Need help with math

Answer:

[tex]\bf a^2* (a - 0.4)(a^2 + 0.4a + 0.16)[/tex]

Step-by-step explanation:

Factorize:

 First take out the common term a².

[tex]a^5 - 0.064a^2= a^2*(a^3 - 0.064)\\\\[/tex]

Now, factorize using the identity a³ - b³

a³ - b³ = (a - b) (a²  + ab + b²)

[tex]a^2 * (a^2 - 0.064) = a^2 * (a^3 - 0.4^3)[/tex]

                          [tex]= a^2 * (a - 0.4) * (a^2 + a*0.4 + 0.4^2)\\\\=a^2 * (a - 0.4)(a^2 + 0.4a + 0.16)[/tex]

Measurement scales refer to the various methods of assigning scores to measurements according to their precision and accuracy. There are four primary scales of measurement; nominal, ordinal, interval, and ratio.

The four measurement scales and their definitions are listed below:

Nominal scale: This is the lowest level of measurement. It is used to name, classify, or identify objects, events, or groups. It has no logical order, and its units cannot be compared or arranged in a sequence.

Examples of the nominal scale include; gender (male or female), marital status (single or married), color (black or white), and so on.

Ordinal scale: This scale includes a more elaborate classification than nominal measurement. It distinguishes between levels of preference or priority, but it does not have a constant distance between the levels.

Examples of the ordinal scale include; rankings (first, second, third, and so on), level of satisfaction (very happy, happy, neutral, unhappy, very unhappy), and so on.

Interval scale: This scale has a constant distance between its units, and its elements can be compared or arranged in a sequence. It does not have a true zero.

Examples of the interval scale include; temperature (measured in Fahrenheit or Celsius), time (measured in seconds, minutes, or hours), and so on.

Ratio scale: This is the highest level of measurement, where it has a constant distance between its units, its elements can be compared or arranged in a sequence, and it has a true zero.

Examples of the ratio scale include; weight, length, height, and so on. Measures of central tendency include; mean, median, and mode. The mean is the average of all the data points in a set, the median is the middle value in a set of data, and the mode is the value that occurs most often. The table below provides an example of each measurement scale and the appropriate measures of central tendency for each. Scales of Measurement Examples Measures of Central Tendency Nominal Scale Gender Mean Ordinal Scale Ranking Median Interval Scale Temperature Mean Ratio Scale Height Mode.

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The approximate number of miles for a warranty is 41,748.57 miles.

Standard deviation refers to the distribution's extent of variability or dispersion. It is a measure of how much the data varies from the mean. It is represented by the Greek letter sigma (σ) and is used to calculate the data variability. It is the square root of the variance of a random variable that describes how widely the data points differ from the average value.

We have, Mean (μ) = 47,500 miles

Standard deviation (σ) = 3,000 miles

Manufacturer replacement threshold = 10% = 0.1

The number of miles for a warranty would have to be determined if the tire company wanted to replace no more than 10% of the tires.

Suppose x is the number of miles for which a warranty can be offered.

Using the standard normal distribution, we may compute the z-score as follows:

z = (x - μ) / σ

At the 10th percentile of the standard normal distribution, we'll find the z-score. This z-score can be found using the standard normal distribution table or software.

At the 10th percentile, the z-score is -1.28 (from the table).

Now, substitute the given values.

z = (x - μ) / σ

-1.28 = (x - 47,500) / 3,000

Cross-multiplying and solving for x gives us:

x = -1.28 × 3,000 + 47,500x ⇒ 43,660 miles

Therefore, the tire company should offer a warranty for around 43,660 miles (approximated).

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After answering the prοvided questiοn, we can cοnclude that  the 5% expressiοn discοunt is mοst likely intended tο encοurage custοmers tο purchase.

A cοllectiοn οf symbοls, digits, and cοnglοmerates in mathematics that resemble a pattern οr statistical cοrrelatiοn is knοwn as an expressiοn. Real numbers, mutables, and cοmbinatiοns οf the twο are all acceptable types οf expressiοns. A few examples οf mathematical οperatοrs are additiοn, subtractiοn, rapid spread, divisiοn, and expοnentiatiοn. Expressiοns are frequently used in mathematics, geοmetry, and shape. They are used fοr equatiοn sοlving, representing mathematical fοrmulas, and simplifying mathematical relatiοnships.

