Express the following English phrase using an algebraic expression. The quotient of some number y and 6 increased by the product of 2 and some number x.

Answer:

[tex]y = \frac{7}{10} e^{3(t - 1)} + \frac{3}{10}e^{-7(t - 1)}[/tex]

Step-by-step explanation:

y′′ + 4y′ − 21y = 0

The auxiliary equation is given by

m² + 4m - 21 = 0

We solve this using the quadratic formula. So

[tex]m = \frac{-4 +/- \sqrt{4^{2} - 4 X 1 X (-21))} }{2 X 1}\\ = \frac{-4 +/- \sqrt{16 + 84} }{2}\\= \frac{-4 +/- \sqrt{100} }{2}\\= \frac{-4 +/- 10 }{2}\\= -2 +/- 5\\= -2 + 5 or -2 -5\\= 3 or -7[/tex]

So, the solution of the equation is

[tex]y = Ae^{m_{1} t} + Be^{m_{2} t}[/tex]

where m₁ = 3 and m₂ = -7.

So,

[tex]y = Ae^{3t} + Be^{-7t}[/tex]

Also,

[tex]y' = 3Ae^{3t} - 7e^{-7t}[/tex]

Since y(1) = 1 and y'(1) = 0, we substitute them into the equations above. So,

[tex]y(1) = Ae^{3X1} + Be^{-7X1}\\1 = Ae^{3} + Be^{-7}\\Ae^{3} + Be^{-7} = 1 (1)[/tex]

[tex]y'(1) = 3Ae^{3X1} - 7Be^{-7X1}\\0 = 3Ae^{3} - 7Be^{-7}\\3Ae^{3} - 7Be^{-7} = 0 \\3Ae^{3} = 7Be^{-7}\\A = \frac{7}{3} Be^{-10}[/tex]

Substituting A into (1) above, we have

[tex]\frac{7}{3}B e^{-10}e^{3} + Be^{-7} = 1 \\\frac{7}{3}B e^{-7} + Be^{-7} = 1\\\frac{10}{3}B e^{-7} = 1\\B = \frac{3}{10} e^{7}[/tex]

Substituting B into A, we have

[tex]A = \frac{7}{3} \frac{3}{10} e^{7}e^{-10}\\A = \frac{7}{10} e^{-3}[/tex]

Substituting A and B into y, we have

[tex]y = Ae^{3t} + Be^{-7t}\\y = \frac{7}{10} e^{-3}e^{3t} + \frac{3}{10} e^{7}e^{-7t}\\y = \frac{7}{10} e^{3(t - 1)} + \frac{3}{10}e^{-7(t - 1)}[/tex]

So the solution to the differential equation is

[tex]y = \frac{7}{10} e^{3(t - 1)} + \frac{3}{10}e^{-7(t - 1)}[/tex]

Answer:

The line passing through the given points is:

[tex]y=-\frac{4}{3}x+\frac{1}{3}[/tex]

in its slope-intercept form

Step-by-step explanation:

Start by finding the slope of the segment that joins the two given points using the slope formula:

[tex]slope=\frac{y_2-y_1}{x_2-x_1}[/tex]

which for our case renders:

[tex]slope=\frac{3-(-5)}{-2-4}=\frac{8}{-6} =-\frac{4}{3}[/tex]

Now we can find the y-intercept by using any one of the given points in the general slope-intercept form of a line with this slope:

[tex]y=-\frac{4}{3} x+b\\3=-\frac{4}{3} (-2)+b\\3=\frac{8}{3}+b\\b=3-\frac{8}{3} \\b=\frac{9-8}{3} \\b=\frac{1}{3}[/tex]

Therefore, the equation of the line becomes:

[tex]y=-\frac{4}{3}x+\frac{1}{3}[/tex]

Answer:

Upstream speed= 6 mile per hour

Downstream speed= 9 miles per hour

Step-by-step explanation:

Distance traveled during upstream

= x miles

Distance traveled during downstream

= y

Time traveling upstream

= 45 minutes.

Time traveling downstream

= 30 minutes

Speed traveling upstream

= a miles/hour

Speed traveling downstream

= b miles/hour

But

b = a+3

Distance= speed*time

X=45a

a= x/45

Y= 30(a+3)

Y= 30a +90

a= (y-90)/30

Equating both a

X/45= (y-90)/30

30x= 45(y-90)

X= 1.5y-135

But x= y

Y= 1.5y-135

135= 0.5y

270 miles = y

Y= x= 270 miles

X= 45a

270/45= a

6 mile/hour = a

b = a+3

b= 6+3

b = 9 miles/hour

Answer:

In the long run, ou expect to  lose $4 per game

Step-by-step explanation:

Suppose we play the following game based on tosses of a fair coin. You pay me $10, and I agree to pay you $n^2 if heads comes up first on the nth toss.

