what are the X and y-intercepts of the equation y = log (7x + 3 ) - 2 round to the nearest hundredth​

By Pythagorean , Eli's home is thus roughly **11.4 miles** from Salem when viewed from a straight line.

A fundamental relationship between a right triangle's three sides in Euclidean geometry is known as the Pythagorean theorem. The hypotenuse is the side that forms the right angle, and the rule says that the square of its length is equal to the sum of the squares of the lengths of the other two sides. In other words, if the hypotenuse is length c and the legs of a right triangle are lengths a and b, then a²+ b² = c² .

Call Yardley Y, Salem S, and Eli's home E. As far as we are aware, Y is located 7 miles from E and 9 miles from S.

We can check that the distance between Eli's house and Salem is the hypotenuse of a right triangle with Yardley as one of the vertices because Eli's house is located due west of Yardley and due south of Salem. The Pythagorean theorem can be used to determine this distance.

Eli's home is 11.4 miles from Salem at a distance of√(7 + 9 ) = √(130).

√(130) = 11.4miles

Eli's home is thus roughly **11.4 miles** from Salem when viewed from a straight line.

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If bout 1.02 X 102 times farther away from the sun than Mercury. The number of kilometers is: C. about 5.91 X 10⁹ kilometers.

If Pluto is 1.02 x 10² times farther away from the Sun than Mercury, we can use the average distance of Mercury from the Sun, which is approximately 57.91 million km (5.791 x 10⁷ km) to calculate the distance of Pluto from the Sun.

Distance of Pluto from the Sun = (1.02 x 10²) x (57.91 x 10⁶ km)

Distance of Pluto from the Sun = 5.91 x 10⁹ km

Therefore, the correct option is: about 5.91 x 10⁹ kilometers.

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In a song the twelve days of christmas, my true love gave to me, then total number of possible gifts that my true love give me all together during the twelve days is equal to the 364.

We have There are twelve days of Christmas in a song and my true love gives me presents every day. We have to calculate total number of gifts my true love give me all together during the twelve days. Now, according to the song, on the first day of Christmas my true love gave me: a partridge on a pear tree 1 gift. On the second day of Christmas, my love really gave me: two doves and a gift. cake etc. Mathematically, gifts by true love are written as 1+2+3+... n gifts per day. Number of gifts per day: 1 to 1, 2 to 3, 3 to 6 and 4 to 10. The amount of mis N (N + 1) (N + 2) / 6. Number of parts This form is called tetrahedral number. Change the value N = 12, then N (N + 1) (N + 2) / 6

= 12 × 13 × 14 / 6

= 364

Hence, 364, the total number of gifts, almost one for each day of the year.

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Answer:

Step-by-step explanation:

6 greater than a number is 24.

This means adding 6 to a number, x, will equal 24.

6 + x = 24

subtract 6 from both sides of the equation, and you are left with x=18

18 is your answer!!

The expected number of flips until getting the first heads is equal to the reciprocal of the probability of getting heads on a single flip, which is 1/p.

The number of independent flips x until the first heads follows a geometric distribution. In general, the probability of getting the first heads on the kth flip is given by [tex](1-p)^(^k^-^1^)[/tex]p. Thus, the expected value or mean of the geometric distribution is E(x) = 1/p.

This means that on average, we need to flip the coin 1/p times until we get the first heads. For example, if the probability of getting heads is 0.5, then on average we need to flip the coin 2 times until we get the first heads. If the probability of getting heads is 0.1, then on average we need to flip the coin 10 times until we get the first heads.

The geometric distribution is a fundamental concept in probability theory and has many applications in fields such as finance, engineering, and computer science.

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$1000 is 71.43 percent of the credit limit.

To lower the debt to credit limit ratio by $1000, you have a few options. The first option is to increase your credit limit by requesting a higher limit from your credit card issuer. Alternatively, you can pay off some of your outstanding balance to reduce your debt. Either way, it's important to make sure you're not maxing out your credit limit, as this can negatively impact your credit score.