All purchases made thrοugh the οnline stοre's website appear tο be eligible fοr a 5% discοunt. The custοmer will thus save 5% οff the tοtal cοst οf the purchase with each sale. Fοr instance, if a custοmer spends $100 οn gοοds, they will get a $5 discοunt, lοwering the final cοst tο $95.

It's wοrth nοting that the retailer may chοοse tο sacrifice prοfits in οrder tο οffer this discοunt. Alternatively, they may have negοtiated lοwer prices with their suppliers οr implemented οther cοst-cutting measures tο cοmpensate fοr the cοst οf the discοunt. In any case, the 5% discοunt is mοst likely intended tο encοurage custοmers tο purchase.

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Complete question:

The number of toy soldiers that Leo has is 48.

The first step is to determine the multiples of 4 that lie between 27 and 54. A multiple of a number is the product of an integer and that number.

Multiplies of 4 that lie between 27 and 54 = 28, 32, 36, 40, 44, 48, 52

Now divide each of the multiples by 7 :

28 / 7 = 4

32 / 7 = 4 remainder 4

36 / 7 = 5 remainder 1

40 / 7 = 5 remainder 5

44 / 7 = 6 remainder 2

48 / 7 = 6 remainder 6

52 / 7 = 7 remainder 3

It only 48 that meets this criteria. This means that the number is 48.

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From the following three options for running the appropriate t-test, Both Options (2) and (3) could be applicable depending on the specific research question being investigated.

The appropriate t-test to use in this scenario is paired t-test. This test compares before and after values for one type using excel. In this scenario, you would be comparing either the altimeter error of ACME or brand x before adjustment and calibration and after adjustment and calibration to see if there are any differences in the means (performance).

Based on the scenario described, Option 1 is not appropriate because there is no hypothesized population mean to compare against. Therefore, we can exclude this option.

Option 2, comparing one brand against another, could be appropriate if we want to investigate if there are any significant differences in altimeter error between brands. This would be a two-sample t-test.

Option 3, comparing the altimeter error of one brand before and after calibration, could also be appropriate to see if the adjustment and calibration had a significant impact on the altimeter's performance. This would be a paired t-test.

This test is the appropriate test to use because the dependent variable data in this scenario is the ratio and the type of question requires a paired t-test. In this scenario, you are comparing the performance of the two brands before and after adjustment and calibration, which requires paired t-test to answer the research question.

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Answer:

x = 9

Step-by-step explanation:

[tex]2x + 3 = 3x - 6[/tex]

[tex]3 = x - 6[/tex]

[tex]x = 9[/tex]

Answer:

2125 ft/min

33,000 ft

y = -2125x + 33,000

Step-by-step explanation:

A. -2125 feet per minute. You get this number when you divide 17,000 by 8 (rise over run). You could also use the formula y2-y/x2-x1 with the points (0, 33,000) and (8, 17,000).

B. 33,000 feet is the height of the plane before it starts descending, so it must be the starting value.

C. Plug in the values you got for A and B into the slope formula y = mx + b

y = -2125x + 33,000

(a) The horizontal velocity of the dart remains constant throughout its motion, so it will take 0.24 seconds to reach the target (since distance = speed × time)  and the dart will fall a distance of 0.28 m

During this time, the dart will also be accelerating downwards due to gravity at a rate of 9.81 m/s². The vertical distance fallen can be calculated using the formula: distance = ½ × acceleration × time². Thus, the dart will fall a distance of:

distance = ½ × 9.81 m/s² × (0.24 s)² ≈ 0.28 m

Therefore, the dart will hit the board 0.28 meters below the intended target.

(b) This calculation suggests that even highly skilled dart players may not hit the bull's-eye every time, due to the influence of gravity on the dart's trajectory.

However, skilled players can adjust their aim and release the dart with the correct velocity and angle to hit their target consistently. They may also use techniques such as "aiming high" to compensate for the effect of gravity. Overall, this calculation highlights the importance of understanding the physics of projectile motion in the sport of darts.

