xavier is a teacher and takes home 90 papers to grade over the weekend. he can grade at a rate of 6 papers per hour. how many papers would xavier have remaining to grade after working for 12 hours?

Since both terms are perfect squares, factor using the difference of squares formula, [tex]a^2-b^2=(a+b)(a-b)[/tex] where [tex]a=x[/tex] and [tex]b=y[/tex]

Answer:

21632

Step-by-step explanation:

20000×1.04=20800

20800×1.04=21632

The second row of AB is indeed the second row of A multiplied on the right by B, which confirms that the statement is true.The correct answer is actually OA.

To justify that the statement is True we consider the following steps:

Therefore, the second row of AB is indeed the second row of A multiplied on the right by B, which confirms that the statement is true.                      

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It should be noted that this approach follows the need to maintain the list's original order of elements.

A function is a relationship between a set of possible outcomes (referred to as the range) and a set of inputs (referred to as the domain), with the property that each input is associated to exactly one output. A function, then, is a mathematical rule that designates a specific output value for each input value. Equations, graphs, and tables are frequently used to represent functions. They are employed to mimic real-world occurrences and to address issues in numerous branches of mathematics, science, engineering, and other disciplines.

given

The value to be deleted from the list is represented by an integer value that the method accepts as an argument.

The method iterates through the list's components using a while loop.

The method determines if the current element equals the requested value while it is in the loop. If so, the procedure uses the ArrayList class's remove method to remove the element from the list. If not, the method increases the index variable and moves on to the next element.

The procedure keeps going over the list until all instances of the given value have been eliminated.

It should be noted that this approach follows the need to maintain the list's original order of elements.

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Using the mean formula we know that the new mean height of 14 people is 159.7 cm.

A mean in math is the average of a data set, calculated by adding all numbers together and then dividing the total of the numbers by the number of numbers.

For illustration, with the dataset containing: 8, 9, 5, 6, 7, the mean is 7, as 8 + 9 + 5 + 6 + 7 = 35, 35/5 = 7.

The mean height given is 159 cm.

Let, x1, x2, x3 ..... x15 be the height of 15 people.

x1+x2+x3 ..... x15/15 = 159

x1+x2+x3 ..... x15 = 159 * 15 = 2385 cm ...(1)

Let, x1 = 148 cm, who leaves the group.

148+x2+x3 ..... x15 = 2385 cm

x2+x3 ..... x15 = 2237 cm

Now, we have 14 people.

New mean = x2+x3 ..... x15/14

= 2237/14

= 159.7 cm

Therefore, using the mean formula we know that the new mean height of 14 people is 159.7 cm.

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The roots of the equation 2x² + 10x + 9 = 2x does not exist i.e no real roots

To find the roots of the given quadratic equation 2x² + 10x + 9 = 2x, we can start by rearranging the equation to the standard form of a quadratic equation

2x² + 10x + 9 - 2x = 0

Simplifying the left-hand side, we get:

2x² + 8x + 9 = 0

Now, we can use the quadratic formula to find the roots of the equation:

x = (-b ± √(b² - 4ac)) / 2a

where a = 2, b = 8, and c = 9.

Substituting these values into the formula, we get:

x = (-8 ± √(8² - 4(2)(9))) / 2(2)

Simplifying the expression under the square root, we get:

x = (-8 ± √-8) / 4

The square root of -8 is not a real number

So, the equation has no real root

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The dilated figure has the following coordinates: A' (-15, 0), B' (-5, 10), C' (10, 10), D' (15, -5), and E' (5, -10).

In this sense, coordinates are the points where a grid system intersects. Latitude and longitude are the traditional ways to express GPS coordinates. Degrees of separation north and south from the equator, which is 0 degrees, are measured by lines of latitude coordinates.

Just multiply the coordinates of each point by 5 to construct a figure about the origin using a scale factor of 5.

