What 1,500 rounded to the nearest hundredth

Answer:

  x = 6

Step-by-step explanation:

You want to know the value of x given a tangent to a circle from an external point has length x, while a secant from that point has length x-2 to the first intersection with the circle, and an additional 5 to the second intersection.

The product of the segments of the secant to the two points of intersection with the circle is equal to the square of the tangent segment.

  (x -2)(x -2 +5) = (x)² . . . . . use given values

  x² +x -6 = x² . . . . . . simplify

  x -6 = 0 . . . . . . subtract x²

  x = 6 . . . . . . add 6

__

Additional comment

There are rules for two secants, for a tangent and secant (as here), and for two chords internal to the circle. These can be easier to remember if you generalize them to a single rule: the product of the distance from the common point to the first circle intersection and the distance to the second circle intersection is the same for each segment.

For a tangent, the two circle intersections are the same point, hence the distance is squared. For chords, the two circle intersections are at the ends of the chords, and the common point is where the chords cross.

The length of the hypotenuse is √5 with sides of 1 cm and 2 cm.

A right triangle's hypotenuse is its longest side, its "opposite" side is the one that faces a certain angle, and its "adjacent" side is the one that faces the angle in question. To describe the sides of right triangles, we utilise specific terminology. The side opposite the right angle is always the hypotenuse of a right triangle. Stretching under is what the Greek word hypoteinousa signifies. In mathematics, the term "hypotenuse" is used to describe the side of a right triangle that is opposite the right angle. The hypotenuse extends below the right angle if the right angle is positioned at the top of the triangle.

Given

Lengths = 1 cm and 2 cm

For length of the hypotenuse,

c = √a² + b²

c = √1² + 2²

c = √ 1 + 4

c = √5

The length of the hypotenuse is √5 with sides of 1 cm and 2 cm.

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Step-by-step explanation:

since it is geometric sequence we will use the formula

[tex]tn = {a \times r}^{n - 1} [/tex]

a = 2

[tex]r = - \frac{1}{4} [/tex]

The first term

T1(a) = 2

The second Term

[tex]t2 = {a \times r}^{2 - 1} = {a \times r}^{1} [/tex]

[tex]t2 = {2 \times - \frac{1}{4} }^{1} = - \frac{1}{2} [/tex]

The third term

[tex]t3 = {a \times r}^{3 - 1} = {a \times r}^{2} [/tex]

[tex]t3 = {2 \times - \frac{1}{4} }^{2} = 2 \times - \frac{1}{16} = \frac{1}{8} [/tex]

The fourth term

[tex]t4 = {a \times r}^{4 - 1} = {a \times r}^{3} [/tex]

[tex]t4 = {2 \times - \frac{1}{4} }^{3} = 2 \times - \frac{1}{64} = - \frac{1}{32} [/tex]

The fifth term

[tex]t5 = {a \times r}^{5 - 1} = {a \times r}^{4} [/tex]

[tex]t5 = {2 \times - \frac{1}{4} }^{4} = 2 \times - \frac{1}{256} = - \frac{1}{128} [/tex]

i hope all these helped

Answer:

[tex]2,\; -\dfrac{1}{2},\; \dfrac{1}{8},\; -\dfrac{1}{32},\; \dfrac{1}{128}[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given:

Substitute the given values of a and r into the formula to create an equation for the nth term:

[tex]a_n=2\left(-\dfrac{1}{4}\right)^{n-1}[/tex]

To find the first 5 terms of the geometric sequence, substitute n = 1 through 5 into the equation.

[tex]\begin{aligned}\implies a_1 & =2\left(-\dfrac{1}{4}\right)^{1-1}\\& =2\left(-\dfrac{1}{4}\right)^{0}\\& =2\left(1\right)\\&=2\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_2 & =2\left(-\dfrac{1}{4}\right)^{2-1}\\& =2\left(-\dfrac{1}{4}\right)^{1}\\& =2\left(-\dfrac{1}{4}\right)\\&=-\dfrac{1}{2}\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_3 & =2\left(-\dfrac{1}{4}\right)^{3-1}\\& =2\left(-\dfrac{1}{4}\right)^{2}\\& =2\left(\dfrac{1}{16}\right)\\&=\dfrac{1}{8}\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_4 & =2\left(-\dfrac{1}{4}\right)^{4-1}\\& =2\left(-\dfrac{1}{4}\right)^{3}\\& =2\left(-\dfrac{1}{64}\right)\\& =-\dfrac{1}{32}\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_5 & =2\left(-\dfrac{1}{4}\right)^{5-1}\\& =2\left(-\dfrac{1}{4}\right)^{4}\\& =2\left(\dfrac{1}{256}\right)\\& =\dfrac{1}{128}\end{aligned}[/tex]