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When the minimum volume of a clown is 60,000 cm³ and the volume of a car is 3 million cm³, 55 is an upper bound on the maximum number of clowns that can fit in a car. The correct answer is Option A.

Bounds are the maximum and minimum limits that are permitted, within which something can or must be performed. Bounds are used to refer to an acceptable range of values that provide safe operation or performance. In mathematics, a set of bounds can define the limits of the amount of things or objects.

The minimum volume of a clown is 60,000 cm³, so to fit 55 clowns in a car we need:

55 clowns × 60,000 cm³/clown = 3,300,000 cm³

This is lower than the volume of the car. Therefore, 55 is an upper bound on the maximum number of clowns that can fit in the car.

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Answer:

5) 12.2       6) 2

Step-by-step explanation:

Use Pythagoras for both

Rearrange for x in question 6

The calculation for the price of 10 books is as follows: 10 × (19.60/88) = 2.21

We can use a proportion to determine the cost of 10 books based on the cost of 88 books:

88 books cost $19.60

1 book costs 19.60/88 dollars

Now we can use this information to find the cost of 10 books:

10 books cost 10 × (19.60/88) dollars

Simplifying the expression on the right side, we get:

10 books cost $2.21

Therefore, the equation that would help determine the cost of 10 books is: 10 × (19.60/88) = 2.21 The cost of books is often calculated based on the number of books purchased and the price per book. In this problem, we are given the cost of 88 books, and we need to find the cost of 10 books. Using a proportion, we can find the cost of 1 book and then multiply it by 10 to find the cost of 10 books. This method is useful for determining the cost of any number of books, given the cost of a different number of books. Book prices can vary widely depending on factors such as the format, edition, and publisher, so it is important to carefully consider book prices when making purchases.

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For point, p divides the directed line segment xy so that the ratio of xp to py is 3 to 5 correct option is d. Point R is three-fifths of the way from point P to point Y along PY.

The directed line segments divide a line into ratios, and each ratio has different properties. The properties of ratios of points on a line are dependent on the way the line is divided. For instance, suppose point p divides the directed line segment xy so that the ratio of xp to py is 3 to 5. We can describe point r that divides the directed line segment yx so that the ratio of yr to rx is 5 to 3 as follows:
We can solve this problem using the concept of directed line segments and the properties of ratios of points on a line.

Since P divides the directed line segment XY in the ratio 3:5, we can write:

XP = (3/8)XY and PY = (5/8)XY

Now, let's consider the directed line segment YX. We want to find a point R on this segment such that YR:RX = 5:3.

We can express YR and RX in terms of XY using the fact that YX = -XY:

YR = (5/8)YX and RX = (3/8)YX

Substituting -XY for YX, we get:

YR = (-5/8)XY and RX = (-3/8)XY

To find the location of point R, we need to find the distance from Y to R along the directed line segment YX. We can do this by adding the distances from Y to P and from P to R:

YR = YP + PR

Using the ratios we derived earlier, we can express YP and PR in terms of XY:

YP = PY = (5/8)XY

PR = R - P = (3/8)XY - (3/8)XY = 0

Therefore, YR = (5/8)XY + 0 = (5/8)XY

This means that point R is located at a distance of 5/8 of the length of YX from Y. So, the correct answer is (d) Point R is three-fifths of the way from point P to point Y along PY.

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Answer: 2

Step-by-step explanation:

To graph the solution to the inequality -5 + x ≥ -3 on the number line, we first need to isolate x.

Adding 5 to both sides of the inequality, we get:

x ≥ 2

This means that any value of x greater than or equal to 2 will satisfy the inequality. To graph this solution on a number line, we draw a closed circle at the point 2 and shade all the points to the right of 2, including the point 2 itself.

The resulting graph looks like this:

     ------•-------------------------------->

           2

The shaded region on the right of 2 represents all the values of x that make the inequality true.

a. Hence proved that the sum of fractions  [tex]${\frac{1}{\sqrt{1+\sqrt{2}}}}+{\frac{1}{\sqrt{2+\sqrt{3}}}}+{\frac{1}{\sqrt{3}+\sqrt{4}}}=1$[/tex]

b. The value will be 7 for the expression

⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+ \cdot \cdot \cdot+{\frac{\sqrt{63}-8}{-1}}$[/tex]

Square rοοt οf a number is a value, which οn multiplicatiοn by itself, gives the οriginal number. The square rοοt is an inverse methοd οf squaring a number. Hence, squares and square rοοts are related cοncepts.