A (-3, 0)

B (-1, 2)

C (2, 2)

D (3, -1)

E (1, -2)

The coordinates of each point are multiplied by 5 to enlarge the image by a scale factor of 5:

A' = (-3 * 5, 0 * 5) = (-15, 0)

B' = (-1 * 5, 2 * 5) = (-5, 10)

C' = (2 * 5, 2 * 5) = (10, 10)

D' = (3 * 5, -1 * 5) = (15, -5)

E' = (1 * 5, -2 * 5) = (5, -10)

The coordinates of the dilated figure are:

A' (-15, 0)

B' (-5, 10)

C' (10, 10)

D' (15, -5)

E' (5, -10)

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20 is the possible outputs for the function defined by c(x)=20x for all of the students that would pay for the field trip

The function that is known to represent what would be the cost of the trip of these students is given as C(x)=20x.

The cost of the trip for one of the students is put at 20 dollars each.

We would have x as the total number of those that would be attending this trip. In order to get the output that would show us the cost of the trip.

We take X to be a whole number.

then x = 1

such that that  c(1) = 20(1) = 20

Then the total is 20 per student

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the cost for an upcoming field trip is 20 dollars the cost of the feild trip c is a function of the number of students x select all the possible outputs for the function defined by c(x)=20x

A 20 B 30 c 50

120 middle schοοl students prefer the dοcumentaries televisiοn genre.

A percentage in mathematics is a number οr ratiο that can be expressed as a fractiοn οf 100. If we need tο calculate a percentage οf a number, we shοuld divide it by 100 and multiply the result. Therefοre, the percentage refers tο a part per hundred. Per 100 is what the wοrd percentage means. The symbοl "%" is used tο represent it.

The tοtal is 100%.  Subtract the οther parts οf the circle tο find the percent fοr spοrts.

100 - 14 -22-20 -20

24

Spοrts is 24%

Multiply the number οf students by the percentage οf students that prefer spοrts

500 *24%

500 *.24

120

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The coffee shop owner does not have sufficient evidence to conclude that the distribution of sales is proportional to the number of facings at a 5% level of significance.

What is proportion ?

A proportion refers to the number or fraction of individuals or items that exhibit a particular characteristic or have a certain attribute, relative to the total number or sample size being considered. It is often expressed as a ratio or percentage.

To test whether the distribution of sales is proportional to the number of facings, we can use the chi-squared goodness of fit test. The null hypothesis for this test is that the observed data follows a specific distribution (in this case, a proportional distribution), while the alternative hypothesis is that the observed data does not follow that distribution.

To conduct the test, we first need to calculate the expected frequency for each category assuming a proportional distribution. We can do this by multiplying the total number of sales (610) by the proportion of facings for each brand:

Starbucks: 610 x 0.3 = 183

Dunkin: 610 x 0.4 = 244

Peet's: 610 x 0.2 = 122

Other: 610 x 0.1 = 61

Next, we calculate the chi-squared statistic using the formula:

χ² = Σ((O - E)² / E)

where O is the observed frequency and E is the expected frequency. The degrees of freedom for this test are (k-1), where k is the number of categories. In this case, k = 4, so the degrees of freedom are 3.

Using the observed and expected frequencies from the table, we get:

χ² = ((130-183)²/183) + ((240-244)²/244) + ((85-122)²/122) + ((155-61)²/61) = 124.36

Looking up the critical value of chi-squared for 3 degrees of freedom and a significance level of 0.05, we get a value of 7.815. Since our calculated χ² value of 124.36 is greater than the critical value of 7.815, we reject the null hypothesis and conclude that the observed distribution of sales is not proportional to the number of facings.

Therefore, the coffee shop owner does not have sufficient evidence to conclude that the distribution of sales is proportional to the number of facings at a 5% level of significance.

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For the fourth-year forward rate: The forward rate is equaI to [(770.89/835.62)(1/(4-3)] - 1 = 0.0289 or 2.89% .