Therefore, the first 5 terms of the given geometric sequence are:

[tex]2,\; -\dfrac{1}{2},\; \dfrac{1}{8},\; -\dfrac{1}{32},\; \dfrac{1}{128}[/tex]

Part (a)

There are 18 red slots out of the total 38, so the probability is [tex]18/38=\boxed{9/19}[/tex]. Note that the previous spins don't impact the result.

Part (b)

Similar to part (a), the answer is [tex]\boxed{9/19}[/tex].

Answer:

Leia needs 3 * 417 = <<3*417=1251>>1251 batteries for her cars.

Step-by-step explanation:

To find out how many batteries Leia needs for her cars, we need to multiply the number of toy cars by the number of batteries needed for each car. Since each car needs 3 batteries and Leia has 417 toy cars, the total number of batteries she needs is 3 * 417 = 1251.

The polynomial function is the product of three terms, each of which is a linear factor with a zero of 2-i, 1/3 respectively.

A polynomial function is a mathematical expression composed of constants and variables, and is used to model a wide variety of phenomena. The polynomial function can be expressed in terms of the factors of its zeros. In this case, the polynomial function of lowest degree with rational coefficients that has the given numbers 2-i, 1/3 as some of its zeros is (x+2)(x-i)(x-1/3). This polynomial can be expressed as the product of three terms, each of which is a linear factor with a zero of 2-i, 1/3 respectively. This means that when x is equal to 2-i or 1/3, the value of the polynomial function is zero. The linear factors are multiplied together to give the polynomial, and this is the simplest form of a polynomial that satisfies the given criteria. The polynomial can then be used to model a variety of phenomena by taking into account the different factors and their effects on the system.

f(x) = (x+2)(x-i)(x-1/3)

f(2-i) = (2-i+2)(2-i-i)(2-i-1/3)

f(2-i) = 0

f(1/3) = (1/3+2)(1/3-i)(1/3-1/3)

f(1/3) = 0

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find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros. 2-i, /3 f(x) = (x+2)(x-i)(x-1/3)

Answer: It is just 4

Step-by-step explanation: That means the absolute value of 4 and it is just 4

By Logical reasoning, missing tile in the given figure is Option C.

Initially find the similarity in each row or column. Get the logic relation between the blocks/tiles. Then try to apply the same logic on the missing tile/block to get the answer.

In the given question there are 9 tiles with 1 missing tile.

Now Check the first row:(As we go to the right one-by-one)

1) Normal circle is changing its square side in anti clockwise direction and also it goes in and out every time we pass a block/tile.

2)Dark circle is at corners of the squares as we pass the block it goes out or in tile by tile and also it changes the corner every time one corner is visited.

3)Triangle is at corners of the square and always inside the square, it changes the corner tile by tile in anti clockwise direction.

Similarly the same logic applies to remaining two rows.

Now for third row:

1) As the normal circle is at right side of square and inside the square, as we pass the tile it goes up(anti clockwise) and to outside of square.

2)The dark circle changes the corner once it visits a corner as it visited the down corner once it changes the corner to top right of square

3)Triangle just changes the corner as we pass, so it goes to bottom left corner of the square.

By Logical reasoning, missing tile in the given figure is Option C.

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The inequality to represent the amount of money, Marquis earned for the fund-raiser is d ≥ 15p

Inequalities are created through the connection of two expressions. It should be noted that the expressions in an inequality aren't always equal. Inequalities implies that the expressions are not equal. They are denoted by the symbols ≥ < > ≤.

Let the cost of each paper be p.

Since Marquis sold gift-wrapping paper for a school fundraiser and he sold at least 15 rolls of paper.

An inequality to represent the amount of money, Marquis earned for the fund-raiser is:

d ≥ (15 × p)

d ≥ 15p

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His chances of correctly predicting an answer are 1/4.

