Find the area of an equilateral triangle with apothem length .
if necessary, write your answer in simplified radical form.

Since the limit of the ratio is less than 1 (1/3 < 1), the series is absolutely convergent (Option A). To determine the convergence of the series, we can use the Ratio Test. The series in question is:

Σ (3n² + 3) / (3n³ + 1)
First, we find the ratio between consecutive terms:
R = (a_(n+1)) / a_n = ((3(n+1)² + 3) / (3(n+1)³ + 1)) / ((3n² + 3) / (3n³ + 1))
R = (3(n+1)² + 3)(3n³ + 1) / ((3n² + 3)(3(n+1)³ + 1))
Now, as n goes to infinity:
lim (n -> ∞) R = lim (n -> ∞) (9n² + 6n + 3)(3n³ + 1) / (9n² + 3)(9n³ + 3n² + 3n + 1)
In this case, the highest power of n in the numerator and denominator is n⁵. To simplify, we can divide both the numerator and denominator by n⁵:
lim (n -> ∞) (9 + 6/n + 3/n²)(3 + 1/n²) / (9 + 3/n)(9 + 3/n² + 3/n + 1/n³)
As n approaches infinity, the terms with n in the denominator approaches zero:
lim (n -> ∞) (9)(3) / (9)(9) = 27 / 81 = 1/3
Since the limit of the ratio is less than 1 (1/3 < 1), the series is absolutely convergent (Option A).

To determine whether the series 3n² +  3n³ +1 is absolutely convergent, conditionally convergent, or divergent, we can use the ratio test.
Using the ratio test, we have:
lim n→∞ |(3(n+1)² + 3)/(3n² + 3n³ +1)
= lim n→∞ |(3n² + 6n + 3)/(3n³ + 3n² +1)
= lim n→∞ |(n² + 2n + 1)/(n³ + n²)
= 0
Since the limit is less than 1, the series is absolutely convergent. Therefore, the answer is A. Absolutely convergent.

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The estimated daily cost of increasing production to 42 televisions is $3717. Therefore, the closest answer choice is B) $3919.

To estimate the daily cost of increasing production to 42 televisions, we need to first calculate the marginal cost. The marginal cost is the derivative of the cost function:

[tex]C'(x) = -0.009x^2 + 0.2x + 71[/tex]

We can then evaluate the marginal cost at x=40 to find the cost of producing one additional television:

[tex]C'(40) = -0.009(40)^2 + 0.2(40) + 71 = $33.40[/tex]

This means that the cost of producing one additional television when 40 are already being produced is $33.40. To estimate the daily cost of increasing production to 42 televisions, we can multiply the marginal cost by 2 (since we want to produce 2 additional televisions):

[tex]2*$33.40 = $66.80[/tex]

Finally, we can add this estimated cost to the current cost of producing 40 televisions:

[tex]C(40) = -0.003(40)^3 + 0.1(40)^2 + 71(40) + 540 = $3650[/tex]

$3650 + $66.80 = $3716.80

Rounding to the nearest dollar, the estimated daily cost of increasing production to 42 televisions is $3717. Therefore, the closest answer choice is B) $3919.

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The amount of profit when there are 0 visitors.

In the context of the given quadratic function, the y-intercept represents the amount of profit when there are 0 visitors.

To explain, the y-intercept is the point at which the function intersects the y-axis. This occurs when the x-value (number of visitors in thousands) is 0. In the given table, the y-intercept corresponds to the data point (0, 0), meaning there is a profit of $0 thousands when there are 0 visitors.

Looking at the table, we can see that when the number of visitors is 0, the profit is also 0. Therefore, the y-intercept of the function represents the amount of profit when there are 0 visitors.

So the correct option is: "the amount of profit when there are 0 visitors".

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To find the volume (cubic inches) of the ball, we first need to find its radius, which is half the diameter.

