Calculus Examples

$\frac{d P}{d t} = \frac{1}{10} P - \frac{1}{5000} P^{2}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\frac{1}{10} P - \frac{1}{5000} P^{2}}$ .

$\frac{1}{\frac{1}{10} P - \frac{1}{5000} P^{2}} \frac{d P}{d t} = \frac{1}{\frac{1}{10} P - \frac{1}{5000} P^{2}} (\frac{1}{10} P - \frac{1}{5000} P^{2})$

Step 1.2

Cancel the common factor of $\frac{1}{10} P - \frac{1}{5000} P^{2}$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{\frac{1}{10} P - \frac{1}{5000} P^{2}} \frac{d P}{d t} = \frac{1}{\frac{1}{10} P - \frac{1}{5000} P^{2}} (\frac{1}{10} P - \frac{1}{5000} P^{2})$

Step 1.2.2

Rewrite the expression.

$\frac{1}{\frac{1}{10} P - \frac{1}{5000} P^{2}} \frac{d P}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{\frac{1}{10} P - \frac{1}{5000} P^{2}} d P = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\frac{1}{10} P - \frac{1}{5000} P^{2}} d P = \int d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Simplify the denominator.

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Step 2.2.1.1.1

Factor $P$ out of $\frac{1}{10} P - \frac{1}{5000} P^{2}$ .

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Step 2.2.1.1.1.1

Factor $P$ out of $\frac{1}{10} P$ .

$\int \frac{1}{P \frac{1}{10} - \frac{1}{5000} P^{2}} d P = \int d t$

Step 2.2.1.1.1.2

Factor $P$ out of $- \frac{1}{5000} P^{2}$ .

$\int \frac{1}{P \frac{1}{10} + P (- \frac{1}{5000} P)} d P = \int d t$

Step 2.2.1.1.1.3

Factor $P$ out of $P \frac{1}{10} + P (- \frac{1}{5000} P)$ .

$\int \frac{1}{P (\frac{1}{10} - \frac{1}{5000} P)} d P = \int d t$

Step 2.2.1.1.2

Combine $P$ and $\frac{1}{5000}$ .

$\int \frac{1}{P (\frac{1}{10} - \frac{P}{5000})} d P = \int d t$

Step 2.2.1.1.3

To write $\frac{1}{10}$ as a fraction with a common denominator, multiply by $\frac{500}{500}$ .

$\int \frac{1}{P (\frac{1}{10} \cdot \frac{500}{500} - \frac{P}{5000})} d P = \int d t$

Step 2.2.1.1.4

Write each expression with a common denominator of $5000$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.1.4.1

Multiply $\frac{1}{10}$ by $\frac{500}{500}$ .

$\int \frac{1}{P (\frac{500}{10 \cdot 500} - \frac{P}{5000})} d P = \int d t$

Step 2.2.1.1.4.2

Multiply $10$ by $500$ .

$\int \frac{1}{P (\frac{500}{5000} - \frac{P}{5000})} d P = \int d t$

Step 2.2.1.1.5

Combine the numerators over the common denominator.

$\int \frac{1}{P \frac{500 - P}{5000}} d P = \int d t$

Step 2.2.1.2

Combine $P$ and $\frac{500 - P}{5000}$ .

$\int \frac{1}{\frac{P (500 - P)}{5000}} d P = \int d t$

Step 2.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$\int 1 \frac{5000}{P (500 - P)} d P = \int d t$

Step 2.2.1.4

Multiply $\frac{5000}{P (500 - P)}$ by $1$ .

$\int \frac{5000}{P (500 - P)} d P = \int d t$

Step 2.2.2

Since $5000$ is constant with respect to $P$ , move $5000$ out of the integral.

$5000 \int \frac{1}{P (500 - P)} d P = \int d t$

Step 2.2.3

Write the fraction using partial fraction decomposition.

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Step 2.2.3.1

Decompose the fraction and multiply through by the common denominator.

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Step 2.2.3.1.1

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $B$ .

$\frac{A}{P} + \frac{B}{500 - P}$

Step 2.2.3.1.2

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $P (500 - P)$ .

$\frac{1 (P (500 - P))}{P (500 - P)} = \frac{A (P (500 - P))}{P} + \frac{(B) (P (500 - P))}{500 - P}$

Step 2.2.3.1.3

Cancel the common factor of $P$ .

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Step 2.2.3.1.3.1

Cancel the common factor.

