a bag contains 19 red,15 yellow and 14 blue marbles. what is the probability of pulling out a blue marble, followed by a red marble, if you replace the blue marble first
-133/384
-11/16
-33/48
-133/1152
-11/384

Answer:

393/1000

Step-by-step explanation:

Answer:

Step-by-step explanation:

393/1000

Answer: a) 21 cans

b) 10.83 square inches.

Step-by-step explanation:

The cans are cylinders of 4 and 1/4 inches tall (or 4.25 in)

and the diameter is 3 inches.

The surface of a cylinder is equal to:

S = pi*r^2 + h*2*pi*r

where r is the radius, half of the diameter, so we have that r = 3in/2 = 1.5 in.

h is the height, h = 4.25 in

pi = 3.14

Then the surface needed for a can is:

S = 3.14*1.5^2 + 4.25*2*3.14*1.5 = 47.1 square inches.

if the sheet is 1000 in^2, we can make an amount of:

N = 1000/47.1 = 21.23  

but we can not do a 0.23 of a can, so we need to round down.

A) we can make 21 cans out of a sheet of metal.

B) the 0.23 of a can that we removed earlier is the amount of metal leftover. The total is 0.23*47.1 in^2 = 10.83 in^2

Answer:

Step-by-step explanation:

To determine the amount of metal needed to make each can, we would determine the total surface area of each can. Since the cans are cylindrical, the formula for determining the total surface area of a cylinder is used. It is expressed as

Total surface area = πr² + 2πr

r = radius of the can

h = height of the can

π = 3.14

From information given,

Diameter = 4.25 inches

Radius = diameter/2 = 4.25/2 = 2.125 inches

Total surface area = 3.14 × 2.125² + 2 × 3.14 × 2.125 = 27.5 in²

1) since 1000 in² sheet material is available, the number of cans that can be made is

1000/27.5240625 = 36 cans

2) The amount of metal sheet left is

1000 - (36 × 27.5) = 10 in²

Answer:

  √37

Step-by-step explanation:

It is helpful to know the squares of small integers. Then you become aware of the approximate magnitudes of square roots.

Point C is between 6 and 7, so the value of C² will be between 6² = 36 and 7² = 49. Since C is closer to 6 than to 7, it represents the root of a number closer to 36 than to 49.

Only one answer choice is in this range: √37, Option 1.

Answer:

a) add a dummy stroke to make the problem as an assignment problem of adding 5 strokes to 5 swimmers. see first attachment.

b) applying the Hungarian method.

   4.8      0     0.9     4.1     2.5

   10.3     0     9.1      1.6     8.7

   4.8      0     10.4    1.9     5.1

   2.8      0     3.2     2.1     4.7

      0      0      0        0       0  

Deduct the smallest element in each column from the other elements of the column.

   2      0     0       2.5    0

   7.5   0     8.2    0       6.2

   2      0     9.5    0.3    2.6

   0      0     2.3    0.5    2.2

   0      0     0       0       0  

Which implies:      

   2            8.2    2.5    6.2

   7.5         9.5    0.3    2.6

   2            2.3    0.5    2.2

33.8 + 34.7 + 28.5 + 29.2 = 126.2

David = Back Stroke

tony = Breast Stroke

Chris = Butterfly

Carl = Free Style

The question is about assigning swimmers in a team to different strokes to achieve the best total time. It's a combinatorial optimization problem which can be solved by considering all the permutations of swimmers' assignments to each stroke and selecting the one with least total time.

The subject of this question is an optimization problem in Mathematics, specifically in the field of Combinatorics. Deciding the arrangement of swimmers to minimize the total time spent can be approached using techniques from this field. Unfortunately, the information provided does not give the exact times of the swimmers, so achieving a detailed solution isn't possible. However, the problem could hypothetically be solved by enumerating all possible assignments of swimmers to strokes and selecting the assignment that has the least total time.

This problem resonates with high school level math, where students begin to tackle optimization problems and permutations.

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Answer:

Type II error will be made if we conclude that the new advertising campaign does not increases the sales when in fact the sales are increased after the advertising campaign.

