Use binomial expansion to solve each exercise. (a) Find the coefficient of xy19 in (3x − 2y) 20.(b) Give a formula for the coefficient of x k in (x+1/ x) n .

Answer:

-3 < x ≤ 6

Step-by-step explanation:

Left side is exclusive, right side is inclusive, hence the < and ≤, respectively.

I'll do the first problem to get you started.

Part (a)

We have a separable equation. Get the y term to the left side and then integrate to get

[tex]\frac{dy}{dt} = ky^{1+c}\\\\\frac{dy}{y^{1+c}} = kdt\\\\\displaystyle \int\frac{dy}{y^{1+c}} = \int kdt\\\\\displaystyle \int y^{-(1+c)}dy = \int kdt\\\\\displaystyle -\frac{1}{c}y^{-c} = kt+D\\\\\displaystyle -\frac{1}{c*y^{c}} = kt+D\\\\[/tex]

I'm using D as the integration constant rather than C since lowercase letter c was already taken.

Let's use initial condition that [tex]y(0) = y_0[/tex]. This means we'll plug in t = 0 and [tex]y = y_0[/tex]. After doing so, solve for D

[tex]\displaystyle -\frac{1}{c*y^{c}} = kt+D\\\\\displaystyle -\frac{1}{c*(y_0)^{c}} = k*0+D\\\\\displaystyle D = -\frac{1}{c*(y_0)^{c}}\\\\[/tex]

Let's plug that in and isolate y

[tex]\diplaystyle -\frac{1}{c}y^{-c} = kt+D\\\\\\\diplaystyle -\frac{1}{c}y^{-c} = kt-\frac{1}{c*(y_0)^{c}}\\\\\\\diplaystyle y^{-c} = -ckt+\frac{1}{(y_0)^{c}}\\\\\\\diplaystyle y^{-c} = \frac{1-c*(y_0)^{c}kt}{(y_0)^{c}}\\\\\\\diplaystyle \frac{1}{y^{c}} = \frac{1-c*(y_0)^{c}kt}{(y_0)^{c}}\\\\\\\diplaystyle y^{c} = \frac{(y_0)^{c}}{1-c*(y_0)^{c}kt}\\\\\\\diplaystyle y = \left(\frac{(y_0)^{c}}{1-c*(y_0)^{c}kt}\right)^{1/c}\\\\\\\diplaystyle y = \frac{y_0}{\left(1-c*(y_0)^{c}kt\right)^{1/c}}\\\\\\[/tex]

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

We end up with [tex]\displaystyle y(t) = \frac{y_0}{\left(1-c*(y_0)^{c}kt\right)^{1/c}}\\\\[/tex] as our final solution. There are likely other forms to express this equation.

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Part (b)

We want y(t) to approach positive infinity.

Based on the solution in part (a), this will happen when the denominator approaches 0 from the left.

So [tex]y(t) \to \infty[/tex] as [tex]1-c*(y_0)^{c}kt \to 0[/tex] in which we can effectively "solve" for t showing that [tex]t \to \frac{1}{c*(y_0)^{c}k}[/tex]

If we define [tex]T = \frac{1}{c*(y_0)^{c}k}[/tex] , then approaching T from the left side will have y(t) approach positive infinity.

This uppercase T value is doomsday. This the time value lowercase t approaches from the left when the population y(t) explodes to positive infinity.

Effectively t = T is the vertical asymptote.

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Part (c)

We're told that the initial condition is y(0) = 5 since at time 0, we have 5 rabbits. This means [tex]y_0 = 5[/tex]

Another fact we know is that y(3) = 35 because after three months, there are 35 rabbits.

Lastly, we know that c = 0.01 since the exponent of dy/dt = ky^(1.01) is 1.01; so we solve 1+c = 1.01 to get c = 0.01

We'll use y(3) = 35, c = 0.01 and [tex]y_0 = 5[/tex] to solve for k

Doing so leads to the following:

[tex]\displaystyle y(t) = \frac{y_0}{\left(1-c*(y_0)^{c}kt\right)^{1/c}}\\\\\\\displaystyle y(3) = \frac{5}{\left(1-0.01*(5)^{0.01}k*3\right)^{1/0.01}}\\\\\\\displaystyle 35 \approx \frac{5}{\left(1-0.0304867k\right)^{100}}\\\\\\\displaystyle 35\left(1-0.0304867k\right)^{100} \approx 5\\\\\\\displaystyle \left(1-0.0304867k\right)^{100} \approx \frac{1}{7}\\\\\\[/tex]