$\frac{1 (P (500 - P))}{P (500 - P)} = \frac{A (P (500 - P))}{P} + \frac{(B) (P (500 - P))}{500 - P}$

Step 2.2.3.1.3.2

Rewrite the expression.

$\frac{1 (500 - P)}{500 - P} = \frac{A (P (500 - P))}{P} + \frac{(B) (P (500 - P))}{500 - P}$

Step 2.2.3.1.4

Cancel the common factor of $500 - P$ .

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Step 2.2.3.1.4.1

Cancel the common factor.

$\frac{1 (500 - P)}{500 - P} = \frac{A (P (500 - P))}{P} + \frac{(B) (P (500 - P))}{500 - P}$

Step 2.2.3.1.4.2

Rewrite the expression.

$1 = \frac{A (P (500 - P))}{P} + \frac{(B) (P (500 - P))}{500 - P}$

Step 2.2.3.1.5

Simplify each term.

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Step 2.2.3.1.5.1

Cancel the common factor of $P$ .

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Step 2.2.3.1.5.1.1

Cancel the common factor.

$1 = \frac{A (P (500 - P))}{P} + \frac{(B) (P (500 - P))}{500 - P}$

Step 2.2.3.1.5.1.2

Divide $A (500 - P)$ by $1$ .

$1 = A (500 - P) + \frac{(B) (P (500 - P))}{500 - P}$

Step 2.2.3.1.5.2

Apply the distributive property.

$1 = A \cdot 500 + A (- P) + \frac{(B) (P (500 - P))}{500 - P}$

Step 2.2.3.1.5.3

Move $500$ to the left of $A$ .

$1 = 500 \cdot A + A (- P) + \frac{(B) (P (500 - P))}{500 - P}$

Step 2.2.3.1.5.4

Rewrite using the commutative property of multiplication.

$1 = 500 \cdot A - A P + \frac{(B) (P (500 - P))}{500 - P}$

Step 2.2.3.1.5.5

Cancel the common factor of $500 - P$ .

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Step 2.2.3.1.5.5.1

Cancel the common factor.

$1 = 500 A - A P + \frac{B (P (500 - P))}{500 - P}$

Step 2.2.3.1.5.5.2

Divide $(B) (P)$ by $1$ .

$1 = 500 A - A P + (B) (P)$

$1 = 500 A - A P + B P$

Step 2.2.3.1.6

Move $500 A$ .

$1 = - A P + B P + 500 A$

Step 2.2.3.2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.2.3.2.1

Create an equation for the partial fraction variables by equating the coefficients of $P$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = - 1 A + B$

Step 2.2.3.2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $P$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = 500 A$

Step 2.2.3.2.3

Set up the system of equations to find the coefficients of the partial fractions.

$0 = - 1 A + B$

$1 = 500 A$

$0 = - 1 A + B$

$1 = 500 A$

Step 2.2.3.3

Solve the system of equations.

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Step 2.2.3.3.1

Solve for $A$ in $1 = 500 A$ .

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Step 2.2.3.3.1.1

Rewrite the equation as $500 A = 1$ .

$500 A = 1$

$0 = - 1 A + B$

Step 2.2.3.3.1.2

Divide each term in $500 A = 1$ by $500$ and simplify.

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Step 2.2.3.3.1.2.1

Divide each term in $500 A = 1$ by $500$ .

$\frac{500 A}{500} = \frac{1}{500}$

$0 = - 1 A + B$

Step 2.2.3.3.1.2.2

Simplify the left side.

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Step 2.2.3.3.1.2.2.1

Cancel the common factor of $500$ .

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Step 2.2.3.3.1.2.2.1.1

Cancel the common factor.

$\frac{500 A}{500} = \frac{1}{500}$

$0 = - 1 A + B$

Step 2.2.3.3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{1}{500}$

$0 = - 1 A + B$

$A = \frac{1}{500}$

$0 = - 1 A + B$

$A = \frac{1}{500}$

$0 = - 1 A + B$

$A = \frac{1}{500}$

$0 = - 1 A + B$

$A = \frac{1}{500}$

$0 = - 1 A + B$

Step 2.2.3.3.2

Replace all occurrences of $A$ with $\frac{1}{500}$ in each equation.

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Step 2.2.3.3.2.1

Replace all occurrences of $A$ in $0 = - 1 A + B$ with $\frac{1}{500}$ .

$0 = - 1 (\frac{1}{500}) + B$

$A = \frac{1}{500}$

Step 2.2.3.3.2.2

Simplify the right side.