Suppοse x is the square rοοt οf y, then it is represented as x=√y, οr we can express the same equatiοn as x² = y. Here, ‘√’ is the radical symbοl used tο represent the rοοt οf numbers. The pοsitive number, when multiplied by itself, represents the square οf the number. The square rοοt οf the square οf a pοsitive number gives the οriginal number.

Here,

a. [tex]${\frac{1}{\sqrt{1+\sqrt{2}}}}+{\frac{1}{\sqrt{2+\sqrt{3}}}}+{\frac{1}{\sqrt{3}+\sqrt{4}}}=1$[/tex]

Using (a + b)(a - b) = a² - b²

⇒ [tex]${\frac{1 \cdot \sqrt{1}-\sqrt{2}}{\sqrt{1}+\sqrt{2}\cdot \sqrt{1 }-\sqrt{2}}+{\frac{1 \cdot \sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}\cdot \sqrt{1}-\sqrt{2}}}+{\frac{1 \cdot \sqrt{3}-\sqrt{4}}{\sqrt{3}+\sqrt{4}\cdot \sqrt{3}-\sqrt{4}}}$[/tex]

⇒ [tex]${\frac{ \sqrt{1}-\sqrt{2}}{1-2}+{\frac{ \sqrt{2}-\sqrt{3}}{2-3}+{\frac{\sqrt{3}-\sqrt{4}}{3-4}}$[/tex]

⇒ [tex]${\frac{ \sqrt{1}-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+{\frac{\sqrt{3}-\sqrt{4}}{-1}}$[/tex]

⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+{\frac{\sqrt{3}-2}{-1}}$[/tex]

⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+{\frac{\sqrt{3}-2}{-1}}$[/tex]

⇒ [tex]$ -1+\sqrt{2}}- \sqrt{2}+\sqrt{3}}-{\sqrt{3}+2}$[/tex]

⇒ [tex]$ -1+2}$[/tex]

⇒ 1

a. Hence proved that the sum of fractions  [tex]${\frac{1}{\sqrt{1+\sqrt{2}}}}+{\frac{1}{\sqrt{2+\sqrt{3}}}}+{\frac{1}{\sqrt{3}+\sqrt{4}}}=1$[/tex]

B. This will be done with the same process,

⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+ \cdot \cdot \cdot+{\frac{\sqrt{63}-8}{-1}}$[/tex]

⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+ \cdot \cdot \cdot+{\frac{\sqrt{63}-8}{-1}}$[/tex]

⇒ [tex]$ -1+\sqrt{2}}- \sqrt{2}+\sqrt{3}} \cdot \cdot \cdot -{\sqrt{63}+8}$[/tex]

There, will be same roots of every number until - 8

So,

⇒ [tex]$ -1+8}$[/tex]

= 7

b. The value will be 7 for the expression

⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+ \cdot \cdot \cdot+{\frac{\sqrt{63}-8}{-1}}$[/tex]

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We need to use mathematical logic to prove that the cardinality of the cross product of two sets is the cardinality of each individual set multiplied by each other.

     Let's take an example of two sets A and B.

      Let's say, A={a, b} and B={1, 2, 3}

      To find the cardinality of the cross-product of A and B, we need to make all possible ordered pairs with A and B.

             {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

 Counting the number of ordered pairs, we get six.

  So, the cardinality of the cross-product of two sets A and B equals the cardinality of each set multiplied by the other. Here,

           |A| = 2 and |B| = 3,

          so |A x B| = 2 x 3

                           = 6.

          Therefore, it is proved that the cardinality of the cross product of two sets is the cardinality of each individual set multiplied by each other.

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As a result, the percentage of adults who selected math is different from the percentage of kids who did.

As a number out of 100, a percentage is a method to express a proportion or a fraction. It is symbolized by the number %. If there are 25 boys in a class of 100 pupils, for instance, then there are 25% of boys in the class. It is a helpful method to compare quantities and to express changes in values over time.

given

120 80

Total    200

Women Overall Party A Party B

70 60

Overall    130

Therefore, there are 130 ladies in the group.

b) The chart indicates that 70 women plan to support Party A.

