Calculus Examples

$\frac{d p}{d t} = \frac{p}{3} - \frac{p^{2}}{2100}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\frac{p}{3} - \frac{p^{2}}{2100}}$ .

$\frac{1}{\frac{p}{3} - \frac{p^{2}}{2100}} \frac{d p}{d t} = \frac{1}{\frac{p}{3} - \frac{p^{2}}{2100}} (\frac{p}{3} - \frac{p^{2}}{2100})$

Step 1.2

Cancel the common factor of $\frac{p}{3} - \frac{p^{2}}{2100}$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{\frac{p}{3} - \frac{p^{2}}{2100}} \frac{d p}{d t} = \frac{1}{\frac{p}{3} - \frac{p^{2}}{2100}} (\frac{p}{3} - \frac{p^{2}}{2100})$

Step 1.2.2

Rewrite the expression.

$\frac{1}{\frac{p}{3} - \frac{p^{2}}{2100}} \frac{d p}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{\frac{p}{3} - \frac{p^{2}}{2100}} d p = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\frac{p}{3} - \frac{p^{2}}{2100}} d p = \int d t$

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

To write $\frac{p}{3}$ as a fraction with a common denominator, multiply by $\frac{700}{700}$ .

$\int \frac{1}{\frac{p}{3} \cdot \frac{700}{700} - \frac{p^{2}}{2100}} d p = \int d t$

Step 2.2.1.1.2

Write each expression with a common denominator of $2100$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.1.2.1

Multiply $\frac{p}{3}$ by $\frac{700}{700}$ .

$\int \frac{1}{\frac{p \cdot 700}{3 \cdot 700} - \frac{p^{2}}{2100}} d p = \int d t$

Step 2.2.1.1.2.2

Multiply $3$ by $700$ .

$\int \frac{1}{\frac{p \cdot 700}{2100} - \frac{p^{2}}{2100}} d p = \int d t$

Step 2.2.1.1.3

Combine the numerators over the common denominator.

$\int \frac{1}{\frac{p \cdot 700 - p^{2}}{2100}} d p = \int d t$

Step 2.2.1.1.4

Factor $p$ out of $p \cdot 700 - p^{2}$ .

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Step 2.2.1.1.4.1

Factor $p$ out of $p \cdot 700$ .

$\int \frac{1}{\frac{p (700) - p^{2}}{2100}} d p = \int d t$

Step 2.2.1.1.4.2

Factor $p$ out of $- p^{2}$ .

$\int \frac{1}{\frac{p (700) + p (- p)}{2100}} d p = \int d t$

Step 2.2.1.1.4.3

Factor $p$ out of $p (700) + p (- p)$ .

$\int \frac{1}{\frac{p (700 - p)}{2100}} d p = \int d t$

Step 2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$\int 1 \frac{2100}{p (700 - p)} d p = \int d t$

Step 2.2.1.3

Multiply $\frac{2100}{p (700 - p)}$ by $1$ .

$\int \frac{2100}{p (700 - p)} d p = \int d t$

Step 2.2.2

Since $2100$ is constant with respect to $p$ , move $2100$ out of the integral.

$2100 \int \frac{1}{p (700 - p)} d p = \int d t$

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 $B$ .

$\frac{A}{p} + \frac{B}{700 - p}$

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 $p (700 - p)$ .

$\frac{1 (p (700 - p))}{p (700 - p)} = \frac{A (p (700 - p))}{p} + \frac{(B) (p (700 - p))}{700 - p}$

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.

$\frac{1 (p (700 - p))}{p (700 - p)} = \frac{A (p (700 - p))}{p} + \frac{(B) (p (700 - p))}{700 - p}$

Step 2.2.3.1.3.2

Rewrite the expression.

$\frac{1 (700 - p)}{700 - p} = \frac{A (p (700 - p))}{p} + \frac{(B) (p (700 - p))}{700 - p}$

Step 2.2.3.1.4

Cancel the common factor of $700 - p$ .