Radius = 12 in / 2 = 6 in

Now we can use the formula for the volume of a sphere:

Volume = (4/3) x π x radius^3 cubic inches

Volume = (4/3) x π x 6^3

Volume ≈ 904.8 cubic inches

So the ball holds approximately 904.8 cubic inches of air. Rounded to the nearest tenth, the answer is 904.8.

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IF the median height is exactly 68 inches, then exactly 50% of the heights are at least 68 inches.

Without seeing the box-and-whisker plot, it is difficult to determine the exact fraction of heights that are at least 68 inches. However, we can make an estimate based on the general characteristics of a box-and-whisker plot.

In a box-and-whisker plot, the "box" represents the middle 50% of the data, with the median (50th percentile) marked by a line inside the box. The "whiskers" represent the minimum and maximum values within 1.5 times the interquartile range (IQR) of the data. Any points outside the whiskers are considered outliers.

Assuming that there are no outliers in the data, we can estimate that at least 50% of the heights are above the median, which is marked by the line inside the box. If the median height is at least 68 inches, then at least 50% of the heights are at least 68 inches.

If we assume that the median height is exactly 68 inches, then exactly 50% of the heights are at least 68 inches.

If the median height is less than 68 inches, then we can estimate that slightly less than 50% of the heights are at least 68 inches.

In summary, without more information about the box-and-whisker plot or the data it represents, we can estimate that at least 50% of the heights are at least 68 inches.

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Considering the following box plot, The interquartile range is a measure of the spread of the middle 50% of the data.

The interquartile range (IQR) is a measure of statistical dispersion that represents the range between the first quartile (Q1) and the third quartile (Q3) in a dataset. It provides a measure of the spread or variability of the middle 50% of the data.

However, explain how to calculate the interquartile range and the percentage of values included in the interquartile range based on a box plot:

(a) To find the interquartile range, you need to calculate the difference between the upper quartile (Q3) and the lower quartile (Q1). In other words, IQR = Q3 - Q1. The interquartile range is a measure of the spread of the middle 50% of the data.

(b) The interquartile range includes 50% of the values in the data set. This means that the other 50% of values lie outside the interquartile range.

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The height of the cone is 3.4cm

A cone is a shape formed by using a set of line segments or the lines which connects a common point, called the apex or vertex.

The volume of a cone is expressed as;

V = 1/3 πr²h

where πr² = base area. therefore the volume can be written as;

V = 1/3 base area × height

base area = 49cm²

height = 45.3 cm³

45 = 1/3 ×40h

135 = 40h

h = 135/40

h = 3.4cm

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Answer:

a) £54

b) £1908

Step-by-step explanation:

a) use £1800 × 3% = £54

b) use £1800 + ( 54 × 2) = £1908

The radius of the circle is 10 inches.

A sector is a part of a given circle which is made from two radii and an arc. It's area can be determined by;

area of a sector = θ/360*πr^2

where θ is the measure of its central angle, and r is its radius.

Then from the given question, we have;

area of a sector = θ/360*πr^2

162 = 180/360 *3.14*r^2

      = 1.57r^2

r^2 = 162/1.57

     = 103.1847

So that;

r = (103.1847)^1/2

 = 10.16

The radius of the circle is approximately 10 inches.

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The volume of the composite figure is 96.3 cm³

First we need to find the volume of the cylinder, and then remove the volume of the rectangular prism.

The radius of the prism is 3cm and the height is 5cm, then the volume is:

V = 3.14*R²*H

V = 3.14*(3cm)²*5cm

V = 141.3 cm³

And the volume of the prism is:

V' = 3cm*3cm*5cm = 45 cm³

The difference gives:

volume = 141.3 cm³ -  45 cm³

volume = 96.3 cm³

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Each part of the circle given it’s equation should be identified as follows;

In Geometry, the standard or general form of the equation of a circle is modeled by this mathematical equation;

(x - h)² + (y - k)² = r²

Where:

Based on the information provided above, we have the following parameters:

Radius, r = √400 units.