He has a 3/4 chance of getting an answer incorrectly.

A random variable is distributed binomially with parameters n, p more precisely if

It is the number of successes over n separate* trials, each of which has a chance of success of p.

Technically, the answer to whether an occurrence is possible is always "yes" or "no," or 1 or 0. Probability is the measure of how probable an event is to occur given the circumstances, assuming that it is feasible.

For instance, the likelihood of the number 5 appearing when we roll the dice is "Yes" (or 1), but the likelihood of the number 8 appearing is "No" (or 0).

Case 1: Guessing exactly 3 correct:

Choosing exactly 3 to get correct = Choosing exactly

3 to get wrong = 6C3 = (6*5*4)/(3*2*1) = 20 ways.

Guess the wrong answer for the first wrong one 3 ways.

Guess the wrong answer for the second wrong one 3 ways.

Guess the wrong answer for the third wrong one 3 ways.

That's 20*3*3*3 = 540 ways

Case 2: Guessing exactly 4 correct:

Choosing exactly 4 to get correct = Choosing exactly

2 to get wrong = 6C2 = (6*5)/(2*1) = 15 ways.

Guess the wrong answer for the first wrong one 3 ways.

Guess the wrong answer for the second wrong one 3 ways.

That's 15*3*3 = 135 ways

Case 3: Guessing exactly 5 correct:

Choosing exactly 5 to get correct = Choosing exactly

1 wrong = 6C1 = 6 ways.

Guess the wrong answer for the wrong one 3 ways.

That's 6*3 = 18 ways

Case 4: Guessing all 6 correct

There is only 1 way to guess them all correct

Total number of ways to pass: 540+135+18+1 = 694 ways to pass

Total number of ways to answer:

(4 ways to answer question 1) times

(4 ways to answer question 2) times

(4 ways to answer question 3) times

(4 ways to answer question 4) times

(4 ways to answer question 5) times

(4 ways to answer question 6) equals 4*4*4*4*4*4 = 46 = 4096

Probability of passing = 694/4096 = 347/2048 = 0.1694335938

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The statement that in the Orthogonal Decomposition Theorem, each term in formula (2) for Å, is itself an orthogonal projection of y onto a subspace of W is true .

The orthogonal decomposition of the vector of is the sum of the vectors of the subspaces of and the vectors of the orthogonal complements of . The orthogonal decomposition theorem states that each vector in can be uniquely described in the form , if W is a subspace of V then ecah vector in can be written in uniquely. Therefore z belongs to W⊥ because it is orthogonal to the spanning set of W. To keep the decomposition y = p + z unique, assume another decomposition y = q + w such that q ∈ W and w ∈ W⊥. Both decompositions are equal to y, so we have p − q = w − z. where p − q is in W and w − z is in W⊥.

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Answer:

Step-by-step explanation:

To find the total length of the pipe needed to lay a pipeline along the boundary of a garden, you will need to know the following:

The shape and size of the garden: The total length of the pipe needed will depend on the shape and size of the garden. For example, if the garden is rectangular, you will need to measure the length and width of the garden to calculate the total length of the pipe needed. If the garden has a more complex shape, you will need to measure the lengths of all the straight sections of the boundary and add them together to find the total length.

The type of pipe being used: The total length of the pipe needed will also depend on the type of pipe being used. Different types of pipes come in different lengths and may require different quantities of fittings and connectors.

The route of the pipeline: The total length of the pipe needed will also depend on the route that the pipeline takes. If the pipeline has to go around obstacles or make sharp turns, it may require more pipe than if it follows a straight path.

The desired width of the pipeline: The total length of the pipe needed will also depend on the desired width of the pipeline. If the pipeline needs to be wider to accommodate multiple pipes or cables, it will require more pipe than if it is only a single pipe.

Any other specific requirements: There may be other specific requirements that will affect the total length of the pipe needed. For example, if the pipeline needs to be buried, you will need to account for the depth of the trench in your calculations.

Answer:

x > 6. graph this on the number line.

Step-by-step explanation:

-2x-12 > -4x

-12 > -2x

12 < 2x

6 < x

Answer:

x > 6

Step-by-step explanation:

The correct solution to the given inequalities is;

a No, because the bike order does not meet the restrictions of 4c + 6a ≤ 120 and 4c + 4a ≤ 100

Let the number of child bikes be denoted as c.