An equation in mathematics is a cIaim that two mathematicaI expressions are equivaIent. The Ieft-hand side (LHS) and the right-hand side (RHS), which are separated by the equaI sign ("="), make up an equation. Equations are a common tooI for probIem-soIving and determining the vaIue of an unknowabIe variabIe since they are used to describe mathematicaI reIationships.

given

I For a bond having a one-year maturity:

[tex]YTM = [(1000/945.90)^{(1/1)}] - 1 = 0.0572 or 5.72%[/tex]

(ii) For a bond having a two-year maturity:

[tex]YTM = [(1000/911.47)^{(1/2)}] - 1 = 0.0474 or 4.74%[/tex]

(iii) For a bond having a three-year maturity:

[tex]YTM = [(1000/835.62)^{(1/3)}] - 1 = 0.0617 or 6.17%[/tex]

(iv) For a bond with a four-year maturity:

[tex]YTM = [(1000/770.89)^{(1/4)}] - 1 = 0.0672 or 6.72%[/tex]

We can use the foIIowing formuIa to determine the forward rates:

Forward rate is equaI to [((Bond Price 1/Bond Price 2)(1/(n2-n1))]]. - 1

where n₂-n₁ is the time period between the maturities, Price of Bond 1 is the price of the bond with maturity n₁, and Price of Bond 2 is the price of the bond with maturity n₂.

We may determine the forward rates using the bonds' current prices by foIIowing these steps:

I For the second-year forward rate:

((911.47/945.90)(1/(2-1))) is the forward rate. - 1 = 0.0379 or 3.79%

(ii) For the third-year forward rate:

The forward rate is equaI to [((835.62/911.47)(1/(3-2))] - 1 = 0.0360 or 3.60%

For the fourth-year forward rate: The forward rate is equaI to [(770.89/835.62)(1/(4-3)] - 1 = 0.0289 or 2.89% .

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The Length of AB in square ABCD is 30 feet.

Since the squares ABCD and EFGH are similar, their corresponding sides are proportional, so we can set up the following relation:

AB/EF = 1/4

We can also use the fact that the ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding sides. Therefore,

AB²/EF² = (Area of square ABCD)/(Area of square EFGH)

Substituting the given values:

AB²/EF² = (Area of square ABCD)/(14400)

Since the areas of squares are proportional to the square of their sides, we can write,

Area of square ABCD/Area of square EFGH = (AB/EF)²

Substituting this into the above equation and solving for AB, we get,

AB²/EF² = (AB/EF)²

AB² = (AB/EF)² * EF²

AB² = (1/4)² * 14400

AB² = 900

AB = 30 feet

Therefore, the length of the side AB of square ABCD is 30 feet.

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Alan can sew 60 feet of fabric in 1.5 minutes.

In mathematics, slope is a measure of the steepness of a line. It is defined as the ratio of the change in the vertical (y) coordinate to the change in the horizontal (x) coordinate between two points on the line. In other words, slope represents the rate at which the line is rising or falling. The slope of a line can be positive, negative, zero or undefined, depending on the direction and steepness of the line. The slope formula is given by:

slope = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line. The slope is often denoted by the letter m.

We can see that the relationship between time and length of fabric is a linear one. Using the two given points (1, 40) and (2, 80), we can find the slope of the line:

slope = (change in y) / (change in x) = (80 - 40) / (2 - 1) = 40

This means that for every 1 minute, Alan can sew 40 feet of fabric. To find how much fabric he can sew in 1.5 minutes, we can use the equation for a line:

y = mx + b

where y is the length of fabric, x is the time in minutes, m is the slope we just found, and b is the y-intercept (which we can find by plugging in one of the points).

Using the point (1, 40), we get:

40 = 40(1) + b

b = 0

So the equation for the line is:

y = 40x

To find the length of fabric Alan can sew in 1.5 minutes, we plug in x = 1.5:

y = 40(1.5) = 60

Therefore, Alan can sew 60 feet of fabric in 1.5 minutes.

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The complete question is:

The triangle DEF is twice the size of the triangle ABC, and the triangles are similar triangles as the sides of the triangles are in correspondence.

Triangles that resemble one another but may not be precisely the same size are said to be comparable triangles. When two objects have the same shape but different sizes, they can be said to be comparable. This shows that when shapes are amplified or demagnified, they superimpose one another. This feature of similar shapes is referred to as "similarity".

According to the figure, AB = 1 DE = 2

Therefore, the triangle DEF is twice as big as the triangle ABC.