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Step 2.2.3.3.2.2.1

Rewrite $- 1 (\frac{1}{500})$ as $- (\frac{1}{500})$ .

$0 = - \frac{1}{500} + B$

$A = \frac{1}{500}$

$0 = - \frac{1}{500} + B$

$A = \frac{1}{500}$

$0 = - \frac{1}{500} + B$

$A = \frac{1}{500}$

Step 2.2.3.3.3

Solve for $B$ in $0 = - \frac{1}{500} + B$ .

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Step 2.2.3.3.3.1

Rewrite the equation as $- \frac{1}{500} + B = 0$ .

$- \frac{1}{500} + B = 0$

$A = \frac{1}{500}$

Step 2.2.3.3.3.2

Add $\frac{1}{500}$ to both sides of the equation.

$B = \frac{1}{500}$

$A = \frac{1}{500}$

$B = \frac{1}{500}$

$A = \frac{1}{500}$

Step 2.2.3.3.4

Solve the system of equations.

$B = \frac{1}{500} A = \frac{1}{500}$

Step 2.2.3.3.5

List all of the solutions.

$B = \frac{1}{500}, A = \frac{1}{500}$

Step 2.2.3.4

Replace each of the partial fraction coefficients in $\frac{A}{P} + \frac{B}{500 - P}$ with the values found for $A$ and $B$ .

$\frac{\frac{1}{500}}{P} + \frac{\frac{1}{500}}{500 - P}$

Step 2.2.3.5

Simplify.

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Step 2.2.3.5.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{500} \cdot \frac{1}{P} + \frac{\frac{1}{500}}{500 - P}$

Step 2.2.3.5.2

Multiply $\frac{1}{500}$ by $\frac{1}{P}$ .

$\frac{1}{500 P} + \frac{\frac{1}{500}}{500 - P}$

Step 2.2.3.5.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{500 P} + \frac{1}{500} \cdot \frac{1}{500 - P}$

Step 2.2.3.5.4

Multiply $\frac{1}{500}$ by $\frac{1}{500 - P}$ .

$5000 \int \frac{1}{500 P} + \frac{1}{500 (500 - P)} d P = \int d t$

Step 2.2.4

Split the single integral into multiple integrals.

$5000 (\int \frac{1}{500 P} d P + \int \frac{1}{500 (500 - P)} d P) = \int d t$

Step 2.2.5

Since $\frac{1}{500}$ is constant with respect to $P$ , move $\frac{1}{500}$ out of the integral.

$5000 (\frac{1}{500} \int \frac{1}{P} d P + \int \frac{1}{500 (500 - P)} d P) = \int d t$

Step 2.2.6

The integral of $\frac{1}{P}$ with respect to $P$ is $\ln (| P |)$ .

$5000 (\frac{1}{500} (\ln (| P |) + C_{1}) + \int \frac{1}{500 (500 - P)} d P) = \int d t$

Step 2.2.7

Since $\frac{1}{500}$ is constant with respect to $P$ , move $\frac{1}{500}$ out of the integral.

$5000 (\frac{1}{500} (\ln (| P |) + C_{1}) + \frac{1}{500} \int \frac{1}{500 - P} d P) = \int d t$

Step 2.2.8

Let $u = 500 - P$ . Then $d u = - d P$ , so $- d u = d P$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.8.1

Let $u = 500 - P$ . Find $\frac{d u}{d P}$ .

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Step 2.2.8.1.1

Rewrite.

$\frac{- 1}{1}$

Step 2.2.8.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 2.2.8.2

Rewrite the problem using $u$ and $d u$ .

$5000 (\frac{1}{500} (\ln (| P |) + C_{1}) + \frac{1}{500} \int \frac{- 1}{u} d u) = \int d t$

Step 2.2.9

Move the negative in front of the fraction.

$5000 (\frac{1}{500} (\ln (| P |) + C_{1}) + \frac{1}{500} \int - \frac{1}{u} d u) = \int d t$

Step 2.2.10

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$5000 (\frac{1}{500} (\ln (| P |) + C_{1}) + \frac{1}{500} (- \int \frac{1}{u} d u)) = \int d t$

Step 2.2.11

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$5000 (\frac{1}{500} (\ln (| P |) + C_{1}) + \frac{1}{500} (- (\ln (| u |) + C_{2}))) = \int d t$

Step 2.2.12

Simplify.

$5000 (\frac{\ln (| P |)}{500} - \frac{\ln (| u |)}{500}) + C_{3} = \int d t$

Step 2.2.13

Replace all occurrences of $u$ with $500 - P$ .