Let the number of adult bikes be denoted as a.

For the building time of bikes, the inequality equation would be;

4c + 6a ≤ 120

For the testing time of bikes, the inequality equation would be;

4c + 4a ≤ 100

Checking the equations, for given 60 child bikes and 6 adult bikes in the week.

For the building of bikes,

( 4 × 60) + ( 6 × 6) ≤ 120

240 + 36 ≤ 120

276 ≥ 120 (Inequality not satisfied)

For the testing of bikes,

( 4 × 60 ) + ( 4 × 6 ) ≤ 100

240 + 24 ≤ 100

264 ≥ 100 (Inequality not satisfied)

Hence, the correct option is A.

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Thus ,after adding  3/4 and 7/10  we get 29/20 where 20 is common denominator

The smallest common multiple among the denominators of a group of fractions is known as the lowest common denominator (least common denominator) in mathematics. Fraction comparison, addition, and subtraction are made easier.

Here,

In order to add 3/4 and 7/10

we have to find LCM of 4 and 10

which comes out to be = 20

Thus,

addding =>  15+`14 /20

=>29/20

Thus ,after adding  3/4 and 7/10  we get 29/20 where 20 is common denominator

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Answer: $14 for 1 ticket

Step-by-step explanation: 84 divided by 6 is 14. hope this helps

Answer:

False.

Step-by-step explanation:

False. The study of the relationships among thermal energy, heat, and work is called thermodynamics, not dynamics. Dynamics is a branch of mechanics that deals with the motion of objects and the forces that cause that motion. Thermodynamics, on the other hand, is the study of the relationships between heat, work, and other forms of energy, and how these relationships can be used to understand the behavior of systems. It is a branch of physics that is concerned with the study of heat and its effects on matter.

The correct statement regarding why f(x) ≠ g(x) is given as follows:

a. f(x) ≠ g(x) because the graphs do not intersect.

Two functions are classified as different when they do not intersect.

The functions in this problem are given as follows:

For them to intersect, it is needed that:

f(x) = g(x).

x² = 2x - 3.

x² - 2x + 3 = 0.

Which is a quadratic function with the coefficients given as follows:

a = 1, b = -2, c = 3.

The number of solutions of the quadratic function depends on the discriminant, given as follows:

Discriminant = b² - 4ac.

Hence the numeric value for the discriminant in this problem is given as follows:

Discriminant = (-2)² - 4 x 1 x 3 = -8.

As the discriminant is negative, the quadratic function has no solutions, meaning that the functions do not intersect and statement a is correct.

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The required number of bracelets that in the given question are 10.

In math, the meaning of a  equation is a numerical explanation that shows that two numerical articulations are equivalent. For example, 3x + 5 = 14 is a condition, where 3x + 5 and 14 are two articulations isolated by an 'equivalent' sign.

Carla divided 28 bracelets equally among 4 friends.

Number of bracelets each friends has = 28/4 = 7 bracelets

Then Carla gave 3 more bracelets to Magan.

So, Total bracelets Magan has = 7 + 3 = 10 bracelets

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Answer: Hence, the time taken by the experienced crew would be 9 hours.

Step-by-step explanation:

The probability that (a and b) or (c and d) or (b and d) are selected is 3/25

To find the probability that a and b or c and d or b and d are selected, we can use the formula for probability:

Probability = Number of favorable outcomes / Total number of outcomes

Probability of a = 1/5

Probability of b = 1/5

Probability of a and b = 1/5 × 1/5 = 1/25

Probability of c = 1/5

Probability of d = 1/5

Probability of c and d = 1/5 × 1/5 = 1/25

Probability of b = 1/5

Probability of d = 1/5

Probability of b and d = 1/5 × 1/5 = 1/25

The probability that (a and b) or (c and d) or (b and d) are selected will be:

Probability of (a and b) or (c and d) or (b and d) = P(a and b) +  P(c and d +  P(b and d)

Probability of (a and b) or  (c and d) or (b and d) = 1/25 + 1/25 + 1/25 = 3/25

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Answer:

  (a)  211,555.72

  (b)  16.64%

Step-by-step explanation:

You want to know the total interest earned on a 200,000 investment at ...

And you want to know the effective rate for the last period.