Center, (h, k) = (9, 4).

By substituting the given radius and center into the equation of a circle, we have;

(x - h)² + (y - k)² = r²

(x - 9)² + (y - 4)² = (√400)²

(x - 9)² + (y - 4)² = 400.

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The mean travel time is 60 minutes, the standard deviation is approximately 6.08 minutes, and the probability that Alice will not be late for her class is 0.50 or 50%.

To find the mean travel time, we need to add up the mean times for each stage of Alice's journey. Let's calculate it step by step:

Step 1: Alice's walking time from home to the bus stop:

Mean walking time = 13 minutes

Step 2: Waiting time for the bus to arrive:

Mean waiting time = 5 minutes

Step 3: Bus journey time from the bus loop:

Mean bus journey time = 24 minutes

Step 4: Walking time from the bus loop to the lecture theatre:

Mean walking time = 18 minutes

Now, let's calculate the total mean travel time:

Mean travel time = Mean walking time + Mean waiting time + Mean bus journey time + Mean walking time

= 13 + 5 + 24 + 18

= 60 minutes

So, the mean travel time is 60 minutes.

To find the standard deviation of Alice's travel time, we need to calculate the variance for each stage and then sum them up. Finally, we take the square root to get the standard deviation. Let's calculate it step by step:

Step 1: Alice's walking time from home to the bus stop:

The variance of walking time = 4 minutes squared

Step 2: Waiting time for the bus to arrive:

The standard deviation of waiting time = 2 minutes

Step 3: Bus journey time from the bus loop:

The standard deviation of bus journey time = 5 minutes

Step 4: Walking time from the bus loop to the lecture theatre:

The variance of walking time = 4 minutes squared

Now, let's calculate the total variance of travel time:

Variance of travel time = Variance of walking time + Variance of waiting time + Variance of bus journey time + Variance of walking time

= 4 + 4 + 25 + 4

= 37 minutes squared

Finally, the standard deviation of travel time is the square root of the variance:

The standard deviation of travel time = [tex]\sqrt(37)[/tex]

≈ 6.08 minutes (rounded to 2 decimal places)

So, the standard deviation of Alice's travel time is approximately 6.08 minutes.

To find the probability that Alice will not be late for her class, we need to calculate the z-score for the desired arrival time and then find the corresponding probability from the standard normal distribution table. Let's calculate it step by step:

Step 1: Calculate the total travel time from home to the lecture theatre:

Total travel time = Mean travel time = 60 minutes

Step 2: Calculate the difference between the desired arrival time and the total travel time:

Time difference = 8 am - 7 am = 1 hour = 60 minutes

Step 3: Calculate the z-score using the formula:

z = (Time difference - Mean travel time) / Standard deviation of travel time

z = [tex]\frac{(60 - 60) }{ 6.08}[/tex]

z = 0

Step 4: Find the probability corresponding to the z-score from the standard normal distribution table.

Since the z-score is 0, the probability is 0.50 (or 50%).

Therefore, the probability (to 2 decimal places) that Alice will not be late for her class is 0.50 or 50%.

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The value of x is -4.

We have,

(3x + 25) and (6x - 2) makes one angle.

So,

(9x + 23) is the angle.

Now,

It is bisected in two angles.

So,

9x + 23 = 1/2 x (6x - 2)

9x + 23 = 3x - 1

9x - 3x = -1 - 23

6x = -24

x = -4

Thus,

The value of x is -4.

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The mean winnings for all of the show's contestants can be estimated with 92% certainty. 35014.48385 is the lower bound, while 40669.38281 is the upper bound.            