$5000 (\frac{\ln (| P |)}{500} - \frac{\ln (| 500 - P |)}{500}) + C_{3} = \int d t$

Step 2.2.14

Simplify.

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Step 2.2.14.1

Combine the numerators over the common denominator.

$5000 \frac{\ln (| P |) - \ln (| 500 - P |)}{500} + C_{3} = \int d t$

Step 2.2.14.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$5000 \frac{\ln (\frac{| P |}{| 500 - P |})}{500} + C_{3} = \int d t$

Step 2.2.14.3

Cancel the common factor of $500$ .

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Step 2.2.14.3.1

Factor $500$ out of $5000$ .

$500 (10) \frac{\ln (\frac{| P |}{| 500 - P |})}{500} + C_{3} = \int d t$

Step 2.2.14.3.2

Cancel the common factor.

$500 \cdot 10 \frac{\ln (\frac{| P |}{| 500 - P |})}{500} + C_{3} = \int d t$

Step 2.2.14.3.3

Rewrite the expression.

$10 \ln (\frac{| P |}{| 500 - P |}) + C_{3} = \int d t$

Step 2.3

Apply the constant rule.

$10 \ln (\frac{| P |}{| 500 - P |}) + C_{3} = t + C_{4}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$10 \ln (\frac{| P |}{| 500 - P |}) = t + C$

Step 3

Solve for $P$ .

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Step 3.1

Divide each term in $10 \ln (\frac{| P |}{| 500 - P |}) = t + C$ by $10$ and simplify.

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Step 3.1.1

Divide each term in $10 \ln (\frac{| P |}{| 500 - P |}) = t + C$ by $10$ .

$\frac{10 \ln (\frac{| P |}{| 500 - P |})}{10} = \frac{t}{10} + \frac{C}{10}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $10$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{10 \ln (\frac{| P |}{| 500 - P |})}{10} = \frac{t}{10} + \frac{C}{10}$

Step 3.1.2.1.2

Divide $\ln (\frac{| P |}{| 500 - P |})$ by $1$ .

$\ln (\frac{| P |}{| 500 - P |}) = \frac{t}{10} + \frac{C}{10}$

Step 3.2

To solve for $P$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{| P |}{| 500 - P |})} = e^{\frac{t}{10} + \frac{C}{10}}$

Step 3.3

Rewrite $\ln (\frac{| P |}{| 500 - P |}) = \frac{t}{10} + \frac{C}{10}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{\frac{t}{10} + \frac{C}{10}} = \frac{| P |}{| 500 - P |}$

Step 3.4

Solve for $P$ .

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Step 3.4.1

Rewrite the equation as $\frac{| P |}{| 500 - P |} = e^{\frac{t}{10} + \frac{C}{10}}$ .

$\frac{| P |}{| 500 - P |} = e^{\frac{t}{10} + \frac{C}{10}}$

Step 3.4.2

Multiply both sides by $| 500 - P |$ .

$\frac{| P |}{| 500 - P |} | 500 - P | = e^{\frac{t}{10} + \frac{C}{10}} | 500 - P |$

Step 3.4.3

Simplify the left side.

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Step 3.4.3.1

Cancel the common factor of $| 500 - P |$ .

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Step 3.4.3.1.1

Cancel the common factor.

$\frac{| P |}{| 500 - P |} | 500 - P | = e^{\frac{t}{10} + \frac{C}{10}} | 500 - P |$

Step 3.4.3.1.2

Rewrite the expression.

$| P | = e^{\frac{t}{10} + \frac{C}{10}} | 500 - P |$

Step 3.4.4

Solve for $P$ .

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Step 3.4.4.1

Rewrite the equation as $e^{\frac{t}{10} + \frac{C}{10}} | 500 - P | = | P |$ .

$e^{\frac{t}{10} + \frac{C}{10}} | 500 - P | = | P |$

Step 3.4.4.2

Divide each term in $e^{\frac{t}{10} + \frac{C}{10}} | 500 - P | = | P |$ by $e^{\frac{t}{10} + \frac{C}{10}}$ and simplify.

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Step 3.4.4.2.1

Divide each term in $e^{\frac{t}{10} + \frac{C}{10}} | 500 - P | = | P |$ by $e^{\frac{t}{10} + \frac{C}{10}}$ .

$\frac{e^{\frac{t}{10} + \frac{C}{10}} | 500 - P |}{e^{\frac{t}{10} + \frac{C}{10}}} = \frac{| P |}{e^{\frac{t}{10} + \frac{C}{10}}}$

Step 3.4.4.2.2

Simplify the left side.