The multiplier of the investment at rate r compounded n times per year for t years is ...

  k = (1 +r/n)^(nt)

Using this multiplier for the rates and periods given the balance of the account at the end of 6 years will be ...

  200,000(1 +.08/12)^(12·2) × (1 +.12/4)^(4·1.5) × (1 +.16/2)^(2·2.5)

  ≈ 411,555.72

The interest earned is the difference between the account balance and the principal invested:

  411,555.72 -200,000 = 211,555.72 . . . . interest earned in 6 years

The annual multiplier for the last term is ...

  (1 +.16/2)^(2·1) = 1.1664

The effective interest rate is 1 less than this:

  16.64% = effective rate during final year

__

Additional comment

The repetitive math can be less tedious if you let a calculator or spreadsheet do it.

No currency units are given in the problem statement.

On solving the provided question, we can say that -  95 confidence interval Z alpha 2 = 1.96, CI = (4.1 , 53.9)

The z critical value is 1.96 for a test with a 95% confidence level (e.g. = 0.05). The z critical value is 5.576 for a test with a 99% confidence level (for instance, with = 0.01).

The two red tails represent the alpha level split by two. Finding the z-score for an alpha level for a two-tailed test is what is meant when a question asks you to determine z alpha/2. The confidence interval may be subtracted from 100% to calculate alpha, which is connected to confidence levels.

for 95% confidence,

[tex]Z_{\frac{\alpha }{2} }[/tex] = 1.96

CI = (5279 - 5250) ± 1.96[tex]\sqrt{\frac{140^2}{395} + \frac{210^2}{395} } \\[/tex]

CI = (4.1 , 53.9)

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The domain is (0, 2), staying the same will be (2, 4), decreasing the fastest will be (6, 10), and the height of the water balloon at 16 seconds the will be 0.

The domain of a function is defined as the set of all the possible input values that are valid for the given function.

The range of a function is defined as the set of all the possible output values that are valid for the given function.

Part A:

We can see from the graphic, increase from 0 to 2 sec.

The domain is equal to (0, 2), where the height of the water balloon increases.

Part B:

The water balloon is the same from 2 to 4 sec.

The field state that the height of the water balloon remains the same (2, 4).

Part C:

Now Height decreasing fasted at 4 to 6 sec.

Since the slope is steepest downward from 4 to 6 sec as comfort to 6 to 10 sec.

The domain is the height of the water balloon decreases rapidly (6, 10).

Part D:

The balloon's height is almost near the ground as resistance will play its role. But will almost touch the ground.

Therefore height of the water balloon will be 0 in 16 seconds.

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Therefore , the term t of each continuously compounded account  is 23.1 years.

Compound interest is the practice of adding interest to the principal of a loan or deposit. It happens when interest is reinvested, added to the lent capital rather than paid out, or when the borrower is forced to pay it, resulting in interest being generated on the principal amount plus any accumulated interest the following period. Compound interest is frequently used in economics and finance.

Here,

A) To determine the account's annual growth rate, we must calculate the impact of continual compounding. Similar to the nominal rate (APR) and actual yield (APY) on a bank's interest-bearing account, the yield will be marginally greater due to compounding and indicate actual growth rather than growth rate. Note e is a very lengthy number that can be entered with as many digits as your calculator allows if you don't have a financial calculator handy. e = 2.71828183

A = P[tex]e^{rt}[/tex]  But for this, we only need to calculate growth or ert; we don't need to worry about P. When rounding in accordance with the instructions, we set t = 1 for a year and have er (or 2.71828183.03), which equals 1.03045, therefore the account increases by 3.05% annually.

B) 3000 = 1500[tex]e^{0.3t}[/tex]      [Divide both sides by 1500]

To isolate t, we must determine the natural logarithm of each side (e's natural logarithm is 1 and 2's natural logarithm is 0.693). 2 = e.03t

0,693 =.03t [Subtract both sides from .03]

23.1 = t

C) Using the formula from section A, first determine the original investment's worth after ten years. A = P A = 1500e.03*10 A = 1500e.3 A = 2,024.79

Next, use the yearly compounding formula to get the value of the second investment.

A = P(1+r)t

A = 1200(1+.045)10

A = 1,863.56

After ten years, the initial investment would be worth more.

Therefore , the term t of each continuously compounded account  is 23.1 years.