Absolutely, there are similarities between the triangles ABC and DEF.

This is due to the fact that triangle ABC's corresponding side is twice as long as DEF's corresponding side and the angle provided are same.

Therefore, Multiplying the coordinates of A by 2 provides coordinates of D.

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Answer:

[tex]\frac{7}{4}[/tex]

Step-by-step explanation:

mixed fraction [tex]\frac{13}{6}[/tex]

sum =[tex]\frac{13}{6}[/tex] +[tex]\frac{1}{2}[/tex] ( LCM)= 12

[tex]\frac{13}{6}[/tex]×[tex]\frac{2}{2}[/tex]  + [tex]\frac{1}{2}[/tex]×[tex]\frac{6}{6}[/tex]

[tex]\frac{26+2}{12}[/tex]

[tex]\frac{28}{12}[/tex] Simplify = [tex]\frac{7}{4}[/tex]

(a) μx=15 and σx=2And, P(15≤x≤17) can be obtained by converting the corresponding x values into z scores.z1=15−15/2=0z2=17−15/2=1P(15≤x≤17) = P(0≤Z≤1) = 0.3413

(b) μx=15 and σx=1.75 And, P(15≤x≤17) can be obtained by converting the corresponding x values into z scores. z1=15−15/1.75=0z 2=17−15/1.75=1.143 P(15≤x≤17) = P(0≤Z≤1.143) = 0.382.

Calculation of µx and σxIf a random sample of size n=49 is drawn from a distribution where µ=15 and σ =14. Then the sample mean is given by the formula;μx=μ=15And, the standard error of the mean (standard deviation of the distribution of the sample means) is given by the formula;σx=σn=1449=2 Thus,μx=15 and σx=2And, P(15≤x≤17) can be obtained by converting the corresponding x values into z scores.z1=15−15/2=0z2=17−15/2=1P(15≤x≤17) = P(0≤Z≤1) = 0.3413

Calculation of µx and σxIf a random sample of size n=64 is drawn from a distribution where µ=15 and σ =14. Then the sample mean is given by the formula;μx=μ=15 And, the standard error of the mean (standard deviation of the distribution of the sample means) is given by the formula;σx=σn=1464=1.75 Thus,μx=15 and σx=1.75 And, P(15≤x≤17) can be obtained by converting the corresponding x values into z scores. z1=15−15/1.75=0z 2=17−15/1.75=1.143 P(15≤x≤17) = P(0≤Z≤1.143) = 0.382

Part (b) is expected to have a higher probability of P(15≤x≤17) than that of Part (a) because the standard deviation is inversely proportional to the sample size n.

Hence, the larger the sample size, the smaller the standard deviation. And, the smaller the standard deviation, the greater the accuracy of the sample mean in estimating the population mean.

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Answer:

Step-by-step explanation:

A is 40

B 3 on the the top after 9 is 27

term is 3

c is 4 on top after 32 is 128

term is 4

d after 300 is 3,000

term is 10

Therefore, with a 95% confidence interval, we can estimate the true mean length of the drive shafts to be between 992.16 mm and 1007.84 mm.

The true mean length of the drive shafts can be estimated using a 95% confidence interval. The margin of error is calculated as 2*1.96*2 = 7.84 mm. The lower bound of the confidence interval is 1000 - 7.84 = 992.16 mm and the upper bound is 1000 + 7.84 = 1007.84 mm.

This confidence interval states that there is a 95% probability that the true mean length of the drive shafts falls within the range of 992.16 mm to 1007.84 mm. The relationship between the two probabilities is that the probability of the true mean length falling within the confidence interval is 95%. If the sample size was increased, the margin of error would decrease, resulting in a tighter range and higher probability that the true mean would fall within the confidence interval.
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The expression that is equivalent to (4 - 2x)(4 + 2x) is equal to 16 - 4x^2 approximately.

To simplify the expression (4 - 2x)(4 + 2x), we can use the FOIL method, which stands for First, Outer, Inner, Last. This method involves multiplying each term in the first factor by each term in the second factor and then combining like terms.