Lower Bound = [tex]X - t(\alpha/2) * s / \sqrt{(n)[/tex]            

Upper Bound = [tex]X + t(\alpha/2) * s / \sqrt{(n)[/tex]

where                

[tex]\alpha/2 = (1 - confidence\: level)/2 = 0.04 \\ X = sample\: mean = 37841.93333 \\ t(\alpha/2) = critical\: t \:for \:the\: confidence\: interval = 1.887496145 \\ s = sample\: standard\: deviation = 5801.688541 n = sample\: size = 15 \\ df = n - 1 = 14[/tex]          

Thus,          

Lower bound =     35014.48385            

Upper bound =     40669.38281            

A lower bound refers to the smallest possible value or limit that a given quantity or parameter can take. In various fields of mathematics and computer science, lower bounds are used to establish limits on the performance of algorithms, the complexity of computational problems, and the amount of resources required to solve a problem. This information can be useful in developing more efficient algorithms or determining the practicality of a given approach.

Lower bounds are useful for understanding the fundamental limits of a system or process. By establishing a lower bound, researchers and practitioners can better understand the potential of a given technology or approach, and can work to optimize it within the constraints imposed by the lower bound.

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Complete Question:-

How much money do winners go home with from the television quiz show Jeopardy? To determine an answer, a random sample of winners was drawn and the amount of money each won was recorded and listed below. Estimate with 92% confidence the mean winning's for all the show's players.

30692 43231 48269 28592 28453

36309 45318 36362 42871  39592

35456 40775 36466 36287 38956

Lower confidence level (LCL) = ?

Upper confidence level (UCL) = ?

a) The number of cities that had only a professional sports team can be found by subtracting the number of cities that had a professional sports team and a symphony, the number of cities that had a professional sports team and a children's museum, and the number of cities that had all three activities from the total number of cities:

33 - (9 + 6 + 3) = 15 cities had only a professional sports team.

b) The number of cities that had a professional sports team and a symphony, but not a children's museum can be found by subtracting the number of cities that had all three activities from the number of cities that had a professional sports team and a symphony:

9 - 3 = 6 cities had a professional sports team and a symphony, but not a children's museum.

c) The number of cities that had a professional sports team or a symphony can be found by adding the number of cities that had a professional sports team, the number of cities that had a symphony, and then subtracting the number of cities that had both:

17 + 15 - 9 + 14 - 6 + 3 = 34 cities had a professional sports team or a symphony.

d) The number of cities that had a professional sports team or a symphony, but not a children's museum can be found by subtracting the number of cities that had all three activities from the answer to part c:

34 - 3 = 31 cities had a professional sports team or a symphony, but not a children's museum.

e) The number of cities that had exactly two of the activities can be found by adding up the number of cities that had a professional sports team and a symphony, the number of cities that had a professional sports team and a children's museum, and the number of cities that had a symphony and a children's museum, and then subtracting twice the number of cities that had all three activities:

9 + 6 + 6 - 2(3) = 15 cities had exactly two of the activities.

The greater number is 455.

There are 197 women in the party.

The geometric ratio of the old and current price of the suit is 25/27.

The first problem requires the application of geometric ratios and algebraic manipulation to determine the greater of the two numbers. Geometric ratios are ratios between two quantities that are constant throughout.

We are also given that their difference is 35, which can be expressed as x - y = 35. We can use algebraic manipulation to solve for the values of x and y.

From the first equation, we can express x in terms of y as x = (13/6)y. Substituting this value of x into the second equation, we get (13/6)y - y = 35. Simplifying this equation, we get y = 210.

To find the value of x, we can substitute y = 210 into the equation x/y = 13/6, giving us x = 455. Therefore, the greater number is 455.

The second problem involves using ratios to find the number of women in a party. We are given that the ratio of men to women is 9 to 7, which can be expressed as 9x/7x, where x is a constant. We are also told that there are 45 men. We can use this information to solve for the number of women.

Therefore, the total number of parts is 45/9 = 5.

We can use this information to find the number of women, which is 7 parts of the ratio, or

=>  7x = (7/16) * 5 * 45 = 196.875.

Since we cannot have a fraction of a person, we round this value up to the nearest whole number, which is 197.

Therefore, there are 197 women in the party.