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Step 2.2.3.1.4.1

Cancel the common factor.

$\frac{1 (700 - p)}{700 - p} = \frac{A (p (700 - p))}{p} + \frac{(B) (p (700 - p))}{700 - p}$

Step 2.2.3.1.4.2

Rewrite the expression.

$1 = \frac{A (p (700 - p))}{p} + \frac{(B) (p (700 - p))}{700 - p}$

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.

$1 = \frac{A (p (700 - p))}{p} + \frac{(B) (p (700 - p))}{700 - p}$

Step 2.2.3.1.5.1.2

Divide $A (700 - p)$ by $1$ .

$1 = A (700 - p) + \frac{(B) (p (700 - p))}{700 - p}$

Step 2.2.3.1.5.2

Apply the distributive property.

$1 = A \cdot 700 + A (- p) + \frac{(B) (p (700 - p))}{700 - p}$

Step 2.2.3.1.5.3

Move $700$ to the left of $A$ .

$1 = 700 \cdot A + A (- p) + \frac{(B) (p (700 - p))}{700 - p}$

Step 2.2.3.1.5.4

Rewrite using the commutative property of multiplication.

$1 = 700 \cdot A - A p + \frac{(B) (p (700 - p))}{700 - p}$

Step 2.2.3.1.5.5

Cancel the common factor of $700 - p$ .

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Step 2.2.3.1.5.5.1

Cancel the common factor.

$1 = 700 A - A p + \frac{B (p (700 - p))}{700 - p}$

Step 2.2.3.1.5.5.2

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

$1 = 700 A - A p + (B) (p)$

$1 = 700 A - A p + B p$

Step 2.2.3.1.6

Simplify the expression.

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Step 2.2.3.1.6.1

Move $A$ .

$1 = 700 A - p A + B p$

Step 2.2.3.1.6.2

Reorder $B$ and $p$ .

$1 = 700 A - p A + B p$

Step 2.2.3.1.6.3

Move $700 A$ .

$1 = - p A + B p + 700 A$

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.

$1 = 700 A$

Step 2.2.3.2.3

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

$0 = - 1 A + B$

$1 = 700 A$

$0 = - 1 A + B$

$1 = 700 A$

Step 2.2.3.3

Solve the system of equations.

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Step 2.2.3.3.1

Solve for $A$ in $1 = 700 A$ .

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Step 2.2.3.3.1.1

Rewrite the equation as $700 A = 1$ .

$700 A = 1$

$0 = - 1 A + B$

Step 2.2.3.3.1.2

Divide each term in $700 A = 1$ by $700$ and simplify.

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Step 2.2.3.3.1.2.1

Divide each term in $700 A = 1$ by $700$ .

$\frac{700 A}{700} = \frac{1}{700}$

$0 = - 1 A + B$

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 $700$ .

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Step 2.2.3.3.1.2.2.1.1

Cancel the common factor.

$\frac{700 A}{700} = \frac{1}{700}$

$0 = - 1 A + B$

Step 2.2.3.3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{1}{700}$

$0 = - 1 A + B$

$A = \frac{1}{700}$

$0 = - 1 A + B$

$A = \frac{1}{700}$

$0 = - 1 A + B$

$A = \frac{1}{700}$

$0 = - 1 A + B$

$A = \frac{1}{700}$

$0 = - 1 A + B$

Step 2.2.3.3.2

Replace all occurrences of $A$ with $\frac{1}{700}$ 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 $\frac{1}{700}$ .

$0 = - 1 (\frac{1}{700}) + B$

$A = \frac{1}{700}$

Step 2.2.3.3.2.2

Simplify the right side.

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Step 2.2.3.3.2.2.1

Rewrite $- 1 (\frac{1}{700})$ as $- (\frac{1}{700})$ .

$0 = - \frac{1}{700} + B$

$A = \frac{1}{700}$

$0 = - \frac{1}{700} + B$

$A = \frac{1}{700}$

$0 = - \frac{1}{700} + B$

$A = \frac{1}{700}$

Step 2.2.3.3.3

Solve for $B$ in $0 = - \frac{1}{700} + B$ .