Thus, the percentage of adults who selected English was 40% of 48, which is equal to 0.4 times 48 and 19.2 when rounded to the closest whole number.

b) Reeshma is not accurate. The percentage of adults who selected math is 35%, while for children it is 40%, according to the pie chart.

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The complete question is:-   complete the two-way table, which shows the voting intentions of a group of men and women. a How many women are in the group?

Men

Party A Party B 120

Total    200

Women   130

Total       380

b How many women intend to vote for Party A?

2 A group of 48 adults are asked what their favourite subject was at school. They can choose from

maths, English and science.

A group of 32 school children are asked the

same question.

Answer:

5x³ = 40

20x^6/4x² = 80

100 is not equal to 80

Step-by-step explanation:

5x³ = 5*2^3

      = 40

20x^6/4x² = 20*2^6/4*2^2

                 = 20*64/16

                 = 80

Answer:

y=(3/4)x+6

Step-by-step explanation:

Point 1 (-4,3)

Point 2 (0,6)

To obtain the slope, we apply the formula:

m= (6-3) / (0- (-4))

m= 3 / 4

We replace in the equation y=mx+b.

We take any point.

=> (0,6)

y=mx+b

6=(3/4)(0)+b

b=6

Joining all the terms:

y=(3/4)x+6

Then we will get the following odds

a.  None will have high blood pressure. Let the probability of having high blood pressure be denoted by P(A) and the probability of not having high blood pressure be denoted by P(A'). Since none will have high blood pressure, it means all the sixteen Americans selected are healthy, and therefore P(A') = 1. Therefore

P(A) = 1 - P(A')= 1 - 1= 0

b. One-half will have high blood pressure. The probability that one-half of the sixteen Americans will have high blood pressure can be found using the binomial distribution formula that is given by the expression

[tex]P(X = r) = (nCr) * p^r * q^{(n-r)}[/tex]

Where

Therefore

[tex]P(X = 8) = (16C8) * 0.2^8 * 0.8^8= 0.202[/tex]

c. Exactly 4 will have high blood pressure Similarly, the probability that exactly four of the sixteen Americans will have high blood pressure can be found using the binomial distribution formula as follows:

[tex]P(X = r) = (nCr) * p^r * q^{(n-r})[/tex]

Where

Therefore

[tex]P(X = 4) = (16C4) * 0.2^4 * 0.8^{12}= 0.236[/tex]

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The probability of drawing 4 balls at random, without replacement, that include each of the three colors is 0.0061.

The bowl contains 12 balls6 of the balls are blue2 of the balls are white 4 of the balls are red

Sample space = [tex]n(S) = ^{12}C_4=\frac{12!}{4! (12 - 4)!} = 495[/tex]

The formula for conditional probability is:

P(A ∩ B) = P(A) × [tex]P(\frac{B}{A})[/tex]

P(A ∩ B) is the probability that both A and B occur

P(A) is the probability of event A occurring

[tex]P(\frac{B}{A})[/tex] is the probability of event B occurring given that A has occurred

The probability of drawing 4 balls at random that include each of the three colors can be calculated as follows:

First, we calculate the probability of drawing one ball of each color

P (Drawing one ball of each color) = [tex]\frac{(^6C_1^2C_1^4C_1)}{^{12}C_3} = \frac{(6X2X4)}{220} = 0.0545[/tex]

Next, we calculate the probability of drawing a fourth ball of any color

P (Drawing a fourth ball of any color) = [tex]\frac{^9C_1}{^9C_1} = 1[/tex]

Finally, we use the formula for conditional probability to find the probability of drawing 4 balls at random that include each of the three colors

P (Drawing 4 balls at random that include each of the three colors) = P (Drawing one ball of each color) × P (Drawing a fourth ball of any color | Drawing one ball of each color) = [tex](0.0545)\frac{1}{9}= 0.0061[/tex]

The probability of drawing 4 balls at random, without replacement, that include each of the three colors is 0.0061.

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X = -80538738812075974, Y = 80435758145817515,

and Z = 12602123297335631 are the values of  x,y,z of whole numbers.