It's also a good idea to review your spending habits and create a budget to ensure you're not overspending and accumulating debt. By making responsible financial decisions and managing your credit wisely, you can improve your credit score and achieve your financial goals.

To calculate the percentage of $1000 in terms of the credit limit of $1400, we need to divide $1000 by $1400 and multiply the result by 100. This gives us,

($1000 / $1400) x 100 = 71.43%

Therefore, $1000 is approximately 71.43% of the credit limit of $1400.

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There are 26,000 distinct license plates that can be issued consisting of one letter followed by a three-digit number.

To find the number of distinct license plates that can be issued consisting of one letter followed by a three-digit number, we need to consider the number of possibilities for each element of the license plate.

For the first element, there are 26 possible letters in the alphabet (assuming only English letters are used).

For the second element, there are 10 possible digits (0 to 9) that can be used.

For the third element, there are also 10 possible digits that can be used.

For the fourth element, there are again 10 possible digits that can be used.

Therefore, to find the total number of distinct license plates, we need to multiply the number of possibilities for each element:

26 × 10 × 10 × 10 = 26,000

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Thus, the  area of the shaded region which is found by subtracting the right triangle's area from the rectangle is 30 sq. in.

Every triangle has inner angles that add up to 180 degrees. A right angle and a right triangle are both formed when one of their internal angles is 90 degrees. The internal 90° angle of right triangles is denoted by a little square in the vertex.

Given data:

Shaded area = area of rectangle - area of right triangle

Shaded area = length * width - 1/2*base*height

Shaded area = 10*5  - 1/2*8*5

Shaded area = 50 - 20

Shaded area = 30 sq. in.

Thus, the  area of the shaded region which is found by subtracting the right triangle's area from the rectangle is 30 sq. in.

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complete question-

A right triangle is removed from a rectangle to create the shaded region shown below. Find the area of the shaded region.

Dimensions of rectangle - 10 in x 5 in

base of right triangle b = 8 in

Duncan is approximately 10.8 kilometers away from his grandparents' house, if necessary, rounding to the nearest tenth.

Based on the given information, we can use the Pythagorean theorem to find the distance between Duncan's house and his grandparents' house. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the distance from Duncan's house to his parents' house (6 km) and the distance from his parents' house to his grandparents' house (9 km) form the two shorter sides of the triangle. To find the distance between Duncan's house and his grandparents' house, we will calculate the hypotenuse:
h² = 6² + 9²
h² = 36 + 81
h² = 117
h ≈ 10.8

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Duncan is approximately 6.7 km from his grandparents' house (rounded to the nearest tenth).

We can use the Pythagorean theorem to find the distance between Duncan's house and his grandparents' house.
Identify the triangle
In this problem, we have a right triangle with legs of 6 km (due north) and an unknown length (due east).

The hypotenuse is the straight-line distance of 9 km from his parents' house to his grandparents' house.
Apply the Pythagorean theorem.
The Pythagorean theorem states that a² + b² = c², where a and b are the legs of the triangle, and c is the hypotenuse. In this case, a = 6 km, b = unknown, and c = 9 km.
Solve for the unknown distance (b)
6² + b² = 9²
36 + b² = 81
b² = 81 - 36
b² = 45
b = √45
b ≈ 6.7 km.

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There are 120 different ways to award the gold, silver, and bronze medals to the 10 women in the 100 meters race.

The number of ways that the gold, silver, and bronze medals can be awarded to 10 women is a combination problem.

We can use the formula for combinations to determine the number of ways to choose 3 women out of 10 for the medals:

[tex]nCr = n! / (r! * (n - r)!)[/tex]

Where n is the total number of women (10) and r is the number of medals (3).

So, the number of ways to award the gold, silver, and bronze medals to the 10 women is:

[tex]10C3 = 10! / (3! * (10 - 3)!) = 120[/tex]

Therefore, there are 120 different ways to award the gold, silver, and bronze medals to the 10 women in the 100 meters race.

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There are 67,108,863 outcomes in which at least one head occurs if a fair coin is tossed 26 times.