Using the FOIL method, we get:

(4 - 2x)(4 + 2x) = 4 × 4 + 4 × 2x - 2x × 4 - 2x × 2x

Simplifying the expression, we get:

16 + 8x - 8x - 4x^2

The two middle terms cancel each other out, leaving us with:

16 - 4x^2

We can also check our answer by factoring the simplified expression back to the original expression. If we factor 16 - 4x^2, we get:

16 - 4x^2 = 4(4 - x^2)

We can then use the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b), to factor further:

4(4 - x^2) = 4(2 + x)(2 - x)

This gives us back the original expression, (4 - 2x)(4 + 2x), confirming that 16 - 4x^2 is equivalent to (4 - 2x)(4 + 2x).

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To find the total cost of purchasing the video game at the local Big Box store with a [tex]10[/tex] % discount and  [tex]6.5[/tex] % sales tax, we can use the following formula:

Total cost = (Sale amount—Discount) x (1 + Tax rate)

Plugging in the values we have:

Total cost [tex]= (49.95–4.995) \times 1.065[/tex]

Total cost  [tex]= 44.955 \times 1.065[/tex]

Total cost [tex]= 47.83[/tex]

Therefore, the total cost of purchasing the video game at the local Big Box store would be $ [tex]47.83[/tex].

At the online store, the game costs $  [tex]44.95[/tex] plus a shipping charge of $ [tex]4.00[/tex] , which comes to a total of $ [tex]48.95.[/tex]

So, purchasing the game at the online store would be slightly cheaper.

To calculate the total cost and interest earned for each principal amount and time period, we can use the following formula:

Total cost = Principal + Interest earned

Interest earned = Principal x Annual rate x Time period (in years)

Plugging in the values we have:

For the first row:

Interest earned [tex]= 2400 \times 0.049 \times0.5[/tex]

Interest earned = $58.80

Total cost [tex]= 2400 + 58.80[/tex]

Total cost = $ [tex]2,458.80[/tex]

For the second row:

Interest earned [tex]= 9460.12 \times 0.022 \times 2[/tex]

Interest earned = $ [tex]415.24[/tex]

Total cost  [tex]= 9460.12 + 415.24[/tex]

Total cost = $ [tex]9,875.36[/tex]

For the third row:

Interest earned = [tex]3923.87 \times 0.02 \times5[/tex]

Interest earned = $ [tex]392.39[/tex]

Total cost  [tex]= 3923.87 + 392.39[/tex]

Total cost = $ [tex]4,316.26[/tex]

Let's assume Jorge's commission for the month was C. We can set up the following equation:

[tex]C = 0.09 \times 89400[/tex]

We know that Harris sold the same amount, but made $447 more in commission. So we can set up another equation:

[tex]C + 447 = \times89400[/tex]

Where x is the commission rate for Harris.

We can substitute the value of C from the first equation into the second equation:

[tex]0.09 \times 89400 + 447 = \times 89400[/tex]

Simplifying:

[tex]8046 + 447 = 89400x[/tex]

[tex]8493 = 89400x[/tex]

[tex]x = 0.095[/tex] or  [tex]9.5[/tex] %

Therefore, Harris' commission rate is [tex]9.5%[/tex]%  .

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The weight of the package is 56 using arithmetic operations.

Mathematicians use multiplication to calculate the product of two or more integers. It is a fundamental operation in mathematics that is frequently used in everyday living. When we need to combine sets of similar sizes, we use multiplication. The fundamental concept of repeatedly adding the same number is represented by the process of multiplication. The results of multiplying two or more numbers are known as the product of those numbers, and the factors that are compounded are referred to as the factors. Repeated adding of the same number is made easier by multiplying the numbers.

let weight of package be= x

x*5/7=40

x=(40*7)/5

x=56

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Answer:

(a) Since there are 428 equally likely pages, the probability of randomly turning to page 45 is 1/428, which is approximately 0.00234 or 0.234%.

(b) There are two possible cases for an even numbered page: either the last digit is 0, 2, 4, 6, or 8, or the page number ends in 00.