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Step 3.4.4.2.2.1

Cancel the common factor of $e^{\frac{t}{10} + \frac{C}{10}}$ .

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Step 3.4.4.2.2.1.1

Cancel the common factor.

$\frac{e^{\frac{t}{10} + \frac{C}{10}} | 500 - P |}{e^{\frac{t}{10} + \frac{C}{10}}} = \frac{| P |}{e^{\frac{t}{10} + \frac{C}{10}}}$

Step 3.4.4.2.2.1.2

Divide $| 500 - P |$ by $1$ .

$| 500 - P | = \frac{| P |}{e^{\frac{t}{10} + \frac{C}{10}}}$

Step 3.4.4.2.3

Simplify the right side.

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Step 3.4.4.2.3.1

Combine the numerators over the common denominator.

$| 500 - P | = \frac{| P |}{e^{\frac{t + C}{10}}}$

Step 3.4.4.3

Rewrite the absolute value equation as four equations without absolute value bars.

$500 - P = \frac{P}{e^{\frac{t + C}{10}}}$

$500 - P = - \frac{P}{e^{\frac{t + C}{10}}}$

$- (500 - P) = \frac{P}{e^{\frac{t + C}{10}}}$

$- (500 - P) = - \frac{P}{e^{\frac{t + C}{10}}}$

Step 3.4.4.4

After simplifying, there are only two unique equations to be solved.

$500 - P = \frac{P}{e^{\frac{t + C}{10}}}$

$500 - P = - \frac{P}{e^{\frac{t + C}{10}}}$

Step 3.4.4.5

Solve $500 - P = \frac{P}{e^{\frac{t + C}{10}}}$ for $P$ .

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Step 3.4.4.5.1

Multiply both sides by $e^{\frac{t + C}{10}}$ .

$(500 - P) e^{\frac{t + C}{10}} = \frac{P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.5.2

Simplify.

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Step 3.4.4.5.2.1

Simplify the left side.

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Step 3.4.4.5.2.1.1

Simplify $(500 - P) e^{\frac{t + C}{10}}$ .

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Step 3.4.4.5.2.1.1.1

Apply the distributive property.

$500 e^{\frac{t + C}{10}} - P e^{\frac{t + C}{10}} = \frac{P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.5.2.1.1.2

Simplify with commuting.

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Step 3.4.4.5.2.1.1.2.1

Reorder $t$ and $C$ .

$500 e^{\frac{C + t}{10}} - P e^{\frac{t + C}{10}} = \frac{P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.5.2.1.1.2.2

Reorder $t$ and $C$ .

$500 e^{\frac{C + t}{10}} - P e^{\frac{C + t}{10}} = \frac{P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.5.2.1.1.2.3

Reorder $500 e^{\frac{C + t}{10}}$ and $- P e^{\frac{C + t}{10}}$ .

$- P e^{\frac{C + t}{10}} + 500 e^{\frac{C + t}{10}} = \frac{P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.5.2.2

Simplify the right side.

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Step 3.4.4.5.2.2.1

Cancel the common factor of $e^{\frac{t + C}{10}}$ .

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Step 3.4.4.5.2.2.1.1

Cancel the common factor.

$- P e^{\frac{C + t}{10}} + 500 e^{\frac{C + t}{10}} = \frac{P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.5.2.2.1.2

Rewrite the expression.

$- P e^{\frac{C + t}{10}} + 500 e^{\frac{C + t}{10}} = P$

Step 3.4.4.5.3

Solve for $P$ .

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Step 3.4.4.5.3.1

Subtract $P$ from both sides of the equation.

$- P e^{\frac{C + t}{10}} + 500 e^{\frac{C + t}{10}} - P = 0$

Step 3.4.4.5.3.2

Move all terms not containing $P$ to the right side of the equation.

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Step 3.4.4.5.3.2.1

Subtract $500 e^{\frac{C + t}{10}}$ from both sides of the equation.

$- P e^{\frac{C + t}{10}} - P = - 500 e^{\frac{C + t}{10}}$

Step 3.4.4.5.3.2.2

Split the fraction $\frac{C + t}{10}$ into two fractions.

$- P e^{\frac{C + t}{10}} - P = - 500 e^{\frac{C}{10} + \frac{t}{10}}$

Step 3.4.4.5.3.3

Factor $P$ out of $- P e^{\frac{C + t}{10}} - P$ .