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Answer:

8-(6+4)-12

8-12-6-4

Step-by-step explanation:

8-(6+4)-12 and 8-12-6-4 are equivalent expressions because they both compute to -6. This is because in both expressions, the 6+4 is calculated first, yielding 10. Then, the 8 is subtracted from 10, yielding -2. Finally, the 12 is subtracted from -2, yielding -6.

6+4)-8-12 is not an equivalent expression because it does not compute to -6. In this expression, the 6+4 is calculated first, yielding 10. Then, the 8 is subtracted from 10, yielding 2. Finally, the 12 is subtracted from 2, yielding -10.

8-6-12+4 is not an equivalent expression because it does not compute to -6. In this expression, the 8 is subtracted from 6, yielding -2. Then, the 12 is subtracted from -2, yielding -14. Finally, 4 is added to -14, yielding -10.

Answer:

[tex]\ln (y) = -0.250 \ln (x)+0.223[/tex]

Step-by-step explanation:

Given table:

[tex]\begin{array}{|c|c|}\cline{1-2} \vphantom{\dfrac12}x & y\\\cline{1-2} \vphantom{\dfrac12} 1 & 1.250\\\cline{1-2} \vphantom{\dfrac12} 2 & 1.051\\\cline{1-2} \vphantom{\dfrac12} 3& 0.950\\\cline{1-2} \vphantom{\dfrac12} 4& 0.884\\\cline{1-2} \vphantom{\dfrac12} 5& 0.836\\\cline{1-2} \vphantom{\dfrac12} 6& 0.799\\\cline{1-2} \end{array}[/tex]

To convert  y = axⁿ  to linear form, take natural logs of both sides and rearrange:

[tex]\begin{aligned}y=ax^n \implies \ln y &= \ln ax^n\\\implies \ln y &= \ln a + \ln x^n\\\implies \ln y &= \ln a + n \ln x\\\implies \ln y &=n \ln x+ \ln a\end{aligned}[/tex]

This is in the straight-line form y = mx + c.

Substitute the first values of x and y from the table into the natural log formula and solve for ln(a):

[tex]\implies \ln 1.250= n \ln 1+\ln a[/tex]

[tex]\implies \ln 1.250= n (0)+\ln a[/tex]

[tex]\implies \ln a = \ln 1.250[/tex]

[tex]\implies \ln a = 0.223\; \sf (3 \; d.p.)[/tex]

Substitute the found value of ln(a) and the last values of x and y from the table into the  natural log formula and solve for n:

[tex]\implies \ln 0.799= n \ln 6+\ln 1.250[/tex]

[tex]\implies n \ln 6=\ln 0.799-\ln 1.250[/tex]

[tex]\implies n =\dfrac{\ln 0.799-\ln 1.250}{\ln 6}[/tex]

[tex]\implies n =-0.250\;\sf (3 \; d.p.)[/tex]

Substitute the found values of n and ln(a) into the formula to create an equation for ln(y):

[tex]\boxed{\ln (y) = -0.250 \ln (x)+0.223}[/tex]

The trigonometric identities values are;

i. sin (A – B) = -0.15

ii. cos (A – B) = 0.5368

iii. tan (A + B) = -3.536

We are given;

cos A = 0.42

sin B = 0.73

Since cos A = 0.42 , then sin A = 0.58

Since sin B = 0.73, then cos B = 0.27

i) sin (A – B)

By trigonometric identities, we know that;

sin (A – B) = sinA cos B - cosA sinB

Thus;

sin (A – B) = (0.58 * 0.27) - (0.42 * 0.73)

sin (A – B) = -0.15

ii)  cos (A – B)

By trigonometric identities, we know that;

cos (A – B) = cosA cosB + sinA sinB

cos (A – B) = (0.42 * 0.27) + (0.58 * 0.73)

cos (A – B) = 0.5368

iii) tan (A + B)

By trigonometric identities, we know that;
(tan A + tan B)/(1 - tanAtanB)

tan A = sinA/cosA = 0.58/0.42 = 1.389

tan B = sinB/cosB = 0.73/0.58 = 1.259

Thus;

tan (A + B) = (1.389 + 1.259)/(1 - (1.389 * 1.259))

tan (A + B) = -3.536

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Answer:

1/8 savings left

Step-by-step explanation:

add 5/8+1/4 which is 7/8 8/8-7/8= 1/8

Answer:

----------------------------

Marcus used:

Remaining part of savings:

Answer:

Step-by-step explanation:

Solve for x:

3 x^4 - 16 x^3 + 4 x + 363 = 0

Eliminate the cubic term by substituting y = x - 4/3:

363 + 4 (y + 4/3) - 16 (y + 4/3)^3 + 3 (y + 4/3)^4 = 0

Expand out terms of the left-hand side:

3 y^4 - 32 y^2 - (476 y)/9 + 3059/9 = 0

Divide both sides by 3:

y^4 - (32 y^2)/3 - (476 y)/27 + 3059/27 = 0

Add 2/3 sqrt(3059/3) y^2 + (32 y^2)/3 + (476 y)/27 to both sides:

y^4 + 2/3 sqrt(3059/3) y^2 + 3059/27 = 2/3 sqrt(3059/3) y^2 + (32 y^2)/3 + (476 y)/27

y^4 + 2/3 sqrt(3059/3) y^2 + 3059/27 = (y^2 + sqrt(3059/3)/3)^2:

(y^2 + sqrt(3059/3)/3)^2 = 2/3 sqrt(3059/3) y^2 + (32 y^2)/3 + (476 y)/27

Add 2 (y^2 + sqrt(3059/3)/3) λ + λ^2 to both sides:

(y^2 + sqrt(3059/3)/3)^2 + 2 λ (y^2 + sqrt(3059/3)/3) + λ^2 = (476 y)/27 + 2/3 sqrt(3059/3) y^2 + (32 y^2)/3 + 2 λ (y^2 + sqrt(3059/3)/3) + λ^2

(y^2 + sqrt(3059/3)/3)^2 + 2 λ (y^2 + sqrt(3059/3)/3) + λ^2 = (y^2 + sqrt(3059/3)/3 + λ)^2:

(y^2 + sqrt(3059/3)/3 + λ)^2 = (476 y)/27 + 2/3 sqrt(3059/3) y^2 + (32 y^2)/3 + 2 λ (y^2 + sqrt(3059/3)/3) + λ^2

(476 y)/27 + 2/3 sqrt(3059/3) y^2 + (32 y^2)/3 + 2 λ (y^2 + sqrt(3059/3)/3) + λ^2 = (2 λ + 32/3 + (2 sqrt(3059/3))/3) y^2 + (476 y)/27 + 2/3 sqrt(3059/3) λ + λ^2:

(y^2 + sqrt(3059/3)/3 + λ)^2 = y^2 (2 λ + 32/3 + (2 sqrt(3059/3))/3) + (476 y)/27 + 2/3 sqrt(3059/3) λ + λ^2

Complete the square on the right-hand side:

(y^2 + sqrt(3059/3)/3 + λ)^2 = (y sqrt(2 λ + 32/3 + (2 sqrt(3059/3))/3) + 238/(27 sqrt(2 λ + 32/3 + (2 sqrt(3059/3))/3)))^2 + (4 (2 λ + 32/3 + (2 sqrt(3059/3))/3) (λ^2 + 2/3 sqrt(3059/3) λ) - 226576/729)/(4 (2 λ + 32/3 + (2 sqrt(3059/3))/3))

To express the right-hand side as a square, find a value of λ such that the last term is 0.

This means 4 (2 λ + 32/3 + (2 sqrt(3059/3))/3) (λ^2 + 2/3 sqrt(3059/3) λ) - 226576/729 = 8/729 (729 λ^3 + 243 sqrt(9177) λ^2 + 3888 λ^2 + 864 sqrt(9177) λ + 165186 λ - 28322) = 0.

Thus the root λ = 1/9 (-sqrt(9177) - 16) + (85 13^(2/3) (i sqrt(3) + 1))/(6 (3 (i sqrt(7766346) - 4023))^(1/3)) + ((-i sqrt(3) + 1) (13 (i sqrt(7766346) - 4023))^(1/3))/(6 3^(2/3)) allows the right-hand side to be expressed as a square.