For the first case, there are 5 possible digits for the units place, and each of these digits can be followed by any one of the 5 possible digits for the tens place. So there are 5x5 = 25 even numbered pages that end in a non-zero digit.

For the second case, there are 4 possible digits for the hundreds place, and each of these digits can be followed by 00. So there are 4 even numbered pages that end in 00.

Therefore, there are 25 + 4 = 29 even numbered pages in total. Since there are 428 pages in the book, the probability of randomly turning to an even numbered page is 29/428, which is approximately 0.06776 or 6.776%.

Step-by-step explanation:

(a) Оскільки існує 428 рівноймовірних сторінок, ймовірність випадкового переходу на сторінку 45 дорівнює 1/428, що становить приблизно 0,00234 або 0,234%.

(b) Існує два можливих випадки, коли сторінка має парний номер: або остання цифра 0, 2, 4, 6 або 8, або номер сторінки закінчується на 00.

У першому випадку є 5 можливих цифр для позначення одиниць, і за кожною з цих цифр може слідувати будь-яка з 5 можливих цифр для позначення десятків. Таким чином, існує 5x5 = 25 парних сторінок, які закінчуються на ненульову цифру.

У другому випадку є 4 можливі цифри для сотень, і за кожною з цих цифр може стояти 00. Отже, є 4 парні сторінки, які закінчуються на 00.

Таким чином, всього є 25 + 4 = 29 парних сторінок. Оскільки в книзі 428 сторінок, ймовірність випадкового переходу на парну сторінку дорівнює 29/428, що становить приблизно 0.06776 або 6.776%.

Answer:

$350

Step-by-step explanation:

with the interest rate you'll be spending more money

The probability of selecting a vowel at random from the word "COVID NINETEEN" is 0.357, and the probability of selecting a consonant at random is 0.643.

The word "COVID NINETEEN" has a total of 14 letters. We can count the number of vowels and consonants in the word to determine the probability of selecting a vowel or a consonant at random.

There are five vowels in the word: O, I, E, E, and E.

Therefore, the probability of selecting a vowel at random is:

P(vowel) = number of vowels / total number of letters

= 5 / 14

= 0.357 or approximately 35.7%    

There are nine consonants in the word: C, V, D, N, T, N, T, N, and N.

Therefore, the probability of selecting a consonant at random is:

P(consonant) = number of consonants / total number of letters

= 9 / 14

= 0.643 or approximately 64.3%

Therefore, the probability of selecting a vowel at random from the word "COVID NINETEEN" is 0.357, and the probability of selecting a consonant at random is 0.643.

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Answer:

Step-by-step explanation:

To find the possible length and breadth of the rectangle, we need to factor the given expression:

x^2 - 6x + 8 = (x - 4)(x - 2)

Therefore, the length and breadth of the rectangle can be any combination of (x-4) and (x-2).

For example, if we choose (x-4) as the length and (x-2) as the breadth, we have:

Length = x - 4

Breadth = x - 2

Conversely, if we choose (x-2) as the length and (x-4) as the breadth, we have:

Length = x - 2

Breadth = x - 4

So, the possible length and breadth of the rectangle are (x-4) and (x-2), and vice versa.

Answer:(x-4) and (x-2)

Step-by-step explanation:

Julie will have driven a total distance of 289 miles if she travels from York to Corby via Derby.

Starting from York, Julie needs to travel to Derby. The distance between York and Derby is given as 89 miles. So, we know that Julie will have driven 89 miles once she reaches Derby.

Next, Julie needs to travel from Derby to Corby, but the given information is a bit tricky here. The distance from Derby to Corby is not given directly. Instead, we are given two distances - Derby to Dory and Dory to Corby.

To find the distance from Derby to Corby, we need to add the distances between Derby and Dory, and Dory and Corby. From the question, we know that the distance between Derby and Dory is 127 miles and the distance between Dory and Corby is 73 miles. Adding these two distances gives us the total distance from Derby to Corby, which is 200 miles.