Step-by-step explanation:

A type II error is a statistical word used within the circumstance of hypothesis testing that defines the error that take place when one is unsuccessful to discard a null hypothesis that is truly false. It is symbolized by β i.e.  

β = Probability of accepting H₀ when H₀ is false.

In this case we need to test the hypothesis whether the new advertising campaign increases the sales or not.

The hypothesis can be defined as:

H₀: The new advertising campaign does not increases the sales.

Hₐ: The new advertising campaign increases the sales.

The confidence level wanted here is 99%.

The type II error will be made if we conclude that the new advertising campaign does not increases the sales when in fact the sales are increased after the advertising campaign.

The type II error could have been made because of the following reasons:

Thus, the type II error might have been committed because of small sample size or small significance level.

Answer:

a) 1/6

b) 3/8

Step-by-step explanation:

Even i struggle with fractions but im sure you will get it one day (✿◡‿◡)

Answer: A. have the same slope

Step-by-step explanation:

b. bisect means right in half but not fully intersecting kinda looks like this _l_ *oh and it asked you to prove that they bisect so clicking on it don't really make sense*

c. perpendicular means there is four equal angels and that is 90 degree angels

d. parallel... well it's obviously not parallel because parallel are two lines that are exactly the same but never intersect

*intersect means touch*

Answer:

The answer is d: LINE D AND LINE E

The other answer is c LINE A AND LINE B

Step-by-step explanation:

Answer:

Step-by-step explanation:

WE are given that [tex]a_n = \frac{(-1)^{n+1}}{5n-4}[/tex]. Then, to now the first for terms, we must replace n by 1,2,3,4 respectively. Then

[tex]a_1 = \frac{(-1)^2}{5(1)-4} = \frac{1}{1}= 1 [/tex]

[tex]a_2 = \frac{(-1)^3}{5(2)-4} = \frac{-1}{6} [/tex]

[tex]a_3 = \frac{(-1)^4}{5(3)-4} = \frac{1}{11}= 1 [/tex]

[tex]a_4 = \frac{(-1)^5}{5(4)-4} = \frac{-1}{16}= 1 [/tex]

Note that as n increase, [tex]a_n[/tex] gets closer to 0. So, the limit of this sequence is 0.

Final answer:

The function in question, f(x) = 3x^2 + 7x + 2, is a quadratic function, and its properties such as graph shape, intercepts, and vertex can be studied. Additionally, the derivative of this function, obtained through power rule differentiation, is f'(x) = 6x + 7.

Explanation:

The question asks about the function f(x) = 3x2 + 7x + 2. This appears to be a quadratic function, which is a fundamental concept in algebra and pre-calculus. Detailing the characteristics of a quadratic function involves finding its graph, which is a parabola, its vertex, axis of symmetry, intercepts, and possibly its extrema (maximum or minimum values).

In mathematics, finding the derivative of a function is a common operation in calculus. Given the information on different functions and their derivatives from the provided reference text, we can deduce that the derivative of f(x) would be found through power rule differentiation: for f(x) = axn, the derivative f'(x) = naxn-1. Applying this to the given function, we find the derivative f'(x) = 6x + 7.

Let x represent amount borrowed at 4.2% and y represent amount invested at 6.8%.

We have been given that Michelle borrows a total of $2500 in student loans from two lenders. We can represent this information in an equation as:

[tex]x+y=2500...(1)[/tex]

[tex]y=2500-x...(1)[/tex]

We are also told that at the end of 3 years, she will owe a total of $354 for the interest from both loans.

Amount of interest earned at a rate of 4.2% in 3 years would be [tex]0.042\cdot 3\cdot x=0.126x[/tex].

Amount of interest earned at a rate of 6.8% in 3 years would be [tex]0.068\cdot 3\cdot y=0.204y[/tex].