The third problem involves finding the geometric ratio of the old and current price of a men's suit. We are given that the suit cost $250,000 last year and that a dozen of these suits cost $3,250,000 this year. We can use the information provided to find the geometric ratio.

Since a dozen of the suits cost $3,250,000, one suit costs $3,250,000/12 = $270,833.33. The ratio of the old price to the new price is 250,000/270,833.33, which simplifies to 25/27.

Therefore, the geometric ratio of the old and current price of the suit is (25/27)¹ = 25/27.

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Complete Question:

The geometric ratio of two numbers is 13/6 and their difference is 35. What is the greater number?

At a party the ratio of men to women is 9 to 7. If 45 men are counted, how many women are there?

A men's suit cost $250,000 last year. This year a dozen of these suits cost $3,250. 000 What is the geometric ratio of the old and current price of the suit?

Answer:

SSA

Step-by-step explanation:

It would be SSA (Side-Side-Angle).  They are congruent where they intersect at B (opposite angles).  Since CF is congruent to GH, and CB is congruent to HB, you have an angle and two sides congruent, in the order SSA.

Answer is: manager can be 95% confident that the true proportion of time spent mowing is between 0.452 and 0.548

We can use the normal approximation to the binomial distribution to estimate the confidence interval for the true proportion of time spent mowing.

The sample proportion of time spent mowing is:

p = 200/400 = 0.5

The standard error of the sample proportion is:

SE = sqrt(p(1-p)/n) = sqrt(0.5 * 0.5 / 400) = 0.025

To find the 95% confidence interval for the true proportion of time spent mowing, we can use the formula:

p ± z*SE

where z* is the critical value from the standard normal distribution corresponding to a 95% confidence level, which is approximately 1.96.

So the 95% confidence interval for the true proportion of time spent mowing is:

0.5 ± 1.96*0.025 = (0.452, 0.548)

Therefore, the manager can be 95% confident that the true proportion of time spent mowing is between 0.452 and 0.548.

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These are answers of asked question.

A. To factor out the Greatest Common Factor (GCF) of the expression 3t^4 - 6t^3 - 9t + 12, we need to identify the highest power of t that can be factored out. In this case, the GCF is 3t. So we can rewrite the expression as follows:

3t^4 - 6t^3 - 9t + 12 = 3t(t^3 - 2t^2 - 3) + 3t(4)

The GCF, 3t, is factored out from the first two terms, leaving us with t^3 - 2t^2 - 3. The last term, 12, is divisible by 3t, so it becomes +3t(4). Therefore, the factored form of the expression is:

3t(t^3 - 2t^2 - 3) + 3t(4)

B. To factor the expression g^2 - 5g - 14 using the Distributive Method, we look for two numbers whose product is -14 and whose sum is -5 (the coefficient of the middle term). In this case, -7 and +2 satisfy these conditions. So we can rewrite the expression as follows:

g^2 - 5g - 14 = (g - 7)(g + 2)

Using the Distributive Property, we multiply (g - 7) by (g + 2) to verify the factoring:

(g - 7)(g + 2) = g(g) + g(2) - 7(g) - 7(2) = g^2 + 2g - 7g - 14 = g^2 - 5g - 14

Therefore, the factored form of the expression is:

(g - 7)(g + 2)

C. To factor the expression r^2 - 64, we can use the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). In this case, a = r and b = 8, since 8^2 = 64. So we can rewrite the expression as follows:

r^2 - 64 = (r + 8)(r - 8)

Using the difference of squares formula, we can multiply (r + 8) by (r - 8) to verify the factoring:

(r + 8)(r - 8) = r(r) - r(8) + 8(r) - 8(8) = r^2 - 8r + 8r - 64 = r^2 - 64

Therefore, the factored form of the expression is:

(r + 8)(r - 8)

D. To factor the expression 9p^2 - 42p + 49, we look for two numbers whose product is 49 and whose sum is -42 (the coefficient of the middle term). In this case, -7 and -7 satisfy these conditions. So we can rewrite the expression as follows:

9p^2 - 42p + 49 = (3p - 7)(3p - 7)

Using the Distributive Property, we multiply (3p - 7) by (3p - 7) to verify the factoring:

(3p - 7)(3p - 7) = 3p(3p) - 3p(7) - 7(3p) - 7(7) = 9p^2 - 21p - 21p +

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The correct way to evaluate the integral of (25x² + 46x + 13)dx is:

∫(25x² + 46x + 13)dx = 4ln|x + 1| + 21ln|x - 1| + C

Here, ln represents the natural logarithm. The arbitrary constant, denoted by C, represents any constant value that could be added to the antiderivative without changing its derivative.
To evaluate the given integral, we will first rewrite it using the correct mathematical notation:

∫(25x² + 46x + 13) dx

Now, we will use integration techniques to find the antiderivative. In this case, we can apply the power rule and linearity of the integral:

∫(25x²) dx + ∫(46x) dx + ∫13 dx

Now we will evaluate each integral separately:

(25/3)x³ + (46/2)x² + 13x + C

Now, we can simplify the expression:

(25/3)x³ + 23x² + 13x + C

So the evaluated integral is:

(25/3)x³ + 23x² + 13x + C

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The distance traveled is 8ft and the displacement is 0ft.

The distance traveled is equal to the total distance that the ball travels, in this case it starts 4ft above the ground, then goes to the ground, and then returns to the initial position which is 4ft above the ground, then the total distance is 4ft + 4ft = 8ft

The displacement is equal to the difference between the final position and the inital one, here we know that both are the same, thus, the displacement is  0ft.

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The required equation for the length of the candle is L = 9-0.75t.

Given that a 9-inch candle burns at the rate of 0.75 in per hour we need to determine the equation for the length L of the candle when it burns for t hours,

So, if the candle is burning at the rate of 0.75 in per hour so after burning for t hour the length decreases by 0.75t, from the original length i.e 9 in

So, we can write the equation as =

L = 9-0.75t

Hence the required equation for the length of the candle is L = 9-0.75t.

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Based on the equation, when Lisa sells 24 copies of Math is Fun, her total pay will be $3140.

The equation relating P to N is:

P = 1700 + 60N

This is because her base salary is $1700, and she earns an additional $60 for each copy of Math is Fun she sells.

In order to find her total pay if she sells 24 copies of Math is Fun, we simply need to substitute N = 24 into the equation:

P = 1700 + 60(24)

P = 1700 + 1440

P = 3140

Therefore, if Lisa sells 24 copies of Math is Fun, her total pay will be $3140.

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The distance between the points (1, 1) and (1, -4) is 5 units and the reflection of the coordinate points is (-4, 6.2).

(a) We need to find the length between the pair of points which are (1, 1) and (1, −4). It can be determined by using the distance formula. The formula to find the distance between two points (x1, y1) and (x2, y2) is given as,

[tex]d = √(x2−x1)²+(y2−y1)²[/tex]

Given data:

The first points are = (1, 1)

second points are =(1, −4)

Substituting the first and second values into the distance formula, we get:

= √(1−1)²+(−4−1)²

= √0+(−5)²

= √25​

= 5

Therefore, The distance between the points (1, 1) and (1, -4) is 5 units.

(b )We need to find the reflection of the coordinate point. According to the reflection rule when a point is reflected across the y-axis, the x-coordinate changes sign while the y-coordinate remains the same. The reflected point will be,

= (x, y) = (-x, y).

Given Data:

Coordinate points = (4, 6.2)

According to the rule, it is given as:

= (4, 6.2)

= (-4, 6.2)

Therefore, the reflection of the coordinate points is (-4, 6.2)

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Therefore, the total area of the forest is approximately 43.10 acres.