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Step 2.2.3.3.3.1

Rewrite the equation as $- \frac{1}{700} + B = 0$ .

$- \frac{1}{700} + B = 0$

$A = \frac{1}{700}$

Step 2.2.3.3.3.2

Add $\frac{1}{700}$ to both sides of the equation.

$B = \frac{1}{700}$

$A = \frac{1}{700}$

$B = \frac{1}{700}$

$A = \frac{1}{700}$

Step 2.2.3.3.4

Solve the system of equations.

$B = \frac{1}{700} A = \frac{1}{700}$

Step 2.2.3.3.5

List all of the solutions.

$B = \frac{1}{700}, A = \frac{1}{700}$

Step 2.2.3.4

Replace each of the partial fraction coefficients in $\frac{A}{p} + \frac{B}{700 - p}$ with the values found for $A$ and $B$ .

$\frac{\frac{1}{700}}{p} + \frac{\frac{1}{700}}{700 - p}$

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.

$\frac{1}{700} \cdot \frac{1}{p} + \frac{\frac{1}{700}}{700 - p}$

Step 2.2.3.5.2

Multiply $\frac{1}{700}$ by $\frac{1}{p}$ .

$\frac{1}{700 p} + \frac{\frac{1}{700}}{700 - p}$

Step 2.2.3.5.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{700 p} + \frac{1}{700} \cdot \frac{1}{700 - p}$

Step 2.2.3.5.4

Multiply $\frac{1}{700}$ by $\frac{1}{700 - p}$ .

$2100 \int \frac{1}{700 p} + \frac{1}{700 (700 - p)} d p = \int d t$

Step 2.2.4

Split the single integral into multiple integrals.

$2100 (\int \frac{1}{700 p} d p + \int \frac{1}{700 (700 - p)} d p) = \int d t$

Step 2.2.5

Since $\frac{1}{700}$ is constant with respect to $p$ , move $\frac{1}{700}$ out of the integral.

$2100 (\frac{1}{700} \int \frac{1}{p} d p + \int \frac{1}{700 (700 - p)} d p) = \int d t$

Step 2.2.6

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

$2100 (\frac{1}{700} (\ln (| p |) + C_{1}) + \int \frac{1}{700 (700 - p)} d p) = \int d t$

Step 2.2.7

Since $\frac{1}{700}$ is constant with respect to $p$ , move $\frac{1}{700}$ out of the integral.

$2100 (\frac{1}{700} (\ln (| p |) + C_{1}) + \frac{1}{700} \int \frac{1}{700 - p} d p) = \int d t$

Step 2.2.8

Let $u = 700 - p$ . 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 $u = 700 - p$ . 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 $d u$ .

$2100 (\frac{1}{700} (\ln (| p |) + C_{1}) + \frac{1}{700} \int \frac{- 1}{u} d u) = \int d t$

Step 2.2.9

Move the negative in front of the fraction.

$2100 (\frac{1}{700} (\ln (| p |) + C_{1}) + \frac{1}{700} \int - \frac{1}{u} d u) = \int d t$

Step 2.2.10

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

$2100 (\frac{1}{700} (\ln (| p |) + C_{1}) + \frac{1}{700} (- \int \frac{1}{u} d u)) = \int d t$

Step 2.2.11

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

$2100 (\frac{1}{700} (\ln (| p |) + C_{1}) + \frac{1}{700} (- (\ln (| u |) + C_{2}))) = \int d t$

Step 2.2.12

Simplify.

$2100 (\frac{\ln (| p |)}{700} - \frac{\ln (| u |)}{700}) + C_{3} = \int d t$

Step 2.2.13

Replace all occurrences of $u$ with $700 - p$ .

$2100 (\frac{\ln (| p |)}{700} - \frac{\ln (| 700 - p |)}{700}) + C_{3} = \int d t$

Step 2.2.14

Simplify.