The probability of getting tails on any given toss is 1/2, so the probability of getting tails on all 26 tosses is (1/2)^26. Therefore, the probability of getting at least one head is

1 - (1/2)^26 ≈ 0.999999999996

So there are almost 100% chance of getting at least one head in 26 coin tosses. To find the number of outcomes that satisfy this condition, we can use the formula for combinations

ⁿCₓ = n! / (x! * (n-x)!)

where n is the total number of trials (26 in this case), x is the number of successes (at least 1 head), and ! denotes the factorial function (e.g., 5! = 54321).

Using this formula, we get

26C1 + 26C2 + ... + 26C26

which simplifies to

2^26 - 1 = 67,108,863

So there are 67,108,863 outcomes in which at least one head occurs in 26 coin tosses.

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The value of length XW and WV are 20 and 25 respectively.

Similar triangles are triangles that have the same shape, but their sizes may vary. The ratio of corresponding sides of similar triangles are equal.

Therefore XW/VW = XY/XZ

= XW/45 = 24/54

54(XW) = 45× 24

54(XW) = 1080

divide both sides by 54

XW = 1080/54

XW = 20

Therefore WV can be found by subtracting XW for VX

= 45-20 = 25

therefore the value of XW and WV are 20 and 25 respectively.

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The difference between the future totals of $2800 invested at 3% simple interest for 4 years and $2800 invested at 3% compound interest, compounded annually for 4 years is $15.4246 .

Suppose a sum of money is taken out on a loan for a period of n years at r% annual rate of interest at compound interest compounded annually. Then after every year, interest accrued in the last year, is not withdrawn or paid back, but it is left with the loanee. so it is considered to be loaned to the loanee. So that for the next year, the total loan amount increases by this amount. So the new principal is P + Pr = P(1+r). Continuing this idea, the total amount after n years is [tex]$A=P{(1+r)}^n$[/tex].  

In our question for the first investment P = $2800, annual rate of interest = 3%, no of periods n = 4.

So using the formula A = P(1+nr) = 2800(1 + (0.03)(4)) = 2800(1.12) = 3136

For the second investment P = $2800. annual rate of interest r = 0.03, no of

periods = 4, and it is invested at compound interest, compounded annually

So [tex]$A = P {(1 + r)}^n = 2800{(1 + 0.03)}^4 = 2800{(1.03)}^4 = 3151.4246$[/tex] .

The difference in the two total amounts = $3151.4246 - $3136 = $15.4246.

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The difference between the future total of 3% simple interest for 4 years and 3% compound interest for 4 years, compounded annually, is:

$349.40 - $336 = $13.40

Simple interest is a type of interest that is calculated only on the original amount of money borrowed or invested, called the principal. It is a linear calculation and does not take into account any interest earned on previously earned interest. The formula for simple interest is:

Simple Interest (SI) = Principal (P) x Rate (R) x Time (T)

Compound interest is a type of interest that is calculated on the initial principal as well as on the accumulated interest from previous periods. It is a compounding calculation that allows for interest to be earned on interest. Compound interest can be compounded annually, semi-annually, quarterly, monthly, or even daily, depending on the frequency of compounding. The formula for compound interest is:

Compound Interest (CI) = [tex]p[/tex][tex](1+R)^{T}[/tex][tex]-p[/tex]

The difference between the future total of 3% simple interest for 4 years on $2800 and 3% compound interest for 4 years, compounded annually, can be calculated as follows:

For simple interest:

The formula for calculating simple interest is:

Simple Interest (SI) = P×R×T

Where:

Principal (P) = $2800

Rate (R) = 3% = 0.03 (as a decimal)

Time (T) = 4 years

Plugging in these values:

SI = $2800 x 0.03 x 4 = $336

For compound interest:

The formula for calculating compound interest is:

Compound Interest (CI) = [tex]P[/tex] [tex](1+R)^{T}[/tex]-[tex]P[/tex]

Where:

Principal (P) = $2800

Rate (R) = 3% = 0.03 (as a decimal)

Time (T) = 4 years

Plugging in these values:

CI = $2800 x (1 + 0.03)⁴ - $2800

CI = $2800 x (1.03)⁴ - $2800

CI = $2800 x 1.1255 - $2800

CI = $3149.40 - $2800

CI = $349.40

The difference between the future total of 3% simple interest for 4 years and 3% compound interest for 4 years, compounded annually, is:

$349.40 - $336 = $13.40

So, the difference is $13.40.

