So, the new length of each block will be 264 feet * 2.5 = 660 feet, and the new width will be 900 feet * 2.5 = 2,250 feet.

Therefore, the new dimensions of each enlarged block in the subdivision planned by the city planner of Mechlinburg will be 660 feet by 2,250 feet. These larger blocks will provide more space for buildings, streets, and public areas, allowing for a potentially larger population and accommodating the city's growth and development plans.

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The original length of the reed is 45.

Given: A reed was broken off a cubit. This reed fitted 60 times along the length of the field. After restoring what was broken off, it fitted 30 times along the width. The area of the field is 525 square nindas

To find: Original length of the reedIn order to solve the problem,

let’s first define the reed length as x. It means the length broken from the reed is x-1. We know that after the broken reed is restored it fits 30 times in the width of the field.

It means;The width of the field = (x-1)/30Next, we know that before breaking the reed it fit 60 times in the length of the field. After breaking and restoring, its length is unchanged and now it fits x times in the length of the field.

Therefore;The length of the field = x/(60/ (x-1))= x (x-1) /60

Now, we can use the formula of the area of the field to calculate the original length of the reed.

Area of the field= length x widthx

(x-1) /60 × (x-1)/30

= 525 2(x-1)2

= 525 × 60x²- 2x -1785

= 0(x-45)(x+39)=0

x= 45 (as x cannot be negative)

Therefore, the original length of the reed is 45. Hence, the answer in 100 words is: The original length of the reed was 45. The width of the field is given as (x-1)/30 and the length of the field is x (x-1) /60, which is obtained by breaking and restoring the reed.

Using the area formula of the field (length × width), we get x= 45.

Thus, the original length of the reed is 45. This is how the original length of the reed can be calculated by solving the given problem.

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By mathematical induction, the equation 1 + 3 + 5 + ... + (2n - 1) = n² is true for every positive integer n.

Using mathematical induction, we can verify that the equation 1 + 3 + 5 + ... + (2n - 1) = n² is true for every positive integer n.
Base case (n=1): 2(1) - 1 = 1, and 1² = 1, so the equation holds for n=1.
Inductive step: Assume the equation is true for n=k, i.e., 1 + 3 + ... + (2k - 1) = k². We must prove it's true for n=k+1.
Consider the sum 1 + 3 + ... + (2k - 1) + (2(k+1) - 1). By the inductive hypothesis, the sum up to (2k - 1) is equal to k². Thus, the new sum is k² + (2k + 1).
Now, let's examine (k+1)²: (k+1)² = k² + 2k + 1.
Comparing the two expressions, we find that they are equal: k^2 + (2k + 1) = k² + 2k + 1. Therefore, the equation holds for n=k+1.
By mathematical induction, the equation 1 + 3 + 5 + ... + (2n - 1) = n² is true for every positive integer n.

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