We can represent this information in an equation as:

[tex]0.126x+0.204y=354...(2)[/tex]

Upon substituting equation (1) in equation (2), we will get:

[tex]0.126x+0.204(2500-x)=354[/tex]

[tex]0.126x+510-0.204x=354[/tex]

[tex]-0.078x+510=354[/tex]

[tex]-0.078x+510-510=354-510[/tex]

[tex]-0.078x=-156[/tex]

[tex]\frac{-0.078x}{-0.078}=\frac{-156}{-0.078}[/tex]

[tex]x=2000[/tex]

Therefore, Michelle borrowed $2000 at 4.2%.

Upon substituting [tex]x=2000[/tex] in equation (1), we will get:

[tex]y=2500-2000=500[/tex]

Therefore, Michelle borrowed $500 at 6.8%.

She borrowed $ 2000 from the 4.2% lender and $ 500 from the 6.8% lender.

Since Michelle borrows a total of $ 2500 in student loans from two lenders, and one charges 4.2% simple interest and the other charges 6.8% simple interest, and she is not required to pay off the principal or interest for 3 yr, but at the end of 3yr, she will owe a total of $ 354 for the interest from both loans, to determine how much did she borrow from each lender, the following calculation must be performed:

Therefore, she borrowed $ 2000 from the 4.2% lender and $ 500 from the 6.8% lender.

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Answer:

Step-by-step explanation

First five multiples of 2030 are: 2, 5, 7, 10 an 14

Answer:

4

Step-by-step explanation:

Let's set up an equation using the formula for the area of a triangle.

Hint #22 / 3

\begin{aligned} \text{Area of a triangle} &= \dfrac12 \cdot \text{base} \cdot \text{height}\\\\ 12&= \dfrac12 \cdot 6 \cdot x \\\\ 12&= 3x \\\\ \dfrac{12}{\blueD{3}}&= \dfrac{3x}{\blueD{3}} ~~~~~~~\text{divide both sides by } {\blueD{ 3}}\\\\ \dfrac{12}{\blueD{3}}&= \dfrac{\cancel{3}x}{\blueD{\cancel{3}}}\\\\ x &=\dfrac{12}{\blueD{3}}\\\\ x &=4\end{aligned}  

Area of a triangle

12

12

3

12

​  

3

12

​  

x

x

​  

=  

2

1

​  

⋅base⋅height

=  

2

1

​  

⋅6⋅x

=3x

=  

3

3x

​  

       divide both sides by 3

=  

3

​  

3

​  

x

​  

=  

3

12

​  

=4

​  

Answer:

55 cm2

Step-by-step explanation:

The area of the new parallelogram is 55 sq.cm, the correct option is C.

A polygon with four sides such that the opposite sides are parallel and equal is called a Parallelogram.

The height of the parallelogram is 4 cm

The base of the parallelogram is 9cm

The height of the parallelogram is increased by 1 cm

New height = 5cm

The base of the parallelogram is increased by 2 cm

New base = 11 cm

Area of a parallelogram is =  Base * Height

Area of parallelogram is  = 5 * 11 = 55 sq.cm

Therefore, the area of the new parallelogram is 55 sq.cm.

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Answer: y = 0.25*f(x - A) + 3

Step-by-step explanation:

Initially we have the graph of y = f(x)

If we do a vertical compression, this means that we multiply the function by the scale factor, in this case the scale factor is 0.25

So now our graph is y = 0.25*f(x)

A translation to the right by A units means that now we valuate the function in x - A, in this case A = 4, so our graph now is:

y = 0.25*f(x - 4)

A vertical translation means that we add a constant to the function, if the constant is positive the tranlsation is upwards, if the constant is negative the translation is downwards.

Here the translation is of 3 units upwards, so our new graph is:

y = 0.25*f(x - A) + 3

Answer:

120

Step-by-step explanation:

add 4 and 2 then multipy by 2 and add a zero

Answer:

We conclude that the true percentage of those in the first group who suffer a second episode is less than or equal to the true percentage of those in the second group who suffer a second episode.

Step-by-step explanation:

We are given that there are 31 people in the first group and this group will be administered the new drug. There are 45 people in the second group and this group will be administered a placebo.

After one year, 11% of the first group has a second episode and 9% of the second group has a second episode.