Assuming X be the toss on which the first head appears.

then the geometric distribution of X is:

X [tex]\sim[/tex] geom(p = 1/2)

the probability function P can be computed as:

[tex]P (X = n) = p(1-p)^{n-1}[/tex]

where

n = 1,2,3 ...

If I agree to pay you $n^2 if heads comes up first on the nth toss.

this implies that , you need to be paid [tex]\sum \limits ^{n}_{i=1} n^2 P(X=n)[/tex]

[tex]\sum \limits ^{n}_{i=1} n^2 P(X=n) = E(X^2)[/tex]

[tex]\sum \limits ^{n}_{i=1} n^2 P(X=n) =Var (X) + [E(X)]^2[/tex]

[tex]\sum \limits ^{n}_{i=1} n^2 P(X=n) = \dfrac{1-p}{p^2}+(\dfrac{1}{p})^2[/tex]        ∵  X [tex]\sim[/tex] geom(p = 1/2)

[tex]\sum \limits ^{n}_{i=1} n^2 P(X=n) = \dfrac{1-p}{p^2}+\dfrac{1}{p^2}[/tex]

[tex]\sum \limits ^{n}_{i=1} n^2 P(X=n) = \dfrac{1-p+1}{p^2}[/tex]

[tex]\sum \limits ^{n}_{i=1} n^2 P(X=n) = \dfrac{2-p}{p^2}[/tex]

[tex]\sum \limits ^{n}_{i=1} n^2 P(X=n) = \dfrac{2-\dfrac{1}{2}}{(\dfrac{1}{2})^2}[/tex]

[tex]\sum \limits ^{n}_{i=1} n^2 P(X=n) =\dfrac{ \dfrac{4-1}{2} }{{\dfrac{1}{4}}}[/tex]

[tex]\sum \limits ^{n}_{i=1} n^2 P(X=n) =\dfrac{ \dfrac{3}{2} }{{\dfrac{1}{4}}}[/tex]

[tex]\sum \limits ^{n}_{i=1} n^2 P(X=n) =\dfrac{ 1.5}{{0.25}}[/tex]

[tex]\sum \limits ^{n}_{i=1} n^2 P(X=n) =6[/tex]

Given that during the game play, You pay me $10 , the calculated expected loss = $10 - $6

= $4

∴

In the long run, you expect to  lose $4 per game

Answer:

The formula of area is Pi×R×R

and then you add all sides and answer of your question is 30cm

Answer:

a = 28

Step-by-step explanation: First you would have to multiply 7 x 8 which equals =56. Then you would have to divide 56 by 2 which is =28 and that would be your area if you are solving for a triangle.

Answer:

A = 60 cm^2

Step-by-step explanation:

The area of a parallelogram is

A = bh

A = 14*5

A = 60 cm^2

[tex] area \: of \: parallelogram \: = \: base \times height[/tex]

[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: a \: \: \: \: \: \: \: \: \: \: \: \: = \: \: \: \: \: \: \: \: \: \: \: 14cm \times 5cm[/tex]

[tex]a \: \: \: \: \: \: \: \: \: \: \: \: \: = \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 70 {cm}^{2} [/tex]

You cannot divide any number by zero since it is meaningless to say that we are dividing a number or object in zero groups or in 0 size parts

Answer:

129

Step-by-step explanation:

120 +9 =129

because it is total and total means add

Answer:

203

Step-by-step explanation:

Square of 101 = 101^2= 10201

Square of 102 = 102 ^2=10404

No. of numbers between both of them =10404-10201=203

Answer:

203

Step-by-step explanation:

✓101 kare = 10.201

✓102 kare = 10.404

✓ 10.404- 10.201 = 203

#Başarılar

#Türkiye

Answer:

n + 3

Step-by-step explanation:

6 is 3 + 3, 11 is 8 + 3 and 12 is 9 + 3 so the output seems to be the input plus 3. Therefore, when the input is n, the output is n + 3.

Answer:

  output = n+3

Step-by-step explanation:

When the input changes by 5 units (from 3 to 8), the output changes by 5 unis (from 6 to 11). So the rate of change of the output with respect to the input is 1. However, each output is 3 more than the input, so ...

for input n

output = n+3

Answer:

the SA is 40

Step-by-step explanation:

the formula for SA of a rectangular prism is 2(lw + lh + wh)

so it would be: 2(2x2+2x2+4x2)

can i get brainliest please? i only need two more

Answer:

The cordinates are:

y = 3

x = 4

Step-by-step explanation:

If you draw a line in the middle of the triangle

It Will go through the base at point V

point V can be marked at y=3 and x=4 the line drawn will intersect 0

Answer:

on [3, 4] = 0.30

on [4, 5] = 0.18

on [5, 6] = 0.12

Step-by-step explanation:

The average rate of change f, of a function f(x) on an interval [a, b] is given by;

[tex]f = \frac{f(b) - f(a)}{b - a}[/tex]              -------------(i)

In our case,

f(x) = log 2(3x - 6)

Now let's get the average rate of change of f(x) on;

(i) [3, 4]

Here, a = 3 and b = 4

f(a) = f(3)        [This is f(x) at x = 3]

=> f(3) = log[2(3(3) - 6)]

=> f(3) = log[2(9 - 6)]

=> f(3) = log[2(3)]

=> f(3) = log[6]

Also,

f(b) = f(4)        [This is f(x) at x = 4]

=> f(4) = log[2(3(4) - 6)]

=> f(4) = log[2(12 - 6)]

=> f(4) = log[2(6)]

=> f(4) = log[12]

Now substitute the values of a, b, f(a) and f(b) into equation (i) as follows;

[tex]f = \frac{log 12 - log 6}{4 - 3}[/tex]             [Remember that log m - log n = log (m / n)]

[tex]f = \frac{log (12 / 6)}{4 - 3}[/tex]

[tex]f = \frac{log (2)}{1}[/tex]

f = log 2 = 0.3010

f = 0.30          [to two decimal places]

∴ The average rate of change on [3, 4] = 0.30

(ii) [4, 5]

Here, a = 4 and b = 5

f(a) = f(4)        [This is f(x) at x = 4]

=> f(4) = log[2(3(4) - 6)]

=> f(4) = log[2(12 - 6)]

=> f(4) = log[2(6)]

=> f(4) = log[12]

Also,

f(b) = f(5)        [This is f(x) at x = 5]

=> f(5) = log[2(3(5) - 6)]

=> f(5) = log[2(15 - 6)]

=> f(5) = log[2(9)]

=> f(5) = log[18]

Now substitute the values of a, b, f(a) and f(b) into equation (i) as follows;

[tex]f = \frac{log 18 - log 12}{5 - 4}[/tex]             [Remember that log m - log n = log (m / n)]

[tex]f = \frac{log (18 / 12)}{5 - 4}[/tex]

[tex]f = \frac{log (1.5)}{1}[/tex]

f = log 1.5 = 0.176

f = 0.18          [to two decimal places]

∴ The average rate of change on [4, 5] = 0.18

(iii) [5, 6]

Here, a = 5 and b = 6

f(a) = f(5)        [This is f(x) at x = 5]

=> f(5) = log[2(3(5) - 6)]

=> f(5) = log[2(15 - 6)]

=> f(5) = log[2(9)]

=> f(5) = log[18]

Also,

f(b) = f(6)        [This is f(x) at x = 6]

=> f(6) = log[2(3(6) - 6)]

=> f(6) = log[2(18 - 6)]

=> f(6) = log[2(12)]

=> f(6) = log[24]

Now substitute the values of a, b, f(a) and f(b) into equation (i) as follows;

[tex]f = \frac{log 24 - log 18}{6 - 5}[/tex]             [Remember that log m - log n = log (m / n)]

[tex]f = \frac{log (24 / 18)}{6 - 5}[/tex]

[tex]f = \frac{log (1.33)}{1}[/tex]

f = log 1.33 = 0.124

f = 0.12         [to two decimal places]

∴ The average rate of change on [5, 6] = 0.12

Answer:CORRECT ANSWER ON PLATO

[3,4]= 1

[4,5]=0.59

[5,6]= 0.41

Answer:=6x3−9c

Step-by-step explanation:

3x3+4x2+3x3−4x2−9c

=3x3+4x2+3x3+−4x2+−9c

Combine Like Terms:

=3x3+4x2+3x3+−4x2+−9c

=(3x3+3x3)+(4x2+−4x2)+(−9c)

=6x3+−9c

Answer:

The answer is 729

Step-by-step explanation:

to get the next term, the sequence multiples the previous number by -3

1 * -3 = -3

-3 * -3 = 9

and so on.

To get the 7th term:

-27 * -3 = 81 (5th term)

81 * -3 = -243 (6th)

-243 * -3 = 729 (7th)

Answer:

The seventh term in the sequence is -729.

Step-by-step explanation:

Notice that the sequence is not increasing linearly. Therefore, this is a geometric sequence.

Recall that the explicit formula for a geometric sequence is given by:

[tex]x_n=a(r)^{n-1}[/tex]

Where a is the first term, r is the common ratio, and n denotes the nth term.

From the sequence, we can see that our first term a is -1.

Because each term is thrice the previous, our common ratio r is 3.

By substitution:

[tex]x_n=-(3)^{n-1}[/tex]

Hence, the seventh term is:

[tex]\displaystyle \begin{aligned} x_7 & = -(3)^{(7)-1} \\ \\ & = -(3)^{6} \\ \\ & = -729\end{aligned}[/tex]

In conclusion, the seventh term is -729.