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100 is part of the solution to the inequality 1.25(x + 16) ≥ 140.75.

What is an inequality equation?

An inequality equation is a mathematical statement that compares two expressions using an inequality symbol such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

To determine if 100 is part of the solution to the inequality 1.25(x + 16) ≥ 140.75, we can substitute 100 for x and see if the inequality holds true.

1.25(x + 16) ≥ 140.75

1.25(100 + 16) ≥ 140.75

1.25(116) ≥ 140.75

145 ≥ 140.75

Since 145 is greater than or equal to 140.75, the inequality holds true when x = 100.

Therefore, 100 is part of the solution to the inequality 1.25(x + 16) ≥ 140.75.

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Carlos can plant 8 bulbs in each row.

Given that, Carlos is having 32 iris bulbs and 40 tulip bulbs he wants to plant them, he chose to plant them in rows with alternate pattern, also wants all of the rows to have the same number of bulbs.

So, we need to find the number of the rows to have the same number of bulbs,

To find the same we will find the GCF of the two number,

GCF = The GCF is also known as the Highest Common Factor (HCF).

Find the GCF of 32, and 40

32 = 2 × 2 × 2 × 2 × 2

40 = 2 × 2 × 2 × 5

The greatest common factor = 2 × 2 × 2 = 8

Hence, he can plant 8 bulbs in each row.

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The chance that a randomly selected individual will buy a car that charges at the least $30,000 is 0.9772, or approximately 97.72%.

To clear up this problem, we need to discover the probability that a randomly selected person will purchase a vehicle that prices as a minimum $30,000, given that the starting income rate of a new automobile on the Monticello vehicle lot is usually dispensed with a mean of $forty,000 and a preferred deviation of $5,000.

We can use the same Standard normal distribution to solve this problem with the aid of changing the beginning sales price of $30,000 to a z-score, and then using a standard normal distribution table or calculator to discover the corresponding opportunity.

The z-rating for a starting sales rate of $30,000 is:

z = (30,000 - 40,000) / 5,000 = -2

Using a standard normal distribution table or calculator, we will locate that the opportunity of a z-score less than or identical to -2 is approximately zero.0228.

But, we need to find the possibility that the starting sales price is at the least $30,000, so we want to subtract this chance from 1:

P(X ≥ 30,000) = 1 - P(X < 30,000)

P(X ≥ 30,000) = 1 - 0.0228

P(X ≥ 30,000) = 0.9772

Therefore, the chance that a randomly selected individual will buy a car that charges at the least $30,000 is 0.9772, or approximately 97.72%.

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If Carmen paid a total of $49.50 for 3 CD's, then the price of each CD before the tax is $15.80.

The "Sales-Tax" is $0.70 per CD, which is added to the original price of each CD to arrive at the total cost of the CDs including sales tax

Let the price of each CD before tax be = "x",

Carmen bought 3 CDs, so the total cost of the CDs without tax would be 3x.

The "Sales-Tax" on each CD is $0.70, and Carmen bought 3 CDs,

The "total-sales" tax will be = 3 × $0.70 = $2.10,

So, the "total-cost" of the CDs including sales tax is "3x + $2.10", and

We know this total cost is = $49.50,

The equation to represent the cost is written as :

⇒ 3x + $2.10 = $49.50

Now, Solving for "x", the price of each CD before tax.

⇒ 3x = $49.50 - $2.10

⇒ 3x = $47.40

⇒ x = ($47.40)/3,

⇒ x ≈ $15.80

Therefore, the price of each CD is $15.80.

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Answer:

The third equation is equal to 2.