Let [tex]p_1[/tex] = true percentage of those in the first group who suffer a second episode.

[tex]p_2[/tex] = true percentage of those in the second group who suffer a second episode.

So, Null Hypothesis, [tex]H_0[/tex] : [tex]p_1-p_2\leq[/tex] 0  or  [tex]p_1\leq p_2[/tex]      {means that the true percentage of those in the first group who suffer a second episode is less than or equal to the true percentage of those in the second group who suffer a second episode}

Alternate Hypothesis, [tex]H_A[/tex] : [tex]p_1-p_2[/tex] > 0  or  [tex]p_1>p_2[/tex]      {means that the true percentage of those in the first group who suffer a second episode is more than the true percentage of those in the second group who suffer a second episode}

The test statistics that will be used here is Two-sample z proportion test statistics;

                              T.S. = [tex]\frac{(\hat p_1-\hat p_2)-(p_1-p_2)}{\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+ \frac{\hat p_2(1-\hat p_2)}{n_2}} }[/tex]  ~ N(0,1)

where, [tex]\hat p_1[/tex] = sample proportion of people in the first group who suffer a second episode = 11%

[tex]\hat p_2[/tex] = sample proportion of people in the second group who suffer a second episode = 9%

[tex]n_1[/tex] = sample of people in first group = 31

[tex]n_2[/tex] = sample of people in second group = 45

So, the test statistics  =  [tex]\frac{(0.11-0.09)-(0)}{\sqrt{\frac{0.11(1-0.11)}{31}+ \frac{0.09(1-0.09)}{45}} }[/tex]

                                     =  0.283

Now, at 0.1 significance level, the z table gives critical value of 1.2816 for right-tailed test. Since our test statistics is less than the critical value of z as 0.283 < 1.2816, so we have insufficient evidence to reject our null hypothesis due to which we fail to reject our null hypothesis.

Therefore, we conclude that the true percentage of those in the first group who suffer a second episode is less than or equal to the true percentage of those in the second group who suffer a second episode.

A hypothesis test can determine whether there is enough evidence to support the claim that the new drug is effective in reducing second episodes of heartburn. It involves defining null and alternative hypotheses, calculating a test statistic, and comparing it to a critical value based on the set significance level.

We start by defining our null and alternative hypotheses. In this case, we are testing against the claim that the true percentage of those in the first group who suffer a second episode is more than the true percentage of those in the second group who suffer a second episode.

So, our null hypothesis (H0) is: The percentage of heart attempts in group 1 is equal to or less than that of group 2.

And, our alternative hypothesis (Ha) is: The percentage of heart attempts in group 1 is greater than that of group 2.

We conduct the hypothesis test using a standard test of proportions. Calculating our test statistic can be done using the formula: Z = (p1 - p2)/sqrt(p(1 - p)[(1/n1) + (1/n2)])

Where, p1 and p2 are the proportions of the two groups, n1 and n2 are the sizes of the two groups, and p is the combined proportion.

Based on the information in the problem, the calculated test statistic value and the critical z-value for a one-tailed test at the significance level 0.1, we can make a decision to reject or fail to reject the null hypothesis. If the calculated absolute z-value is greater than the critical z-value, we reject the null hypothesis and conclude that there is enough evidence at the 0.1 level. Otherwise, we fail to reject the null hypothesis and conclude that there is not enough evidence to support the claim.

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Answer:

z-score

Step-by-step explanation:

Z-score gives the relative value of any data population in relation to its mean. It depicts at what distance any data is from the mean. In technical terms it can be termed as the number of times a data is data away from the mean. Positive value of z score means value is more than mean while a negative value signifies data is less than that of mean.

Z – score is primarily used in qualitative analysis of numerical data by the statistician after data is arranged in normal distribution form. Z score of 0 means the value is same as mean while z-score of value 1 means data is one standard deviation away from the mean.

The z-score is the measure that determines the distance of a value from the mean in terms of standard deviations which is option c.