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Step 2.2.14.1

Combine the numerators over the common denominator.

$2100 \frac{\ln (| p |) - \ln (| 700 - p |)}{700} + C_{3} = \int d t$

Step 2.2.14.2

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

$2100 \frac{\ln (\frac{| p |}{| 700 - p |})}{700} + C_{3} = \int d t$

Step 2.2.14.3

Cancel the common factor of $700$ .

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Step 2.2.14.3.1

Factor $700$ out of $2100$ .

$700 (3) \frac{\ln (\frac{| p |}{| 700 - p |})}{700} + C_{3} = \int d t$

Step 2.2.14.3.2

Cancel the common factor.

$700 \cdot 3 \frac{\ln (\frac{| p |}{| 700 - p |})}{700} + C_{3} = \int d t$

Step 2.2.14.3.3

Rewrite the expression.

$3 \ln (\frac{| p |}{| 700 - p |}) + C_{3} = \int d t$

Step 2.3

Apply the constant rule.

$3 \ln (\frac{| p |}{| 700 - p |}) + C_{3} = t + C_{4}$

Step 2.4

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

$3 \ln (\frac{| p |}{| 700 - p |}) = t + C$

Step 3

Solve for $p$ .

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Step 3.1

Divide each term in $3 \ln (\frac{| p |}{| 700 - p |}) = t + C$ by $3$ and simplify.

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Step 3.1.1

Divide each term in $3 \ln (\frac{| p |}{| 700 - p |}) = t + C$ by $3$ .

$\frac{3 \ln (\frac{| p |}{| 700 - p |})}{3} = \frac{t}{3} + \frac{C}{3}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $3$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{3 \ln (\frac{| p |}{| 700 - p |})}{3} = \frac{t}{3} + \frac{C}{3}$

Step 3.1.2.1.2

Divide $\ln (\frac{| p |}{| 700 - p |})$ by $1$ .

$\ln (\frac{| p |}{| 700 - p |}) = \frac{t}{3} + \frac{C}{3}$

Step 3.2

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

$e^{\ln (\frac{| p |}{| 700 - p |})} = e^{\frac{t}{3} + \frac{C}{3}}$

Step 3.3

Rewrite $\ln (\frac{| p |}{| 700 - p |}) = \frac{t}{3} + \frac{C}{3}$ 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 $b^{y} = x$ .

$e^{\frac{t}{3} + \frac{C}{3}} = \frac{| p |}{| 700 - p |}$

Step 3.4

Solve for $p$ .

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Step 3.4.1

Rewrite the equation as $\frac{| p |}{| 700 - p |} = e^{\frac{t}{3} + \frac{C}{3}}$ .

$\frac{| p |}{| 700 - p |} = e^{\frac{t}{3} + \frac{C}{3}}$

Step 3.4.2

Multiply both sides by $| 700 - p |$ .

$\frac{| p |}{| 700 - p |} | 700 - p | = e^{\frac{t}{3} + \frac{C}{3}} | 700 - p |$

Step 3.4.3

Simplify the left side.

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Step 3.4.3.1

Cancel the common factor of $| 700 - p |$ .

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Step 3.4.3.1.1

Cancel the common factor.

$\frac{| p |}{| 700 - p |} | 700 - p | = e^{\frac{t}{3} + \frac{C}{3}} | 700 - p |$

Step 3.4.3.1.2

Rewrite the expression.

$| p | = e^{\frac{t}{3} + \frac{C}{3}} | 700 - p |$

Step 3.4.4

Solve for $p$ .

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Step 3.4.4.1

Rewrite the equation as $e^{\frac{t}{3} + \frac{C}{3}} | 700 - p | = | p |$ .

$e^{\frac{t}{3} + \frac{C}{3}} | 700 - p | = | p |$

Step 3.4.4.2

Divide each term in $e^{\frac{t}{3} + \frac{C}{3}} | 700 - p | = | p |$ by $e^{\frac{t}{3} + \frac{C}{3}}$ and simplify.