[tex]\displaystyle \left(1-0.0304867k\right)^{100} \approx 7^{-1}\\\\\\\displaystyle 1-0.0304867k \approx \left(7^{-1}\right)^{1/100}\\\\\\\displaystyle 1-0.0304867k \approx 7^{-0.01}\\\\\\\displaystyle k \approx \frac{7^{-0.01}-1}{-0.0304867}\\\\\\\displaystyle k \approx 0.63211155281122\\\\\\\displaystyle k \approx 0.632112\\\\\\[/tex]

We can now compute the doomsday time value

[tex]T = \frac{1}{c*(y_0)^c*k}\\\\\\T \approx \frac{1}{0.01*(5)^{0.01}*0.632112}\\\\\\T \approx \frac{1}{0.00642367758836}\\\\\\T \approx 155.674064621806\\\\\\T \approx 155.67\\\\\\[/tex]

The answer is approximately 155.67 months

Answer:

$3.50

Step-by-step explanation:

5.25/3 = 1.75

1.75 x 2 = 3.5

Answer:

7/25

Step-by-step explanation:

From randomly guessing, it depends on how many you think you got right and how many you really don't know. You could get all 25 wrong, but more likely at random get 7 right.

Answer:

The minimum score you can get is 0.

Step-by-step explanation:

Answer:

<M = 50

Step-by-step explanation:

The sum of the angles of a triangle is 180

52+ 78+x = 180

Combine like terms

130 +x = 180

Subtract 130 from each side

130+x-130 =180-130

x = 50

Answer:

50°

Step-by-step explanation:

180 is the total of all the angles added together.

so if you subtract the 2 you already know from the total, you’ll have the size of the angle M.

180 - (52+78) = 180 - 130 = 50°

Answer:

[tex]y'\approx -0.41[/tex]

Step-by-step explanation:

Implicit Derivatives

When it's not possible to express one variable as an explicit function of the other, we use implicit derivatives and solve for y'.

Find y' in the equation given below:

xe^y - 4y = 3x - 14 + 2e^2

Taking derivatives with respect to x, recalling y'=dy/dx, and dx/dx=1:

(xe^y)' - (4y)' = (3x)' - (14 + 2e^2 )'

Using the product rule for the first derivative, and simple rules for the rest:

e^y + xe^yy' - 4y' = 3 - 0

Recall the derivative of a constant is zero.

Group terms with y' in the left side and the rest in the right side:

xe^yy' - 4y' = 3 - e^y

Factoring y':

y'(xe^y - 4) = 3 - e^y

Solving:

[tex]\displaystyle y'=\frac{3 - e^y}{xe^y - 4}[/tex]

Evaluating for x=2, y=2:

[tex]\displaystyle y'=\frac{3 - e^2}{2e^2 - 4}[/tex]

Calculating:

[tex]\mathbf{y'\approx -0.41}[/tex]

Answer:

4:5 is the ratio in it simplest form

Answer:

Step-by-step explanation:

Reflection of points across y-axis doesn't change the distance between the points so AB = A'B' = 10 units

Let the coordinates be:

Reflection:

Lets compare distances:

and

As we see the results are same

Answer:

Reflection of points across y-axis doesn't change the distance between the points so AB = A'B' = 10 units

Step-by-step explanation:

Answer:

155

Step-by-step explanation:

Answer:

D

Step-by-step explanation:

I took the quiz

Answer: X =5/2, 15/2   x=2.5, 7.5

Step-by-step explanation:

Answer:

(10x−7)^2

Step-by-step explanation:

(10x+7)(10x-7)

i stead of making it a "100" since they are both the same  but positive and negative () and bring it to the power of 2

Answer:

5x − 4 = 3x + 8

5x - 4 + 4 = 3x + 8 + 4

5x = 3x + 12

5x - 3x = 3x + 12 - 3x

2x = 12

2x / 2 = 12 / 2

x = 6

Answer:

x=6

Step-by-step explanation:

Let's solve your equation step-by-step.

5x−4=3x+8

Step 1: Subtract 3x from both sides.

5x−4−3x=3x+8−3x

2x−4=8

Step 2: Add 4 to both sides.

2x−4+4=8+4

2x=12

Step 3: Divide both sides by 2.