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Step 3.4.4.5.3.3.1

Factor $P$ out of $- P e^{\frac{C + t}{10}}$ .

$P (- 1 e^{\frac{C + t}{10}}) - P = - 500 e^{\frac{C}{10} + \frac{t}{10}}$

Step 3.4.4.5.3.3.2

Factor $P$ out of $- P$ .

$P (- 1 e^{\frac{C + t}{10}}) + P \cdot - 1 = - 500 e^{\frac{C}{10} + \frac{t}{10}}$

Step 3.4.4.5.3.3.3

Factor $P$ out of $P (- 1 e^{\frac{C + t}{10}}) + P \cdot - 1$ .

$P (- 1 e^{\frac{C + t}{10}} - 1) = - 500 e^{\frac{C}{10} + \frac{t}{10}}$

Step 3.4.4.5.3.4

Rewrite $- 1 e^{\frac{C + t}{10}}$ as $- e^{\frac{C + t}{10}}$ .

$P (- e^{\frac{C + t}{10}} - 1) = - 500 e^{\frac{C}{10} + \frac{t}{10}}$

Step 3.4.4.5.3.5

Divide each term in $P (- e^{\frac{C + t}{10}} - 1) = - 500 e^{\frac{C}{10} + \frac{t}{10}}$ by $- e^{\frac{C + t}{10}} - 1$ and simplify.

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Step 3.4.4.5.3.5.1

Divide each term in $P (- e^{\frac{C + t}{10}} - 1) = - 500 e^{\frac{C}{10} + \frac{t}{10}}$ by $- e^{\frac{C + t}{10}} - 1$ .

$\frac{P (- e^{\frac{C + t}{10}} - 1)}{- e^{\frac{C + t}{10}} - 1} = \frac{- 500 e^{\frac{C}{10} + \frac{t}{10}}}{- e^{\frac{C + t}{10}} - 1}$

Step 3.4.4.5.3.5.2

Simplify the left side.

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Step 3.4.4.5.3.5.2.1

Cancel the common factor of $- e^{\frac{C + t}{10}} - 1$ .

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Step 3.4.4.5.3.5.2.1.1

Cancel the common factor.

$\frac{P (- e^{\frac{C + t}{10}} - 1)}{- e^{\frac{C + t}{10}} - 1} = \frac{- 500 e^{\frac{C}{10} + \frac{t}{10}}}{- e^{\frac{C + t}{10}} - 1}$

Step 3.4.4.5.3.5.2.1.2

Divide $P$ by $1$ .

$P = \frac{- 500 e^{\frac{C}{10} + \frac{t}{10}}}{- e^{\frac{C + t}{10}} - 1}$

Step 3.4.4.5.3.5.3

Simplify the right side.

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Step 3.4.4.5.3.5.3.1

Combine the numerators over the common denominator.

$P = \frac{- 500 e^{\frac{C + t}{10}}}{- e^{\frac{C + t}{10}} - 1}$

Step 3.4.4.5.3.5.3.2

Move the negative in front of the fraction.

$P = - \frac{500 e^{\frac{C + t}{10}}}{- e^{\frac{C + t}{10}} - 1}$

Step 3.4.4.5.3.5.3.3

Factor $- 1$ out of $- e^{\frac{C + t}{10}}$ .

$P = - \frac{500 e^{\frac{C + t}{10}}}{- e^{\frac{C + t}{10}} - 1}$

Step 3.4.4.5.3.5.3.4

Rewrite $- 1$ as $- 1 (1)$ .

$P = - \frac{500 e^{\frac{C + t}{10}}}{- e^{\frac{C + t}{10}} - 1 (1)}$

Step 3.4.4.5.3.5.3.5

Factor $- 1$ out of $- e^{\frac{C + t}{10}} - 1 (1)$ .

$P = - \frac{500 e^{\frac{C + t}{10}}}{- (e^{\frac{C + t}{10}} + 1)}$

Step 3.4.4.5.3.5.3.6

Simplify the expression.

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Step 3.4.4.5.3.5.3.6.1

Rewrite $- (e^{\frac{C + t}{10}} + 1)$ as $- 1 (e^{\frac{C + t}{10}} + 1)$ .

$P = - \frac{500 e^{\frac{C + t}{10}}}{- 1 (e^{\frac{C + t}{10}} + 1)}$

Step 3.4.4.5.3.5.3.6.2

Move the negative in front of the fraction.