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Step 3.4.4.2.1

Divide each term in $e^{\frac{t}{3} + \frac{C}{3}} | 700 - p | = | p |$ by $e^{\frac{t}{3} + \frac{C}{3}}$ .

$\frac{e^{\frac{t}{3} + \frac{C}{3}} | 700 - p |}{e^{\frac{t}{3} + \frac{C}{3}}} = \frac{| p |}{e^{\frac{t}{3} + \frac{C}{3}}}$

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 $e^{\frac{t}{3} + \frac{C}{3}}$ .

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Step 3.4.4.2.2.1.1

Cancel the common factor.

$\frac{e^{\frac{t}{3} + \frac{C}{3}} | 700 - p |}{e^{\frac{t}{3} + \frac{C}{3}}} = \frac{| p |}{e^{\frac{t}{3} + \frac{C}{3}}}$

Step 3.4.4.2.2.1.2

Divide $| 700 - p |$ by $1$ .

$| 700 - p | = \frac{| p |}{e^{\frac{t}{3} + \frac{C}{3}}}$

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.

$| 700 - p | = \frac{| p |}{e^{\frac{t + C}{3}}}$

Step 3.4.4.3

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

$700 - p = \frac{p}{e^{\frac{t + C}{3}}}$

$700 - p = - \frac{p}{e^{\frac{t + C}{3}}}$

$- (700 - p) = \frac{p}{e^{\frac{t + C}{3}}}$

$- (700 - p) = - \frac{p}{e^{\frac{t + C}{3}}}$

Step 3.4.4.4

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

$700 - p = \frac{p}{e^{\frac{t + C}{3}}}$

$700 - p = - \frac{p}{e^{\frac{t + C}{3}}}$

Step 3.4.4.5

Solve $700 - p = \frac{p}{e^{\frac{t + C}{3}}}$ for $p$ .

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Step 3.4.4.5.1

Multiply both sides by $e^{\frac{t + C}{3}}$ .

$(700 - p) e^{\frac{t + C}{3}} = \frac{p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

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 $(700 - p) e^{\frac{t + C}{3}}$ .

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Step 3.4.4.5.2.1.1.1

Apply the distributive property.

$700 e^{\frac{t + C}{3}} - p e^{\frac{t + C}{3}} = \frac{p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

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 $C$ .

$700 e^{\frac{C + t}{3}} - p e^{\frac{t + C}{3}} = \frac{p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

Step 3.4.4.5.2.1.1.2.2

Reorder $t$ and $C$ .

$700 e^{\frac{C + t}{3}} - p e^{\frac{C + t}{3}} = \frac{p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

Step 3.4.4.5.2.1.1.2.3

Reorder $700 e^{\frac{C + t}{3}}$ and $- p e^{\frac{C + t}{3}}$ .

$- p e^{\frac{C + t}{3}} + 700 e^{\frac{C + t}{3}} = \frac{p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

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 $e^{\frac{t + C}{3}}$ .

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Step 3.4.4.5.2.2.1.1

Cancel the common factor.

$- p e^{\frac{C + t}{3}} + 700 e^{\frac{C + t}{3}} = \frac{p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

Step 3.4.4.5.2.2.1.2

Rewrite the expression.

$- p e^{\frac{C + t}{3}} + 700 e^{\frac{C + t}{3}} = p$

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.

$- p e^{\frac{C + t}{3}} + 700 e^{\frac{C + t}{3}} - p = 0$

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 $700 e^{\frac{C + t}{3}}$ from both sides of the equation.

$- p e^{\frac{C + t}{3}} - p = - 700 e^{\frac{C + t}{3}}$

Step 3.4.4.5.3.2.2

Split the fraction $\frac{C + t}{3}$ into two fractions.

$- p e^{\frac{C + t}{3}} - p = - 700 e^{\frac{C}{3} + \frac{t}{3}}$

Step 3.4.4.5.3.3

Factor $p$ out of $- p e^{\frac{C + t}{3}} - p$ .