A z-score determines how far a particular value is from the mean relative to the data set's standard deviation. Given an experimental value, X, the mean, μ (mu), and the standard deviation, σ (sigma), the z-score is calculated using the formula Z = (X - μ) / σ. A z-score represents the number of standard deviations an experimental value is above or below the mean. For example, if a data value has a z-score of 2, it is two standard deviations above the mean. Contrarily, a z-score of -1.5 indicates that the value is one and a half standard deviations below the mean. Z-scores are used across various fields to compare different values within a data set or among different data sets with different means and standard deviations.

Answer:

a) The volume of S is 34.13

b) The volume of S is 14.8

c) The volume of S is 5.17

d) The volume of S is 11.33

Step-by-step explanation:

a) The cross section area is equal to:

[tex]A=a^{2} =((5x-x^{2})-x)^{2} =(4x-x^{2} )^{2}[/tex]

The volume of S is equal to:

[tex]Vol_{S} =\int\limits^4_0 {A(x)} \, dx =\int\limits^4_0 {(4x-x^{2})^{2} } \, dx =34.13[/tex]

b) The cross section area is equal to:

[tex]A=\frac{a^{2}\sqrt{3} }{4} =\frac{\sqrt{3} }{4} ((5x-x^{2} )-x)^{2} =\frac{\sqrt{3} }{4} (4x-x^{2} )^{2}[/tex]

The volume of S is equal to:

[tex]Vol_{S} =\int\limits^4_0 {A(x)} \, dx =\frac{\sqrt{3} }{4} \int\limits^4_0 {(4x-x^{2})^{2} } \, dx =14.8[/tex]

c)

[tex]y=5x-x^{2} \\\frac{dy}{dx} =0\\5x-x^{2} =0\\x=5/2\\y(5/2)=25/4\\y=5x-x^{2} \\x^{2} -5x+y=0\\x=\frac{5+-\sqrt{25-4y} }{2}[/tex]

The cross section area is equal to:

[tex]A_{1} =\frac{1}{2} \pi r_{1}^{2} =\frac{1}{2} \pi (\frac{1}{2} (\frac{5+\sqrt{25-4y} }{2} -\frac{5-\sqrt{25-4y} }{2} ))^{2} =\frac{1}{8} \pi (25-4y)\\A_{2} =\frac{1}{2} \pi r_{2}^{2}=\frac{1}{2}\pi (\frac{1}{2} (y-\frac{5-\sqrt{25-4y} }{2} ))^{2} =\frac{1}{32} \pi (2y-5+\sqrt{25-4y} )^{2}[/tex]

The volume of S is equal to:

[tex]Vol_{S} =\int\limits^a_b {A_{1}(y) } \, dy+\int\limits^4_0 {A_{2}(y) } \, dy ,where-a=25/4,b=4\\Vol_{S} =\int\limits^a_b {\frac{1}{8}\pi (25-4y)} \, dy +\int\limits^a_b {\frac{1}{32}\pi (2y-5+\sqrt{25-4y} )^{2} } \, dy =5.17[/tex]

d) The cross section area is:

[tex]A_{1} =\frac{1}{2}ab=\frac{1}{2} a^{2} =\frac{1}{2} (\frac{5+\sqrt{25-4y} }{2}-\frac{5-\sqrt{25-4y}}{2} )^{2} =\frac{1}{2} (25-4y)\\A_{1}=\frac{1}{2}ab=\frac{1}{2} a^{2} =\frac{1}{2}(y-\frac{5-\sqrt{25-4y}}{2}} )^{2} =\frac{1}{8} (2y-5+\sqrt{25-4y}})^{2}[/tex]

The volume of S is equal to:

[tex]Vol_{S} =\int\limits^a_b {A_{1}(y) } \, dy +\int\limits^4_0 {A_{2}(y) } \, dy ,where-a=25/4,b=4\\Vol_{S}=\int\limits^a_b {\frac{1}{2}(25-4y) } \, dy +\int\limits^4_0 {\frac{1}{8}(2y-5+\sqrt{25-4y})^{2} } \, dy =11.33[/tex]

Answer:

T(n) = cn +(a-c)

Step-by-step explanation:

Note that T(1) = a, then T(2) = a+c, T(3) = (a+c)+c = a+2c, T(4) = (a+2c)+c = a+3c. Thus, our hypotheis is that T(n) = a+(n-1)c. We will prove this by strong induction.