(This value will be substituted later):

(y^2 + sqrt(3059/3)/3 + λ)^2 = (y sqrt(2 λ + 32/3 + (2 sqrt(3059/3))/3) + 238/(27 sqrt(2 λ + 32/3 + (2 sqrt(3059/3))/3)))^2

Take the square root of both sides:

y^2 + sqrt(3059/3)/3 + λ = y sqrt(2 λ + 32/3 + (2 sqrt(3059/3))/3) + 238/(27 sqrt(2 λ + 32/3 + (2 sqrt(3059/3))/3)) or y^2 + sqrt(3059/3)/3 + λ = -y sqrt(2 λ + 32/3 + (2 sqrt(3059/3))/3) - 238/(27 sqrt(2 λ + 32/3 + (2 sqrt(3059/3))/3))

Solve using the quadratic formula:

y = 1/6 (sqrt(2) sqrt(9 λ + 48 + sqrt(9177)) + sqrt(2) sqrt(48 - sqrt(9177) - 9 λ + 238 sqrt(2) 1/sqrt(9 λ + 48 + sqrt(9177)))) or y = 1/6 (sqrt(2) sqrt(9 λ + 48 + sqrt(9177)) - sqrt(2) sqrt(48 - sqrt(9177) - 9 λ + 238 sqrt(2) 1/sqrt(9 λ + 48 + sqrt(9177)))) or y = 1/6 (sqrt(2) sqrt(48 - sqrt(9177) - 9 λ - 238 sqrt(2) 1/sqrt(9 λ + 48 + sqrt(9177))) - sqrt(2) sqrt(9 λ + 48 + sqrt(9177))) or y = 1/6 (-sqrt(2) sqrt(9 λ + 48 + sqrt(9177)) - sqrt(2) sqrt(48 - sqrt(9177) - 9 λ - 238 sqrt(2) 1/sqrt(9 λ + 48 + sqrt(9177)))) where λ = 1/9 (-sqrt(9177) - 16) + (85 13^(2/3) (i sqrt(3) + 1))/(6 (3 (i sqrt(7766346) - 4023))^(1/3)) + ((-i sqrt(3) + 1) (13 (i sqrt(7766346) - 4023))^(1/3))/(6 3^(2/3))

Substitute λ = 1/9 (-sqrt(9177) - 16) + (85 13^(2/3) (i sqrt(3) + 1))/(6 (3 (i sqrt(7766346) - 4023))^(1/3)) + ((-i sqrt(3) + 1) (13 (i sqrt(7766346) - 4023))^(1/3))/(6 3^(2/3)) and approximate:

y = -2.83639 - 2.06535 i or y = -2.83639 + 2.06535 i or y = 2.83639 - 1.07606 i or y = 2.83639 + 1.07606 i

Substitute back for y = x - 4/3:

x - 4/3 = -2.83639 - 2.06535 i or y = -2.83639 + 2.06535 i or y = 2.83639 - 1.07606 i or y = 2.83639 + 1.07606 i

Add 4/3 to both sides:

x = -1.50306 - 2.06535 i or y = -2.83639 + 2.06535 i or y = 2.83639 - 1.07606 i or y = 2.83639 + 1.07606 i

Substitute back for y = x - 4/3:

x = -1.50306 - 2.06535 i or x - 4/3 = -2.83639 + 2.06535 i or y = 2.83639 - 1.07606 i or y = 2.83639 + 1.07606 i

Add 4/3 to both sides:

x = -1.50306 - 2.06535 i or x = -1.50306 + 2.06535 i or y = 2.83639 - 1.07606 i or y = 2.83639 + 1.07606 i

Substitute back for y = x - 4/3:

x = -1.50306 - 2.06535 i or x = -1.50306 + 2.06535 i or x - 4/3 = 2.83639 - 1.07606 i or y = 2.83639 + 1.07606 i

Add 4/3 to both sides:

x = -1.50306 - 2.06535 i or x = -1.50306 + 2.06535 i or x = 4.16972 - 1.07606 i or y = 2.83639 + 1.07606 i

Substitute back for y = x - 4/3:

x = -1.50306 - 2.06535 i or x = -1.50306 + 2.06535 i or x = 4.16972 - 1.07606 i or x - 4/3 = 2.83639 + 1.07606 i

Add 4/3 to both sides:

Answer:  x = -1.50306 - 2.06535 i or x = -1.50306 + 2.06535 i or x = 4.16972 - 1.07606 i or x = 4.16972 + 1.07606 i