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Step 3.4.4.5.3.3.1

Factor $p$ out of $- p e^{\frac{C + t}{3}}$ .

$p (- 1 e^{\frac{C + t}{3}}) - p = - 700 e^{\frac{C}{3} + \frac{t}{3}}$

Step 3.4.4.5.3.3.2

Factor $p$ out of $- p$ .

$p (- 1 e^{\frac{C + t}{3}}) + p \cdot - 1 = - 700 e^{\frac{C}{3} + \frac{t}{3}}$

Step 3.4.4.5.3.3.3

Factor $p$ out of $p (- 1 e^{\frac{C + t}{3}}) + p \cdot - 1$ .

$p (- 1 e^{\frac{C + t}{3}} - 1) = - 700 e^{\frac{C}{3} + \frac{t}{3}}$

Step 3.4.4.5.3.4

Rewrite $- 1 e^{\frac{C + t}{3}}$ as $- e^{\frac{C + t}{3}}$ .

$p (- e^{\frac{C + t}{3}} - 1) = - 700 e^{\frac{C}{3} + \frac{t}{3}}$

Step 3.4.4.5.3.5

Divide each term in $p (- e^{\frac{C + t}{3}} - 1) = - 700 e^{\frac{C}{3} + \frac{t}{3}}$ by $- e^{\frac{C + t}{3}} - 1$ and simplify.

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Step 3.4.4.5.3.5.1

Divide each term in $p (- e^{\frac{C + t}{3}} - 1) = - 700 e^{\frac{C}{3} + \frac{t}{3}}$ by $- e^{\frac{C + t}{3}} - 1$ .

$\frac{p (- e^{\frac{C + t}{3}} - 1)}{- e^{\frac{C + t}{3}} - 1} = \frac{- 700 e^{\frac{C}{3} + \frac{t}{3}}}{- e^{\frac{C + t}{3}} - 1}$

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 $- e^{\frac{C + t}{3}} - 1$ .

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Step 3.4.4.5.3.5.2.1.1

Cancel the common factor.

$\frac{p (- e^{\frac{C + t}{3}} - 1)}{- e^{\frac{C + t}{3}} - 1} = \frac{- 700 e^{\frac{C}{3} + \frac{t}{3}}}{- e^{\frac{C + t}{3}} - 1}$

Step 3.4.4.5.3.5.2.1.2

Divide $p$ by $1$ .

$p = \frac{- 700 e^{\frac{C}{3} + \frac{t}{3}}}{- e^{\frac{C + t}{3}} - 1}$

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.

$p = \frac{- 700 e^{\frac{C + t}{3}}}{- e^{\frac{C + t}{3}} - 1}$

Step 3.4.4.5.3.5.3.2

Move the negative in front of the fraction.

$p = - \frac{700 e^{\frac{C + t}{3}}}{- e^{\frac{C + t}{3}} - 1}$

Step 3.4.4.5.3.5.3.3

Factor $- 1$ out of $- e^{\frac{C + t}{3}}$ .

$p = - \frac{700 e^{\frac{C + t}{3}}}{- e^{\frac{C + t}{3}} - 1}$

Step 3.4.4.5.3.5.3.4

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

$p = - \frac{700 e^{\frac{C + t}{3}}}{- e^{\frac{C + t}{3}} - 1 (1)}$

Step 3.4.4.5.3.5.3.5

Factor $- 1$ out of $- e^{\frac{C + t}{3}} - 1 (1)$ .

$p = - \frac{700 e^{\frac{C + t}{3}}}{- (e^{\frac{C + t}{3}} + 1)}$

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 $- (e^{\frac{C + t}{3}} + 1)$ as $- 1 (e^{\frac{C + t}{3}} + 1)$ .

$p = - \frac{700 e^{\frac{C + t}{3}}}{- 1 (e^{\frac{C + t}{3}} + 1)}$

Step 3.4.4.5.3.5.3.6.2

Move the negative in front of the fraction.