Step-by-step explanation:

6 times one third is equal to 6 divided by three, which equals two. Using order of operations, 5 times 2 equals 10, and that divided by 5 is 2, so that equation is equal to 2.

The slower vehicle is the car, and it would take (11.67 - 10.5) = 1.17 minutes longer to travel 7 miles compared to the subway train.

To find out how much longer it would take to travel 7 miles in the slower vehicle, we need to calculate the speed of

each vehicle first.

The car travels 3 miles in 5 minutes, which means it travels at a speed of 3/5 miles per minute.

The subway train travels 4 miles in 6 minutes, which means it travels at a speed of 4/6 miles per minute. Simplifying

this, we get a speed of 2/3 miles per minute.

Now, we can calculate how long it would take each vehicle to travel 7 miles:

The car travels at a speed of 3/5 miles per minute, so it would take (7 / (3/5)) = 11.67 minutes to travel 7 miles.

The subway train travels at a speed of 2/3 miles per minute, so it would take (7 / (2/3)) = 10.5 minutes to travel 7 miles.

Therefore, the slower vehicle is the car, and it would take (11.67 - 10.5) = 1.17 minutes longer to travel 7 miles compared to the subway train.

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The solution to the equation is x = 16/11.

what is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas. It involves the study of variables, expressions, equations, and functions.

Diego's mistake is in the step where he subtracts 3x from both sides of the equation. The correct subtraction should be 28, not 12.

Here's the correct solution:

4(7 - 2x) = 3(x + 4)

28 - 8x = 3x + 12

28 - 12 = 3x + 8x

16 = 11x

x = 16/11

Therefore, the solution to the equation is x = 16/11.

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The result for the appropriate measure, relative risk, is approximately 0.515.

This indicates that the risk of death within 5 years is about 51.5% lower in the new procedure group compared to the standard procedure group.

This suggests a strong association between the surgical procedure and the reduction in the risk of death.

To determine the strength of the association between the surgical procedures and death, we can calculate the relative risk.
Identify the numbers given in the question.
- Total participants: 550
- New procedure group: 275 (half of 550)
- Deaths in new procedure group: 98
- Standard procedure group: 275 (half of 550)
- Deaths in standard procedure group: 190
Calculate the death rate (proportion of deaths) in each group.
- Death rate in new procedure group = (Number of deaths in new procedure group) / (Total in new procedure group) = 98/275 ≈ 0.356
- Death rate in standard procedure group = (Number of deaths in standard procedure group) / (Total in standard procedure group) = 190/275 ≈ 0.691
Calculate the relative risk.
- Relative risk = (Death rate in new procedure group) / (Death rate in standard procedure group) = 0.356/0.691 ≈ 0.515.

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The best prediction possible for the number of times the spinner will land on green or blue is given as follows:

30 spins.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

Out of six regions, three are green and two are blue, hence the probability of one spin resulting in green or blue is given as follows:

p = (3 + 2)/6

p = 5/6.

Thus the expected number out of 36 trials of spins resulting in green or blue is given as follows:

E(X) = 5/6 x 36

E(X) = 30 spins.

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The minimum value of V occurs at x ≈ 0.93, which means that the volume of the box is smallest when the height is about 0.93 units.

To find the extreme values of V, we need to take the derivative of V and set it equal to zero. So, let's begin:

[tex]V(x) = x(10-2x)(16-2x)[/tex]
Taking the derivative with respect to x:
[tex]V'(x) = 10x - 4x^2 - 32x + 12x^2 + 320 - 48x[/tex]
Setting V'(x) = 0 and solving for x:
[tex]10x - 4x^2 - 32x + 12x^2 + 320 - 48x = 0\\8x^2 - 30x + 320 = 0[/tex]
Solving for x using the quadratic formula:
[tex]x = (30 ± \sqrt{(30^2 - 4(8)(320))) / (2(8))\\x = (30 ± \sqrt{(1680)) / 16\\x = 0.93 or x =5.07[/tex]
So, the extreme values of V occur at x ≈ 0.93 and x ≈ 5.07. To determine whether these are maximum or minimum values, we need to examine the second derivative of V. If the second derivative is positive, then the function has a minimum at that point. If the second derivative is negative, then the function has a maximum at that point. If the second derivative is zero, then we need to use a different method to determine whether it's a maximum or minimum.