Note that T(1) = a = a+(1-1)c. So the base case is proved. Assume that the result is true for all k<n. Then

T(n) = T(n-1)+c = (a+(n-2)c)+c = a+(n-1)c= cn+ (a-c).

So, by induction, the result holds.

Note that if a=1 and c = 3 then T(n) = 1+(n-1)3 = 3n-3+1 = 3n-2, which invalidates option b)

If a=3 and c=5 then we have that T(n) = 5n+(3-5) = 5n-2, which invalidates c) and d).

Then the formula is correct.

Answer:

(a) is correct

[tex]T(n) = a+(n-1)c[/tex]

Step-by-step explanation:

Notice that according to the information that you are given

[tex]T(1)=a \\T(2)=T(1)+c = a+c\\T(3)=T(2)+c = a+c+c = a+2c[/tex]

If you think about it there is a clear pattern, it would be

[tex]T(n) = a+(n-1)c[/tex]

Now notice that (a) is correct  if we set a=1 and c=3 we get

[tex]T(n) = 1+3(n-1) = 3n-2[/tex]

Answer: The answer is p = 343

Given that 'p' is inversely proportional to the square of 'q', we first found the constant of proportionality (k) by substituting the given 'p' and 'q' values. With 'k' known, we substituted the new value of 'q' to find the corresponding value of 'p', which turned out to be 343 when q=2.

The given question describes an inverse proportionality. Specifically, it states that p is inversely proportional to the square of q. To express this mathematically, we write it as p = k/(q^2), where k is the constant of proportionality. For finding this constant, we use the given values of p and q, so 28 = k/(7^2), which means k = 28*49 = 1372.

Now, we substitute the value of k and the new value of q into the equation to find the corresponding value of p. Hence, when q = 2, p = 1372/(2^2) = 1372/4 = 343. Therefore, when q = 2, p equals 343.

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Answer:

11/5 =x

Step-by-step explanation:

3(4 – 2x) = -x+1

Distribute

12 -6x = -x+1

Add 6x to each side

12 -6x+6x = 6x-x +1

12 = 5x +1

Subtract 1 from each side

12-1 = 5x+1-1

11= 5x

Divide each side by 5

11/5 =5x/5

11/5 =x

Steps to solve:

3(4 - 2x) = -x + 1

~Distribute

(3 * 4) + (3 * -2x) = -x + 1

~Simplify

12 - 6x = -x + 1

~Subtract 12 to both sides

12 - 12 - 6x = -x + 1 - 12

~Simplify

-6x = -x - 11

~Add x to both sides

-6x + x = -x + x - 11

~Simplify

-5x = -11

~Divide -5 to both sides

-5x/-5 = -11/-5

~Simplify

x = 11/5

Best of Luck!

Answer:

The answer is d (13.26)

Step-by-step explanation:

set it up like this: (95/360) times (2 times pi times 8)

after plugging this equation into a calculator you get 13.26450232 and round to 13.26

Final answer:

The arc length of a circle with an 8-inch radius and a central angle of 95 degrees is 13.26 inches when we use 3.14 for π and round to the nearest hundredth.

Explanation:

To find the arc length of a circle, we use the formula arc length (Δs) = rΘ, where 'r' is the radius and Θ is the central angle in radians. Since there are 2π radians in a full 360-degree rotation, we can find the radian measure of 95 degrees by using the conversion ratio π radians/180 degrees. The radian measure is (95/180)π.

Using 3.14 for π and the given radius of 8 inches, the calculation becomes: Δs = 8 * (95/180) * 3.14. Simplifying this equation gives the arc length as 13.26 inches when rounded to the nearest hundredth

Answer: The answer is y = 18(1.15)^x

Step-by-step explanation:

Answer:

y = (18) * (1.15)^x

Step-by-step explanation:

A limited edition poster increases in value each year with an initial value of $18. After 1 year and an increase of 15% per year, the poster is worth $20. 70. Which equation can be used to find the y value after x years.

