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Find the general solution of the following equation: y'(t) = 3y -5

Answers

Answer 1

Answer:

Answer:

The general solution of the equation is y = $(A)/(3)e^(3t)$ + 5

Step-by-step explanation:

Since the differential equation is given as y'(t) = 3y -5

The differential equation is re-written as

dy/dt = 3y - 5

separating the variables, we have

dy/(3y - 5) = dt

integrating both sides, we have

∫dy/(3y - 5) = ∫dt

∫3dy/[3(3y - 5)] = ∫dt

(1/3)∫3dy/(3y - 5) = ∫dt

(1/3)㏑(3y - 5) = t + C

㏑(3y - 5) = 3t + 3C

taking exponents of both sides, we have

exp[㏑(3y - 5)] = exp(3t + 3C)

3y - 5 = $e^(3t)e^(3C)$

3y - 5 = $Ae^(3t)$ $A = e^(3C)$

3y = $Ae^(3t)$ + 5

dividing through by 3, we have

y = $(A)/(3)e^(3t)$ + 5

So, the general solution of the equation is y = $(A)/(3)e^(3t)$ + 5

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Answers

$(2 \sin ^(2) \alpha-1)/(\sin \alpha+\cos \alpha) = sin \alpha - cos \alpha$

Solution:

Given that we have to simplify:

$(2 \sin ^(2) \alpha-1)/(\sin \alpha+\cos \alpha)$ ---- eqn 1

We know that,

$sin^2 x = 1 - cos^2 x$

Substitute the above identity in eqn 1

$(2\left(1-\cos ^(2) \alpha\right)-1)/(\sin \alpha+\cos \alpha)$

Simplify the above expression

$(2-2 \cos ^(2) \alpha-1)/(\sin \alpha+\cos \alpha)$

$(1-2 \cos ^(2) \alpha)/(\sin \alpha+\cos \alpha)$ ------- eqn 2

By the trignometric identity,

$(sin x + cos x)(sin x - cos x) = 1-2cos^2 x$

Substitute the above identity in eqn 2

$((\sin \alpha+\cos \alpha)(\sin \alpha-\cos \alpha))/(\sin \alpha+\cos \alpha)$

Cancel the common factors in numerator and denominator

$((\sin \alpha+\cos \alpha)(\sin \alpha-\cos \alpha))/(\sin \alpha+\cos \alpha)=\sin \alpha-\cos \alpha$

Thus the simplified expression is:

$(2 \sin ^(2) \alpha-1)/(\sin \alpha+\cos \alpha) = sin \alpha - cos \alpha$

Use the exponential growth model, A = A0 e^kt to show that the time is takes a population to double (to frow from A0 to 2 A0) is given by t = ln 2/k.

Answers

Answer:

Proof below

Step-by-step explanation:

Exponential Grow Model

The equation to model some time dependant event as an exponential is

$A=A_oe^(kt)$

Where Ao is the initial value, k is a constant and t is the time. With the value of Ao and k, we can compute the value of A for any time

We are required to find the time when the population being modeled doubles from Ao to 2 Ao. We need to solve the equation

$2A_o=A_oe^(kt)$

Simplifying by Ao

$2=e^(kt)$

Taking logarithms in both sides

$ln2=lne^(kt)$

By properties of logarithms and since lne=1

$ln2=kt\cdot lne=kt$

Solving for t

$\displaystyle t=(ln2)/(k)$

Hence proven

2.) What amount presently must be invested earning 5.25% compounded continuouslyso that it will grow up to be worth $25,000 12 years from now?

Answers

The amount A resulting from a principal amount P being invested at rate r compounded continuously for time t is given by

... A = P·e^(rt)

FIll in your given values and solve for P.

... 25000 = P·e^(0.0525·12) = P·e^0.63

... P = 25000/e^0.63 ≈ 13314.80 . . . . . divide by the coefficient of P

The amount that must be invested is $13,314.80.

An initial investment (P) compounded continuously with a rate of interest (r) in time (t) will grow to amount (Q) is given by:

Q = P * e^(rt)

Q=25000, r=0.0525, t=12

25000 = P * e^(0.0525*12)

1.8776P = 25000

P = 13314.8

Choices :
A. 40.1 meters
B. 12.3 meters
C. 19.2 meters
D. 47 meters

Answers

Answer:

25^2=16^2 +d^2

d^2=625-256

d=19.2

option C

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The formula for the volume of a cylinder is V 7r²h, where r is the radius of the cylinder and his the
height of the cylinder. Rewrite the formula to solve for h in terms of r and V.
G

Answers

Final answer:

To solve for the height of a cylinder in terms of its radius and volume, rewrite the formula as h = V/(7r²).

Explanation:

To rewrite the formula to solve for h in terms of r and V, we need to isolate h on one side of the equation. The given formula is V = 7r²h. To isolate h, we divide both sides of the equation by 7r². This gives us the formula: h = V/(7r²). This formula allows us to calculate the height of a cylinder given its radius and volume.

Learn more about Volume of a cylinder here:

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Point R is on line segment QS. Given RS = 2 and QS = 10, determine the length QR

Answers

Answer: QR=12

Step-by-step explanation:

10+2=12

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