Taking the second derivative of V:
V''(x) = 10 - 8x - 24x + 24x + 96
V''(x) = -8x + 106

Plugging in x = 0.93 and x = 5.07:
V''(0.93) ≈ 98.36 > 0, so V has a minimum at x ≈ 0.93.
V''(5.07) ≈ -56.56 < 0, so V has a maximum at x ≈ 5.07.

Now, to interpret these values in terms of the volume of the box, we need to remember that V(x) represents the volume of a box with length 2x, width 2x, and height x. So, the maximum value of V occurs at x ≈ 5.07, which means that the volume of the box is greatest when the height is about 5.07 units. The minimum value of V occurs at x ≈ 0.93, which means that the volume of the box is smallest when the height is about 0.93 units.

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a) The extreme values of V are:

Minimum value: V(0) = 0

Relative maximum value: V(3) = 216

Absolute maximum value: V(4) = 128

b) The absolute maximum value of V at x = 4 represents the case where the box has a square base of side length 4 units, height 2 units, and width 8 units, which has a volume of 128 cubic units.

a) To find the extreme values of V, we first need to find the critical points of the function. This means we need to find where the derivative of the function equals zero or is undefined.

Taking the derivative of V(x), we get:

[tex]V'(x) = 48x - 36x^2 - 4x^3[/tex]

Setting this equal to zero and solving for x, we get:

[tex]48x - 36x^2 - 4x^3 = 0[/tex]
4x(4-x)(3-x) = 0

So the critical points are x = 0, x = 4, and x = 3.

We now need to test these critical points to see which ones correspond to maximum or minimum values of V.

We can use the second derivative test to do this. Taking the derivative of V'(x), we get:

[tex]V''(x) = 48 - 72x - 12x^2[/tex]

Plugging in the critical points, we get:

V''(0) = 48 > 0 (so x = 0 corresponds to a minimum value of V)
V''(4) = -48 < 0 (so x = 4 corresponds to a maximum value of V)
V''(3) = 0 (so we need to do further testing to see what this critical point corresponds to)

To test the critical point x = 3, we can simply plug it into V(x) and compare it to the values at x = 0 and x = 4:

V(0) = 0
V(3) = 216
V(4) = 128

So x = 3 corresponds to a relative maximum value of V.
b) In terms of the volume of the box, the function V(x) represents the volume of a rectangular box with a square base of side length x and height (10-2x) and width (16-2x).
The minimum value of V at x = 0 represents the case where the box has no dimensions (i.e. it's a point), so the volume is zero.
The relative maximum value of V at x = 3 represents the case where the box is a cube with side length 3 units, which has a volume of 216 cubic units.
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a. the simplified expression in standard form is -40p² + 6200p - 153000.

b.  the monthly profit is $87250.

What is profit?

The revenue generated from selling a good is referred to as the profit, and it must exceed the cost of the good. In other terms, a gain from any business activity constitutes a profit.

Here, we have

Given: The relationship between monthly profit P, monthly sales n, and unit price p is P = n(p−U)−F. For an online business advice service, the unit cost is $30 per hour-long session and the monthly fixed cost is $3000.

a.

P = n(p−U) − F.....(1)

n = 5000−40p, U = 30, F = $3000

We substitute the all values in equation(1) and we get

P = (5000−40p)(p-30) - 3000

P = 5000p - 150000 - 40p² + 1200p - 3000

P = -40p² + 6200p - 153000

Hence, the simplified expression in standard form is -40p² + 6200p - 153000.

b. When unit price, p, is $77.50.

= -40(77.5)² + 6200(77.5) - 153000

= -240250 + 480500 - 153000

= 87250

Hence, the monthly profit is $87250.

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The volume of the given rectangular prism is 27.6 cubic cm.