2x/2=12/2

x=6

Answer:

x=6

Answer:

yes they are congruent

Step-by-step explanation:

they both the same shape and size

Answer: x=-9

Step-by-step explanation:

18/x=-2

multiply x

18=-2x

divide by -2

-9=x

x=-9

Answer:

[tex]18 \div x = - 2 \\ \frac{18}{x} = - 2 \\ x = \frac{18}{ (- 2)} \\ \boxed{x = - 9}[/tex]

Answer:

The minimum sample required = 1296.65

Step-by-step explanation:

Given that:

Variance [tex]\sigma^2 = 2.89[/tex]

Standard deviation [tex]\sigma = \sqrt{2.89}[/tex]

Standard deviation [tex]\sigma = 1.7[/tex]

Margin of error = 0.11

Confidence Interval = 98%

Level of significance =  1 - 0.98 = 0.02

The critical value = [tex]Z _{\alpha//2} = Z_{0.02/2} = Z_{0.01}[/tex]

= 2.33  

Thus, the minimum sample size is given by the formula:

[tex]n = \bigg ( \dfrac{Z_{\alpha/2} \times \sigma }{E} \bigg)^2[/tex]

[tex]n = \bigg (\dfrac{2.33 \times 1.7 }{0.11} \bigg)^2[/tex]

n = 1296.65

Answer:

No, every student will have 3.2 minutes

Answer:

Each student has approximately 3 minutes and 12 seconds.

Step-by-step explanation:

You divide the total amount of minutes by the total amount of students. 48/15=3.2

After you get your answer, you convert to minutes. There are 3 minutes and the .2 converts to 12 seconds

Answer:

616

Step-by-step explanation:

Answer:

Hmm graph 3, i think.

Step-by-step explanation:

Answer:

I think it is 5.

Step-by-step explanation:

5/8 in decimal form is 0.625. 0.625 x 8 = 5. I hope this helps/ sorry if it didn't.

Answer:

x=8 and y=21

Step-by-step explanation:

12x-28=7x+12 (the 2 angles are equivalent)

12x-7x=12+28

5x=40

x=8

9y-77 + 12x-28 =180 (the sum of those 2 angles is a flat angle (180°))

9y-77+12*8-28=180

9y=180+28+77-96

9y=189

y=21

Answer:

25%

Step-by-step explanation:

25% of 60 = 15

Answer:

Brittany needs another $3.7405.              

Step-by-step explanation:

Per pound Cost of turkey = $5.96 per pound

The amount Brittany buys the turkey = 2.55 pounds

Brittany's cost for turkey = 2.55 × $5.96 = $15.198

Per pound cost for cheese = $3.35 per pound

The amount Brittany buys the cheese = 3.7 pounds

Brittany's cost for cheese = 2.55 × $3.35 = $8.5425

So,

Brittany's total cost = Turkey cost + Cheese cost

                                = $15.198 + $8.5425

                                = $23.7405

As brittany gave the clerk 20 dollars.

So, the amount she further needs will be:

$23.7405 - $20 =  $3.7405

Therefore, Brittany needs another $3.7405.                                                                      

4/9 and 1/11

product means multiplying

4 times 1

4*1=4

9 times 11

9*11=99

answer: 4/99

Answer:

A = 50 square units

Step-by-step explanation:

Right Triangles

A right triangle is identified because it has one internal angle of 90°.

The longest side is called hypotenuse and the other two sides are called legs. Being c the hypotenuse and a and b the legs, the Pythagora's theorem relates the with the equation:

[tex]c^2=a^2+b^2[/tex]

If the triangle is also isosceles, then both legs have the same measure or a=b:

[tex]c^2=a^2+a^2=2a^2[/tex]

Since we know the hypotenuse has a measure of 10\sqrt{2}:

[tex](10\sqrt{2})^2=2a^2[/tex]

Operating:

[tex]100*2=2a^2[/tex]

Dividing by 2:

[tex]a^2=100~~\Rightarrow a=\sqrt{100}[/tex]

a = 10 units

The area of the triangle is:

[tex]\displaystyle A=\frac{a.b}{2}[/tex]

[tex]\displaystyle A=\frac{10*10}{2}[/tex]

A = 50 square units

Answer:

Since the instruction in the question indicates that the % symbol should not be used, the weighted average cost of capital (WACC) for Easy Car Corp is therefore 12.07.