$p = - - \frac{700 e^{\frac{C + t}{3}}}{e^{\frac{C + t}{3}} + 1}$

Step 3.4.4.5.3.5.3.6.3

Multiply $- 1$ by $- 1$ .

$p = 1 \frac{700 e^{\frac{C + t}{3}}}{e^{\frac{C + t}{3}} + 1}$

Step 3.4.4.5.3.5.3.6.4

Multiply $\frac{700 e^{\frac{C + t}{3}}}{e^{\frac{C + t}{3}} + 1}$ by $1$ .

$p = \frac{700 e^{\frac{C + t}{3}}}{e^{\frac{C + t}{3}} + 1}$

Step 3.4.4.6

Solve $700 - p = - \frac{p}{e^{\frac{t + C}{3}}}$ for $p$ .

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Step 3.4.4.6.1

Multiply both sides by $e^{\frac{t + C}{3}}$ .

$(700 - p) e^{\frac{t + C}{3}} = - \frac{p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

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 $(700 - p) e^{\frac{t + C}{3}}$ .

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Step 3.4.4.6.2.1.1.1

Apply the distributive property.

$700 e^{\frac{t + C}{3}} - p e^{\frac{t + C}{3}} = - \frac{p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

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 $C$ .

$700 e^{\frac{C + t}{3}} - p e^{\frac{t + C}{3}} = - \frac{p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

Step 3.4.4.6.2.1.1.2.2

Reorder $t$ and $C$ .

$700 e^{\frac{C + t}{3}} - p e^{\frac{C + t}{3}} = - \frac{p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

Step 3.4.4.6.2.1.1.2.3

Reorder $700 e^{\frac{C + t}{3}}$ and $- p e^{\frac{C + t}{3}}$ .

$- p e^{\frac{C + t}{3}} + 700 e^{\frac{C + t}{3}} = - \frac{p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

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 $e^{\frac{t + C}{3}}$ .

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Step 3.4.4.6.2.2.1.1

Move the leading negative in $- \frac{p}{e^{\frac{t + C}{3}}}$ into the numerator.

$- p e^{\frac{C + t}{3}} + 700 e^{\frac{C + t}{3}} = \frac{- p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

Step 3.4.4.6.2.2.1.2

Cancel the common factor.

$- p e^{\frac{C + t}{3}} + 700 e^{\frac{C + t}{3}} = \frac{- p}{e^{\frac{t + C}{3}}} e^{\frac{t + C}{3}}$

Step 3.4.4.6.2.2.1.3

Rewrite the expression.

$- p e^{\frac{C + t}{3}} + 700 e^{\frac{C + t}{3}} = - p$

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.

$- p e^{\frac{C + t}{3}} + 700 e^{\frac{C + t}{3}} + p = 0$

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 $700 e^{\frac{C + t}{3}}$ from both sides of the equation.

$- p e^{\frac{C + t}{3}} + p = - 700 e^{\frac{C + t}{3}}$

Step 3.4.4.6.3.2.2

Split the fraction $\frac{C + t}{3}$ into two fractions.

$- p e^{\frac{C + t}{3}} + p = - 700 e^{\frac{C}{3} + \frac{t}{3}}$

Step 3.4.4.6.3.3

Factor $p$ out of $- p e^{\frac{C + t}{3}} + p$ .

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Step 3.4.4.6.3.3.1

Factor $p$ out of $- p e^{\frac{C + t}{3}}$ .

$p (- 1 e^{\frac{C + t}{3}}) + p = - 700 e^{\frac{C}{3} + \frac{t}{3}}$

Step 3.4.4.6.3.3.2

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

$p (- 1 e^{\frac{C + t}{3}}) + p = - 700 e^{\frac{C}{3} + \frac{t}{3}}$

Step 3.4.4.6.3.3.3

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

$p (- 1 e^{\frac{C + t}{3}}) + p \cdot 1 = - 700 e^{\frac{C}{3} + \frac{t}{3}}$

Step 3.4.4.6.3.3.4

Factor $p$ out of $p (- 1 e^{\frac{C + t}{3}}) + p \cdot 1$ .