Answer:

see below (I hope this helps!)

Step-by-step explanation:

∠VZY and ∠VXW form a linear pair and are supplementary, and since m∠VZY = 90°, m∠VXW = 180 - 90 = 90°. Since m∠WXU + m∠UXV = ∠VXW, we can write the following equation:

44 + 5x - 4 = 90

5x + 40 = 90

5x = 50

x = 10°

Therefore, m∠UXV = 5 * 10 - 4 = 46°

Answer:

x = 10 m(UXV)=46°

Step-by-step explanation:

m(UXV) + m(UXW) = 90° because m(VXY)= 90°

m(UXW) = 44°

m(UXV) + 44° = 90°

m(UXV) = 46°

5x - 4 = 46°

5x = 50

5x/5 = 50/5

x = 10

Hope this helps ^-^

Answer:

The ratio will be 48/109

Step-by-step explanation:

Answer:2

Step-by-step explanation: 15x5=75

75/26=2.886

Answer:

0.200

Step-by-step explanation:

∑ₓ₌₁⁴⁰ (1 + i)⁻ˣ = 5

∑ₓ₌₁⁴⁰ ((1 + i)⁻¹)ˣ = 5

This is a geometric series.  The sum of the first n terms of a geometric series is:

S = a₁ (1 − rⁿ) / (1 − r)

where a₁ is the first term and r is the common ratio.

Here, a₁ = (1 + i)⁻¹, r = (1 + i)⁻¹, and n = 40.  For simplicity, let's write both a₁ and r in terms of r.  So a₁ = r and r = r.

5 = r (1 − r⁴⁰) / (1 − r)

5 (1 − r) = r (1 − r⁴⁰)

5 − 5r = r − r⁴¹

r⁴¹ − 6r + 5 = 0

Solving with a calculator, r ≈ 0.8334.

Therefore:

(1 + i)⁻¹ = 0.8334

1 + i = 1.200

i = 0.200

Answer:

3

Step-by-step explanation:

As per table, the mass is:

So the atom 3 has the highest mas of 132

Answer choice 3 is correct

Answer:

3

Step-by-step explanation:

Add protons and neutrons.

Answer:

Distribute the 3

Step-by-step explanation:

order of operations mate

Answer:

Step-by-step explanation:

From the given information:

The present value of the house = 128000

interest rate compounded monthly r = 7.8% = 0.078

number of months in a year n= 12

duration of time t = 30 years

To find their regular monthly payment, we have:

[tex]PV = P \begin {bmatrix} \dfrac{1 - (1 + \dfrac{r}{n})^{-nt}}{\dfrac{r}{n}} \end {bmatrix}[/tex]

[tex]128000 = P \begin {bmatrix} \dfrac{1 - (1 + \dfrac{0.078}{12})^{- 12*30}}{\dfrac{0.078}{12}} \end {bmatrix}[/tex]

128000 = 138.914 P

P = 128000/138.914

P = $921.433

∴ Their regular monthly payment P = $921.433

To find the unpaid balance when they begin paying the $1400.

when they begin the payment ,

t = 30 year - 5years

t= 25 years

[tex]PV= 921.433 \begin {bmatrix} \dfrac{1 - (1 - \dfrac{0.078}{12})^{25*30}}{\dfrac{0.078}{12}} \end {bmatrix}[/tex]

PV = $121718.2714

C) In order to estimate how many payments of $1400  it will take to pay off the loan, we have:

[tex]121718.2714 = \begin {bmatrix} \dfrac{1300 (1 - \dfrac{12.078}{12}))^{-nt}}{\dfrac{0.078}{12}} \end {bmatrix}[/tex]

[tex]121718.2714 = 200000 \begin {bmatrix} (1 - \dfrac{12.078}{12}))^{-nt} \end {bmatrix}[/tex]

[tex]\dfrac{121718.2714}{200000 } = \begin {bmatrix} (1 - \dfrac{12.078}{12}))^{-nt} \end {bmatrix}[/tex]

[tex]0.60859 = \begin {bmatrix} (1 - \dfrac{12}{12.078}))^{nt} \end {bmatrix}[/tex]

[tex]0.60859 = (0.006458)^{nt}[/tex]

[tex]nt = \dfrac{0.60859}{0.006458}[/tex]

nt = 94.238 payments is required to pay off the loan.

How much interest will they save by paying the loan using the number of payments from part (c)?

The total amount of interest payed on $921.433 = 921.433 × 30(12) years

= 331715.88

The total amount paid using 921.433 and 1300 = (921.433 × 60 )+( 1300 + 94.238)

= 177795.38

The amount of interest saved = 331715.88  - 177795.38

The amount of interest saved = $153920.5