Length of the prism = 4⅗ cm

Width of the prism = 3 cm

Height of the prism = 2 cm

Volume of the rectangular prism = length × width × height

= 4⅗ × 3 × 2

= 23/5 × 6

= 138/5

= 27.6 cm³

In conclusion, the rectangular prism has the volume 27.6 cm³

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Sample size and sample standard deviation are the key information needed for a single-sample t-test.

In the event that you need to compare the test cruel with the populace cruel, and you perform a single-sample t-test rather than a z-test, the vital piece of data that will be lost is the populace standard deviation.

Within the z-test, the populace standard deviation is known and the standard mistake of the cruel is calculated utilizing the populace standard deviation.

In a single-sample t-test, the populace standard deviation is obscure, and the standard mistake of the cruel is evaluated from the test standard deviation. 

Therefore, sample size and sample standard deviation are the key information needed for a single-sample t-test. 

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Answer:

[tex] \frac{(3n + 4)(1 - n)}{ {n}^{2} } = \frac{ - 3 {n}^{2} - n + 4 }{ {n}^{2} } [/tex]

As n approaches infinity, this quotient approaches -3.

A is the correct answer.

The  reduced exact values for the root of f(x) = x² + 10·x - 5, obtained using the quadratic formula are; x = (-5 + √(30)) and x = (-5 - √(30))

The quadratic formula is a formula that is used to find the roots of a quadratic equation, by plugging in the coefficients of a quadratic equation into the formula.

The roots of the quadratic function, f(x) = x² + 10·x - 5, can be obtained using the quadratic formula as follows;

The quadratic formula, which can be used to find the roots of the quadratic function, f(x) = a·x² + b·x + c is; x = (-b ± √(b² - 4·a·c))/(2·a)

The function f(x) = x² + 10·x - 5, indicates;

a = 1, b = 10, and c = -5, therefore;

x = (-10 ± √(10² - 4 × 1 × (-5)))/(2 × 1) = (-10 ± √(120))/2

(-10 ± √(120))/2 = (-5 × 2 ± 2 × √(30))/2

(-5 × 2 ± 2 × √(30))/2 =  (-5 ± √(30))/2

The roots of the quadratic function are;

x = (-5 + √(30))/2, and x = (-5 - √(30))/2

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The property used to simplify the expression is the Commutative Property of Addition, which states that changing the order of addends does not change the sum.

The Commutative Property is a property of operations that states that the order in which two numbers are added or multiplied does not affect the result. In other words, the property says that changing the order of the terms being added or multiplied will not change the final answer.

The property that was used to simplify the expression is the Commutative Property of Addition. This property states that the order in which we add two numbers does not affect the result. In other words, if we have two numbers a and b, then a + b is equal to b + a.

In the given expression, we have two terms, 3c and 4c, that are being added together. By applying the Commutative Property of Addition, we can rearrange the terms to get 4c + 3c. This gives us the simplified expression 7c + 9.

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Two conditions are not met, the AST cannot be used to determine the convergence or divergence of the series.

The Alternating Series Test (AST) is a tool that can be used to determine whether a series converges or not.

In order to apply the AST, the series must meet two conditions:
The terms of the series must alternate in sign. In other words, the terms should switch between positive and negative values.

Mathematically, this can be represented as [tex](-1)^n \times a_n or ( -1 ) ^(n+1) \times a_n[/tex] for some sequence a_n.
The terms of the series must be decreasing in magnitude, and the limit of the sequence a_n must be 0 as n approaches infinity.

In mathematical notation, this can be written as: [tex]a_(n+1) \leq  a_n[/tex]for all n, and [tex]\lim_{n \to \infty} a_n   = 0[/tex].
If a series meets both of these conditions, then the AST can be applied, and the series converges.

Either of these conditions is not met, the AST cannot be used to determine the convergence or divergence of the series.
To recap, the Alternating Series Test can be used to determine if a series converges if and only if:
The series alternates in sign (terms switch between positive and negative values).
The terms of the series are decreasing in magnitude, and the limit of the sequence [tex]a_n[/tex] is 0 as n approaches infinity.

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