To find this equation, this is an exponential equation, meaning that the number increases at a rapid rate. In this case the post increases each year by 15% and started at $18. The equation you would use to find this is: y = a * b^x. We can fill in the a and b values based off the given information. Since the value is increasing by 15% we will add 1 to 15% to get 1.15. This will be the b value. The a value is our initial value which in this case is $18. Now we can plug everything in to get: y = (18) * (1.15)^x.

Answer:

a

Step-by-step explanation:

The order of four steps are Assessing needs and wants, Considering opportunity costs, Assessing risks and returns and Setting short- and long-term goals, Option C is correct.

Finance is the study and discipline of money, currency and capital assets.

The correct order of the four steps in solving one's personal financial challenges is indeed:

Assessing needs and wants

Considering opportunity costs

Assessing risks and returns

Setting short- and long-term goals

Hence, the order of four steps are Assessing needs and wants, Considering opportunity costs, Assessing risks and returns and Setting short- and long-term goals, Option C is correct.

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Answer: These would be my two thoughts.

Step-by-step explanation:

ANSWER #1.    15-8= 7

ANSWER #2.    3-8= -5

Answer:

  x = 9

Step-by-step explanation:

The perimeter is the sum of the lengths of the base and the two equal legs:

  51 = x + 2(3x -6)

  51 = 7x -12 . . . . . eliminate parentheses

  63 = 7x . . . . . . . . add 12

  9 = x . . . . . . . . . . divide by 7

Answer:

The solution to the differential equation y'(y + 7)y'' = (y')²

y = Ae^(Kx) - 7

Step-by-step explanation:

Given the differential equation

y'(y + 7)y'' = (y')² ..................(1)

We want to solve using the substitution u = y'.

Let u = y'

The u' = y''

Using these, (1) becomes

u(y + 7)u' = u²

u' = u²/u(y + 7)

u' = u/(y + 7)

But u' = du/dy

So

du/dy = u/(y + 7)

Separating the variables, we have

du/u = dy/(y + 7)

Integrating both sides, we have

ln|u| = ln|y + 7| + ln|C|

u = e^(ln|y + 7| + ln|C|)

= K(y + 7)

But u = y' = dy/dx

dy/dx = K(y + 7)

Separating the variables, we have

dy/(y + 7) = Kdx

Integrating both sides

ln|y + 7| = Kx + C1

y + 7 = e^(Kx + C1) = Ae^(Kx)

y = Ae^(Kx) - 7

Final answer:

To solve the given differential equations by using the substitution u = y', substitute u for y' and find the values of u. Then, solve the resulting first order ordinary differential equation by separating variables and integrating to determine the solution.

Explanation:

To solve the given differential equations by using the substitution u = y', we need to substitute u for y' and find the values of u. Let's take the first equation as an example:

Start by substituting u for y' in the equation: (y + 7)y'' = (y')^2

Replace y' with u in the equation: (y + 7)u' = u^2

Then, we can solve this first order ordinary differential equation by separating variables and integrating:

Divide both sides by (y + 7): u' = (u^2) / (y + 7)

Separate the variables: (y + 7)dy = (u^2)du

Integrate both sides: (1/2)(y^2 + 14y) = (1/3)u^3 + C (where C is the constant of integration)

Solve for y by rearranging the equation: y^2 + 14y = (2/3)u^3 + 2C

This is the solution to the given differential equation.

Answer: B

Step-by-step explanation:

Can't really explain it. A tenth is .1, a hundredth is .01

Answer:

y = 1/2x -1

Step-by-step explanation:

to find the y-intercept you find where the line crosses the y axis

for the slope you need to use the equation

change of y2 - change of y1

change of x2 - change of x1

y=ax+b

(-2; -2); (0;-1); (2; 0)

-1=a*0+b => b=-1

0=a*2+b

0=a*2-1

a*2=1 => a=1/2

y=x/2 -1