Finally, we can add up the distances traveled between each location to find the total distance traveled by Julie. Adding the distances of each leg of the journey, we get:

89 miles (York to Derby) + 200 miles (Derby to Corby via Dory) = 289 miles

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Complete Question:

If Julie drives from York to Corby via Dory how many miles will she have driven?

York 89

Derby 127 73

Corby

The axis should be labeled x-axis and y-axis respectively.

A graph of triangle ABC with the points A (-3, 0), B (-2, 4), and C (1, -1) is shown below.

The coordinates of triangle A'B'C' are A' (0, 3), B' (4, 2), and C' (-1, -4).

The coordinates of triangle A"B"C" are A" (0, 0), B" (4, -1), and C" (-1, -4)).

In Mathematics and Geometry, the rotation of a point 90° about the center (origin) in a clockwise direction would produce a point that has these coordinates (y, -x).

By applying a rotation of 90° clockwise about the center (origin), the coordinates of triangle A'B'C' are as follows;

(x, y)                               →            (y, -x)

Coordinate A = (-3, 0) → Coordinate A' = (0, -(-3)) = (0, 3)

Coordinate B = (-2, 4) → Coordinate B' = (4, -(-2)) = (4, 2)

Coordinate C = (1, -1) → Coordinate C' = (-1, -(1)) = (-1, -1)

Next, we would translate A'B'C' 3 units down:

(x, y)                               →            (x, y - 3)

Coordinate A' =  (0, 3) = A" (0, 0)

Coordinate B' = (4, 2) = B" (4, -1)

Coordinate C' = C" (-1, -4).

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Answer:

y = - 2x + 4

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

y = [tex]\frac{1}{2}[/tex] x - 9 ← is in slope- intercept form

with slope m = [tex]\frac{1}{2}[/tex]

given a line with slope m then the slope of a line perpendicular to it is

[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{\frac{1}{2} }[/tex] = - 2 , then

y = - 2x + c ← is the partial equation

to find c substitute (3, - 2 ) into the partial equation

- 2 = - 2(3) + c = - 6 + c ( add 6 to both sides )

4 = c

y = - 2x + 4 ← equation of perpendicular line

The derivative of the given function [tex]y = 5x^2 sec^{(-1)(2x-3)^2}[/tex] is [tex]dy/dx=-20x\sqrt{((2x-3)^2-1)}[/tex]

It can be derived as:

We can use the chain rule and the derivative of [tex]sec^{(-1)x}[/tex] which is [tex]-1/(x*\sqrt{(x^2-1)})[/tex]

First, we apply the chain rule to the function.

Let  [tex]u = (2x-3)^2[/tex], then:

[tex]y = 5x^2 sec^{(-1)u}[/tex]

[tex]dy/dx = d/dx [5x^2 sec^{(-1)u}][/tex]

[tex]dy/dx = d/dx [5x^2 sec^{(-1)[(2x-3)^2]}][/tex]

[tex]dy/dx= 5x^2 d/dx[sec^{(-1)u}][/tex]     (Using the chain rule)

Now, let [tex]v = u^{(1/2)} = (2x-3)[/tex].

Then:

[tex]dy/dx = 5x^2 d/dv [sec^{(-1)v}] dv/dx[/tex]          (Using the chain rule again)

We have:

[tex]d/dv [sec^{(-1)v}] = -1/(v*\sqrt{(v^2-1)}) = -1/[(2x-3)*\sqrt{((2x-3)^2-1)}][/tex]

Also, [tex]dv/dx = 2[/tex]

Substituting these back into the equation:

[tex]dy/dx = 5x^2 d/dv [sec^{(-1)v}] dv/dx[/tex]

[tex]dy/dx= 5x^2 (-1/[(2x-3)*\sqrt{((2x-3)^2-1)}] (2)[/tex]

Simplifying this expression gives:

[tex]dy/dx = -20x (2x-3)/[(2x-3)*\sqrt{((2x-3)^2-1)}][/tex]

[tex]dy/dx = -20x\sqrt{((2x-3)^2-1)}[/tex]

Therefore, the derivative of y with respect to x is:

[tex]dy/dx = -20x\sqrt{((2x-3)^2-1)}[/tex]

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