$P = - - \frac{500 e^{\frac{C + t}{10}}}{e^{\frac{C + t}{10}} + 1}$

Step 3.4.4.5.3.5.3.6.3

Multiply $- 1$ by $- 1$ .

$P = 1 \frac{500 e^{\frac{C + t}{10}}}{e^{\frac{C + t}{10}} + 1}$

Step 3.4.4.5.3.5.3.6.4

Multiply $\frac{500 e^{\frac{C + t}{10}}}{e^{\frac{C + t}{10}} + 1}$ by $1$ .

$P = \frac{500 e^{\frac{C + t}{10}}}{e^{\frac{C + t}{10}} + 1}$

Step 3.4.4.6

Solve $500 - P = - \frac{P}{e^{\frac{t + C}{10}}}$ for $P$ .

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Step 3.4.4.6.1

Multiply both sides by $e^{\frac{t + C}{10}}$ .

$(500 - P) e^{\frac{t + C}{10}} = - \frac{P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.6.2

Simplify.

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Step 3.4.4.6.2.1

Simplify the left side.

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Step 3.4.4.6.2.1.1

Simplify $(500 - P) e^{\frac{t + C}{10}}$ .

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Step 3.4.4.6.2.1.1.1

Apply the distributive property.

$500 e^{\frac{t + C}{10}} - P e^{\frac{t + C}{10}} = - \frac{P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.6.2.1.1.2

Simplify with commuting.

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Step 3.4.4.6.2.1.1.2.1

Reorder $t$ and $C$ .

$500 e^{\frac{C + t}{10}} - P e^{\frac{t + C}{10}} = - \frac{P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.6.2.1.1.2.2

Reorder $t$ and $C$ .

$500 e^{\frac{C + t}{10}} - P e^{\frac{C + t}{10}} = - \frac{P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.6.2.1.1.2.3

Reorder $500 e^{\frac{C + t}{10}}$ and $- P e^{\frac{C + t}{10}}$ .

$- P e^{\frac{C + t}{10}} + 500 e^{\frac{C + t}{10}} = - \frac{P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.6.2.2

Simplify the right side.

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Step 3.4.4.6.2.2.1

Cancel the common factor of $e^{\frac{t + C}{10}}$ .

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Step 3.4.4.6.2.2.1.1

Move the leading negative in $- \frac{P}{e^{\frac{t + C}{10}}}$ into the numerator.

$- P e^{\frac{C + t}{10}} + 500 e^{\frac{C + t}{10}} = \frac{- P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.6.2.2.1.2

Cancel the common factor.

$- P e^{\frac{C + t}{10}} + 500 e^{\frac{C + t}{10}} = \frac{- P}{e^{\frac{t + C}{10}}} e^{\frac{t + C}{10}}$

Step 3.4.4.6.2.2.1.3

Rewrite the expression.

$- P e^{\frac{C + t}{10}} + 500 e^{\frac{C + t}{10}} = - P$

Step 3.4.4.6.3

Solve for $P$ .

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Step 3.4.4.6.3.1

Add $P$ to both sides of the equation.

$- P e^{\frac{C + t}{10}} + 500 e^{\frac{C + t}{10}} + P = 0$

Step 3.4.4.6.3.2

Move all terms not containing $P$ to the right side of the equation.

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Step 3.4.4.6.3.2.1

Subtract $500 e^{\frac{C + t}{10}}$ from both sides of the equation.

$- P e^{\frac{C + t}{10}} + P = - 500 e^{\frac{C + t}{10}}$

Step 3.4.4.6.3.2.2

Split the fraction $\frac{C + t}{10}$ into two fractions.

$- P e^{\frac{C + t}{10}} + P = - 500 e^{\frac{C}{10} + \frac{t}{10}}$

Step 3.4.4.6.3.3

Factor $P$ out of $- P e^{\frac{C + t}{10}} + P$ .

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Step 3.4.4.6.3.3.1

Factor $P$ out of $- P e^{\frac{C + t}{10}}$ .

$P (- 1 e^{\frac{C + t}{10}}) + P = - 500 e^{\frac{C}{10} + \frac{t}{10}}$

Step 3.4.4.6.3.3.2

Raise $P$ to the power of $1$ .

$P (- 1 e^{\frac{C + t}{10}}) + P = - 500 e^{\frac{C}{10} + \frac{t}{10}}$

Step 3.4.4.6.3.3.3

Factor $P$ out of $P^{1}$ .

