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A large corporation starts at time t = 0 to invest part of its receipts continuously at a rate of P dollars per year in a fund f

Andrews [41]

Answer:

$A = \frac{P}{r}\left( e^{rt} -1 \right)$

Step-by-step explanation:

This is <em>a separable differential equation</em>. Rearranging terms in the equation gives

$\frac{dA}{rA+P} = dt$

Integration on both sides gives

$\int \frac{dA}{rA+P} = \int dt$

where $c$ is a constant of integration.

The steps for solving the integral on the right hand side are presented below.

$\int \frac{dA}{rA+P} = \begin{vmatrix} rA+P = m \implies rdA = dm\end{vmatrix} \\\\\phantom{\int \frac{dA}{rA+P} } = \int \frac{1}{m} \frac{1}{r} \, dm \\\\\phantom{\int \frac{dA}{rA+P} } = \frac{1}{r} \int \frac{1}{m} \, dm\\\\\phantom{\int \frac{dA}{rA+P} } = \frac{1}{r} \ln |m| + c \\\\&\phantom{\int \frac{dA}{rA+P} } = \frac{1}{r} \ln |rA+P| +c$

Therefore,

$\frac{1}{r} \ln |rA+P| = t+c$

Multiply both sides by $r.$

$\ln |rA+P| = rt+c_1, \quad c_1 := rc$

By taking exponents, we obtain

$e^{\ln |rA+P|} = e^{rt+c_1} \implies |rA+P| = e^{rt} \cdot e^{c_1} rA+P = Ce^{rt}, \quad C:= \pm e^{c_1}$

Isolate $A$ .

$rA+P = Ce^{rt} \implies rA = Ce^{rt} - P \implies A = \frac{C}{r}e^{rt} - \frac{P}{r}$

Since $A = 0$ when $t=0$ , we obtain an initial condition $A(0) = 0$ .

We can use it to find the numeric value of the constant $c$ .

Substituting $0$ for $A$ and $t$ in the equation gives

$0 = \frac{C}{r}e^{0} - \frac{P}{r} \implies \frac{P}{r} = \frac{C}{r} \implies C=P$

Therefore, the solution of the given differential equation is

$A = \frac{P}{r}e^{rt} - \frac{P}{r} = \frac{P}{r}\left( e^{rt} -1 \right)$

4 0

3 years ago

A triangle has an area of 357 square inches. The height of the triangle is 23.8 inches. What is the length of the base of the tr

Nitella [24]

You lay out the triangle formula for area which would be A=(1/2)(b)(h)
plug in for variables
375=(1/2)(23.8)(b)
solve for b

3 0

2 years ago

Please help this is for a friend with co vid- 19

postnew [5]

Answer:

am not sure but

Step-by-step explanation:

since they are similar they must have similar ratio

so 12/4 =3/1

so 3:1

4*3=12

for other sides I think

9*3 =27

and

6*3=18

y=27

x=18

5 0

2 years ago

Noah currently has an account balance of $2,976.45. He opened the account 11 years ago with a deposit of $1,963.45. If the inter

Vesnalui [34]

What we know:
Acct. balance A (11)= $2,976.45
Pricipal=$1,963.45
Compounded (n)=quarterly (4)
rate=r
Compound interest formula: A (t)=p (1+r/n)^nt

What we need toi find:
If the interest compounds quarterly, what is the interest rate on the account? Find r.
A (t)=p (1+r/n)^nt
2976.45=1963.45 (1+r/4)^(4×11)
2976.45=1963.45 (1+r/4)^44
2976.45/1963.45=1963.45/1963.45 (1+r/4)^44
1.5159=(1+r/4)^44
(1.5159)^(1/44)=((1+r/4)^44)^(1/44)

1.0095=1+r/4
1.0095-1=1-1+r/4
.0095=r/4
.0095(4)=r/4 (4)
.038=r

Rate = 3.8%

5 0

3 years ago

Read 2 more answers

What is the true sales amount if the tax is 8.25% and the total with tax is $6440.25

Novay_Z [31]

Answer:

i think the answer is $5,949.42

5 0

2 years ago