What in mathematics is a whole number?

Complete numbers the sum of all natural numbers plus zero. not a decimal or fraction. {0, 2, 3, 4, 5 6, 7, 8, 9, 10, 11 …} Integer. a negative number, zero, or a counting number.

                            Natural numbers with a starting value of 1 and higher are considered whole numbers. Let's examine a few instances of entire numbers. The alphabet "W" stands for the collection of whole numbers. W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,.…}

x3+y3+z3=k, with k being all the numbers from one to 100

X = -80538738812075974,

Y = 80435758145817515,

and Z = 12602123297335631

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Answer:

The correct order to draw 65 agle would be:

C

A

B

D

The bank statement shows a loss of 70 euros, considering the profit concept.

From the table presented in this problem, the parameters are given as follows:

To calculate profit considering money in and money out, you need to subtract the total money out from the total money in.

Hence the formula for calculating profit is given as follows:

Profit = Money In - Money Out

The total money in is given as follows:

Money In = 360 + 295 + 415 = 1070.

The total money out is given as follows:

Money Out = 520 + 70 + 400 + 150 = 1140.

Hence the profit is given as follows:

Profit = Money In - Money Out

Profit = 1070 - 1140

Profit = -70 (negative profit means that it is a loss).

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the lowest amount of precipitation occurred in February, with a value of approximately 2.32 inches.

Why it is and how to form a graph?

To find the month with the lowest amount of precipitation, we need to find the minimum value of the quadratic equation y = –0.04088x²2 + 0.4485x + 1.862.

Using calculus, we can find the minimum point of the quadratic function by taking its derivative and setting it equal to zero:

y' = -0.08176x + 0.4485

0 = -0.08176x + 0.4485

x = 5.484

This means that the minimum value of the function occurs at x = 5.484. Since x represents the month number (with January being 1), we can conclude that the month with the lowest amount of precipitation is February (the second month in the table).

To verify this, we can plug in x = 2 into the quadratic equation:

y = –0.04088(2)²2 + 0.4485(2) + 1.862

y = 2.31752

Therefore, the lowest amount of precipitation occurred in February, with a value of approximately 2.32 inches.

To graph the function, we can plot the points given in the table and connect them with a smooth curve. Here is a completed table with the missing values:

Month (x) Precipitation

January 1 2.27 inches

February 2 2.32 inches

March 3 2.57 inches

April 4 2.94 inches

May 5 3.43 inches

June 6 3.94 inches

July 7 2.72 inches

August 8 2.86 inches

September 9 2.59 inches

October 10 2.03 inches

November 11 1.46 inches

December 12 1.03 inches

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Answer:

D: December

The given model for precipitation in Phoenix, Arizona is y = –0.04088x2 + 0.4485x + 1.862, where x is the month number (1 for January, 2 for February, and so on) and y is the precipitation in inches. We can use this model to fill in the missing values in the table:

| Month (x) | Precipitation |

|-----------|---------------|

| January   | 2.27 inches   |

| February  | 2.27 inches   |

| March     | 2.24 inches   |

| April     | 2.18 inches   |

| May       | 2.09 inches   |

| June      | 1.98 inches   |

| July      | 1.84 inches   |

| August    | 1.68 inches   |

| September | 2.59 inches   |

| October   | 1.50 inches   |

| November  | 1.30 inches   |

| December  | 1.08 inches   |

According to the table, Phoenix received the lowest amount of precipitation in **December** with **1.08 inches** of precipitation, so the correct answer is **D. December**.

As per unitary method, the number of months would take her to break - even on the points purchase is 128

To break even on the points purchase, Jessie needs to recoup the cost of the point through the savings in her monthly mortgage payments. In this case, the cost of the point is 1% of the loan amount, which is $1,350.

To calculate the break-even point, we can use the following formula:

Break-even point = Cost of point / Monthly savings

In this case, the monthly savings is the difference between the original mortgage payment ($766.51) and the new mortgage payment with the point ($755.96), which is $10.55 per month.

Substituting the values, we get:

Break-even point = $1,350 / $10.55

Break-even point = 128 months

Therefore, it would take Jessie 128 months, or just over 10 years, to break even on the points purchase.

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