Step-by-step explanation:

This can be calculated using the following steps:

Step 1: Calculation of the current bond price

Semiannual coupon amount = Bond face value * Semiannual coupon rate = $1000 * (10% / 2) = $50

Semiannual coupon discount factor = ((1 - (1 / (1 + r))^n) / r) .......... (1)

Where;

r = Semiannual yield to maturity (YTM) = 8% / 2 = 0.08 / 2 = 0.04

n = number of semiannuals = 23 years * 2 = 46

Substituting the values into equation (1), we have:

Semiannual coupon discount factor = ((1-(1/(1 + 0.04))^46) / 0.04) = 20.8846535613106

Present value of coupon = (Semiannual coupon amount * Semiannual coupon discount factor) = $50 * 20.8846535613106 = $1,044.23

Present value of the face value of the bond = Face value / (1 + r)^n = $1,000 / (1 + 0.04)^46 = $164.61

Therefore, we have:

Current bond price = Present value coupon + Present value of the face value of the bond = $1,044.23 + $164.61 = $1,208.84

Step 2: Calculation of weights of each finance source

Market value of common shares outstanding = Number common shares outstanding * Current price per share = 2,000,000 * $13 = $26,000,000.

Market value of bond = Number of bonds * Current bond price = 30,000 * $1,208.84 = $36,265,200

Total financing market value = Market value of common shares outstanding + Market value of bond = $26,000,000 + $36,265,200 = $62,265,200

Weight of Market value of common shares outstanding = Market value of common shares outstanding / Total financing market value = $26,000,000 / $62,265,200 = 0.42

Weight of Market value of bond = Market value of bond / Total financing market value = $36,265,200 / $62,265,200 = 0.58

Step 3: Calculation of return on equity

Current year dividend = Last year dividend * (1 + Dividend growth rate) = $2 * (1 + 4.0%) = $2.08

Next year dividend = Current year dividend * (1 + Dividend growth rate) = $2.08 * (1 + 4.0%) = $2.1632

The return on equity can now be calculated using the following formula:

Current share price = Next year dividend / (Return on equity – Dividend growth rate) ………………….. (2)

Where;

Current share price = $13

Next year dividend = $2.1632

Return on equity = ?

Dividend growth rate = 4.0%, or 0.04

Substituting the values into equation (2) and solve for return on equity, we have:

13 = 2.1632 / (Return on equity - 0.04)

13 * (Return on equity - 0.04) = 2.1632

(13 * Return on equity) – (13 * 0.04) = 2.1632

(13 * Return on equity) – 0.52 = 2.1632

13 * Return on equity = 2.1632 + 0.52

Return on equity = 2.6832 / 13

Return on equity = 0.21

Step 4: Calculation of Weighted average cost of capital

Weighted average cost of capital = (WS * CE) + (WD * CD * (1 – T)) ………………… (4)

Where;

WS = Weight of Market value of common shares outstanding = 0.42

WD = Weight of debt = Weight of Market value of bond = 0.58

CE = Cost of equity = Return on equity = 0.21

CD = Cost of debt = YTM = 8%, or 0.08

T = Tax rate = 30%, or 0.30

Substituting the values into equation (3), we have:

Weighted average cost of capital = (0.42 * 0.21) + (0.58 * 0.08 * (1 - 0.30)) = 0.12068, or 12.068%

Rounding to two decimal places, we have:

Weighted average cost of capital = 12.07%

Since the instruction in the question indicates that the % symbol should not be used, the weighted average cost of capital (WACC) for Easy Car Corp is therefore 12.07.

Answer:

a = 18

b = 6√3

Step-by-step explanation:

WE can see in the diagram that there are two right angled triangles formed.

One with 45° angle as other interior angle and other with 60° interior angle.

So we will take the triangles one by one to find the required values.

As a is used in both triangles, first we will find the value of a using the left triangle.

In the triangle,

Hypotenuse = h = 18√2

θ = 45°

Using trigonometric ratio:

[tex]sin\ 45 =\frac{Perpendicular}{Hypotenuse}\\\frac{1}{\sqrt{2}}= \frac{a}{18\sqrt{2}}\\a = \frac{1}{\sqrt{2}} * 18\sqrt{2}\\a = 18[/tex]

Now in the right side triangle

θ1 = 60°

Perpendicular = a = 18

Base = b = ?

So,

[tex]tan\ 60 = \frac{perpendicular}{base}\\\sqrt{3} = \frac{\sqrt{18}}{b}\\b = \frac{18}{\sqrt{3}}\\b = \frac{6*3}{\sqrt{3}}\\b = \frac{6*\sqrt{3}*\sqrt{3}}{\sqrt{3}}\\b = 6\sqrt{3}[/tex]

Hence,

a = 18

b = 6√3

Answer:

c= -20

x= 38

Explanation:

Solve for x and c by simplifying both sides of the equation, then isolating the variable.  

hope this helps!