Area is the measure of the size of a two-dimensional surface or region. It is typically measured in square units, such as square meters (m²) or square feet (ft²). The area of a shape can be calculated by multiplying the length and the width of the shape. Different shapes have different formulas for calculating their area, such as the formula for the area of a rectangle, which is length x width, or the formula for the area of a circle, which is πr², where r is the radius of the circle and π is a mathematical constant approximately equal to 3.14159.

Here,

Let's call the total area of the forest "T".

We know that 25 acres of the forest is covered by coniferous trees. Therefore, the remaining area of the forest is:

T - 25

We also know that the shrubs and bushes constitute 42% of the total area of the forest. This means that:

0.42 x T = area covered by shrubs and bushes

Putting it all together, we can set up the following equation:

T = 25 + (0.42 x T)

Simplifying, we can solve for T:

0.58 x T = 25

T = 25 / 0.58

T = 43.10 acres (rounded to two decimal places)

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Mario can afford to buy a maximum of 12 hamburgers from the local store.

Let's assume Mario can buy "h" hamburgers.

The cost of each hamburger is $2.90, and Mario wants to buy "h" hamburgers, so the total cost of hamburgers would be 2.90h.

He is also buying one packet of hamburger rolls, which costs $4.75.

Therefore, the total amount he can spend at the store must be less than or equal to his budget of $39.55.

Putting it all together, the inequality representing this situation is:

2.90h + 4.75 ≤ 39.55

To find out how many hamburgers Mario can afford to buy, we need to solve the inequality:

2.90h + 4.75 ≤ 39.55

Subtracting 4.75 from both sides of the inequality:

2.90h ≤ 34.80

Next, divide both sides of the inequality by 2.90:

h ≤ [tex]\frac{34.80 }{ 2.90}[/tex]

Simplifying the right side:

h ≤ 12

Therefore, Mario can afford to buy a maximum of 12 hamburgers from the local store.

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The length of the diagonal of the house is 23.43 meters based on the dimensions of the John's guest house.

The diagonal of the house will be calculated using Pythagoras theorem. We know that diagonal will be hypotenuse and length and width will be base and perpendicular.

Diagonal = ✓length² + width²

Keep the values in formula

Diagonal = ✓18² + 15²

Taking square

Diagonal = ✓324 + 225

Adding the values

Diagonal = ✓549

Taking square root

Diagonal = 23.43 meters

Hence, the diagonal of the house is 23.43 meters.

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The measure of angle ABC is 30 degrees.

To find the measure of angle ABC, we need to use the fact that the sum of angles in a triangle is always 180 degrees. Since we know that angle ABD is a right angle (90 degrees) and angle CBD is 60 degrees, we can find the measure of angle ABC as follows:

angle ABD + angle CBD + angle ABC = 180 degrees

Substituting the given values, we get:

90 degrees + 60 degrees + angle ABC = 180 degrees

Simplifying, we get:

150 degrees + angle ABC = 180 degrees

Now we can solve for angle ABC by subtracting 150 degrees from both sides:

angle ABC = 180 degrees - 150 degrees

angle ABC = 30 degrees

Therefore, angle ABC measures 30 degrees.

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The width of the gravel pathway is 7 meters.

The length of the rectangular garden is 50m and the width is 13m. Let's assume the width of the gravel pathway to be w meters.

The length of the rectangular garden including the two widths of the pathway would be 50+2w meters, and the width including the two widths of the pathway would be 13+2w meters.

The area of the rectangular garden including the pathway is the product of the length and the width:

(50+2w)(13+2w)

We can now set up an equation using the area of the garden and the amount of gravel available:

(50+2w)(13+2w) - 50*13 = 80

Simplifying this equation gives:

4w^2 + 126w - 3196 = 0

This is a quadratic equation that we can solve for w using the quadratic formula:

w = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 4, b = 126, and c = -3196.

Plugging in these values and solving for w gives:

w = 7 or w = -22.75

Since the width of the pathway cannot be negative, the only valid solution is w = 7.

Therefore, the width of the gravel pathway is 7 meters.

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