$P (- 1 e^{\frac{C + t}{10}}) + P \cdot 1 = - 500 e^{\frac{C}{10} + \frac{t}{10}}$

Step 3.4.4.6.3.3.4

Factor $P$ out of $P (- 1 e^{\frac{C + t}{10}}) + P \cdot 1$ .

$P (- 1 e^{\frac{C + t}{10}} + 1) = - 500 e^{\frac{C}{10} + \frac{t}{10}}$

Step 3.4.4.6.3.4

Rewrite $- 1 e^{\frac{C + t}{10}}$ as $- e^{\frac{C + t}{10}}$ .

$P (- e^{\frac{C + t}{10}} + 1) = - 500 e^{\frac{C}{10} + \frac{t}{10}}$

Step 3.4.4.6.3.5

Divide each term in $P (- e^{\frac{C + t}{10}} + 1) = - 500 e^{\frac{C}{10} + \frac{t}{10}}$ by $- e^{\frac{C + t}{10}} + 1$ and simplify.

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Step 3.4.4.6.3.5.1

Divide each term in $P (- e^{\frac{C + t}{10}} + 1) = - 500 e^{\frac{C}{10} + \frac{t}{10}}$ by $- e^{\frac{C + t}{10}} + 1$ .

$\frac{P (- e^{\frac{C + t}{10}} + 1)}{- e^{\frac{C + t}{10}} + 1} = \frac{- 500 e^{\frac{C}{10} + \frac{t}{10}}}{- e^{\frac{C + t}{10}} + 1}$

Step 3.4.4.6.3.5.2

Simplify the left side.

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Step 3.4.4.6.3.5.2.1

Cancel the common factor of $- e^{\frac{C + t}{10}} + 1$ .

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Step 3.4.4.6.3.5.2.1.1

Cancel the common factor.

$\frac{P (- e^{\frac{C + t}{10}} + 1)}{- e^{\frac{C + t}{10}} + 1} = \frac{- 500 e^{\frac{C}{10} + \frac{t}{10}}}{- e^{\frac{C + t}{10}} + 1}$

Step 3.4.4.6.3.5.2.1.2

Divide $P$ by $1$ .

$P = \frac{- 500 e^{\frac{C}{10} + \frac{t}{10}}}{- e^{\frac{C + t}{10}} + 1}$

Step 3.4.4.6.3.5.3

Simplify the right side.

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Step 3.4.4.6.3.5.3.1

Combine the numerators over the common denominator.

$P = \frac{- 500 e^{\frac{C + t}{10}}}{- e^{\frac{C + t}{10}} + 1}$

Step 3.4.4.6.3.5.3.2

Move the negative in front of the fraction.

$P = - \frac{500 e^{\frac{C + t}{10}}}{- e^{\frac{C + t}{10}} + 1}$

Step 3.4.4.6.3.5.3.3

Factor $- 1$ out of $- e^{\frac{C + t}{10}}$ .

$P = - \frac{500 e^{\frac{C + t}{10}}}{- e^{\frac{C + t}{10}} + 1}$

Step 3.4.4.6.3.5.3.4

Rewrite $1$ as $- 1 (- 1)$ .

$P = - \frac{500 e^{\frac{C + t}{10}}}{- e^{\frac{C + t}{10}} - 1 (- 1)}$

Step 3.4.4.6.3.5.3.5

Factor $- 1$ out of $- e^{\frac{C + t}{10}} - 1 (- 1)$ .

$P = - \frac{500 e^{\frac{C + t}{10}}}{- (e^{\frac{C + t}{10}} - 1)}$

Step 3.4.4.6.3.5.3.6

Simplify the expression.

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Step 3.4.4.6.3.5.3.6.1

Rewrite $- (e^{\frac{C + t}{10}} - 1)$ as $- 1 (e^{\frac{C + t}{10}} - 1)$ .

$P = - \frac{500 e^{\frac{C + t}{10}}}{- 1 (e^{\frac{C + t}{10}} - 1)}$

Step 3.4.4.6.3.5.3.6.2

Move the negative in front of the fraction.

$P = - - \frac{500 e^{\frac{C + t}{10}}}{e^{\frac{C + t}{10}} - 1}$

Step 3.4.4.6.3.5.3.6.3

Multiply $- 1$ by $- 1$ .

$P = 1 \frac{500 e^{\frac{C + t}{10}}}{e^{\frac{C + t}{10}} - 1}$

Step 3.4.4.6.3.5.3.6.4

Multiply $\frac{500 e^{\frac{C + t}{10}}}{e^{\frac{C + t}{10}} - 1}$ by $1$ .