$p (- 1 e^{\frac{C + t}{3}} + 1) = - 700 e^{\frac{C}{3} + \frac{t}{3}}$

Step 3.4.4.6.3.4

Rewrite $- 1 e^{\frac{C + t}{3}}$ as $- e^{\frac{C + t}{3}}$ .

$p (- e^{\frac{C + t}{3}} + 1) = - 700 e^{\frac{C}{3} + \frac{t}{3}}$

Step 3.4.4.6.3.5

Divide each term in $p (- e^{\frac{C + t}{3}} + 1) = - 700 e^{\frac{C}{3} + \frac{t}{3}}$ by $- e^{\frac{C + t}{3}} + 1$ and simplify.

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Step 3.4.4.6.3.5.1

Divide each term in $p (- e^{\frac{C + t}{3}} + 1) = - 700 e^{\frac{C}{3} + \frac{t}{3}}$ by $- e^{\frac{C + t}{3}} + 1$ .

$\frac{p (- e^{\frac{C + t}{3}} + 1)}{- e^{\frac{C + t}{3}} + 1} = \frac{- 700 e^{\frac{C}{3} + \frac{t}{3}}}{- e^{\frac{C + t}{3}} + 1}$

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 $- e^{\frac{C + t}{3}} + 1$ .

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Step 3.4.4.6.3.5.2.1.1

Cancel the common factor.

$\frac{p (- e^{\frac{C + t}{3}} + 1)}{- e^{\frac{C + t}{3}} + 1} = \frac{- 700 e^{\frac{C}{3} + \frac{t}{3}}}{- e^{\frac{C + t}{3}} + 1}$

Step 3.4.4.6.3.5.2.1.2

Divide $p$ by $1$ .

$p = \frac{- 700 e^{\frac{C}{3} + \frac{t}{3}}}{- e^{\frac{C + t}{3}} + 1}$

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.

$p = \frac{- 700 e^{\frac{C + t}{3}}}{- e^{\frac{C + t}{3}} + 1}$

Step 3.4.4.6.3.5.3.2

Move the negative in front of the fraction.

$p = - \frac{700 e^{\frac{C + t}{3}}}{- e^{\frac{C + t}{3}} + 1}$

Step 3.4.4.6.3.5.3.3

Factor $- 1$ out of $- e^{\frac{C + t}{3}}$ .

$p = - \frac{700 e^{\frac{C + t}{3}}}{- e^{\frac{C + t}{3}} + 1}$

Step 3.4.4.6.3.5.3.4

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

$p = - \frac{700 e^{\frac{C + t}{3}}}{- e^{\frac{C + t}{3}} - 1 (- 1)}$

Step 3.4.4.6.3.5.3.5

Factor $- 1$ out of $- e^{\frac{C + t}{3}} - 1 (- 1)$ .

$p = - \frac{700 e^{\frac{C + t}{3}}}{- (e^{\frac{C + t}{3}} - 1)}$

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 $- (e^{\frac{C + t}{3}} - 1)$ as $- 1 (e^{\frac{C + t}{3}} - 1)$ .

$p = - \frac{700 e^{\frac{C + t}{3}}}{- 1 (e^{\frac{C + t}{3}} - 1)}$

Step 3.4.4.6.3.5.3.6.2

Move the negative in front of the fraction.

$p = - - \frac{700 e^{\frac{C + t}{3}}}{e^{\frac{C + t}{3}} - 1}$

Step 3.4.4.6.3.5.3.6.3

Multiply $- 1$ by $- 1$ .

$p = 1 \frac{700 e^{\frac{C + t}{3}}}{e^{\frac{C + t}{3}} - 1}$

Step 3.4.4.6.3.5.3.6.4

Multiply $\frac{700 e^{\frac{C + t}{3}}}{e^{\frac{C + t}{3}} - 1}$ by $1$ .

$p = \frac{700 e^{\frac{C + t}{3}}}{e^{\frac{C + t}{3}} - 1}$

Step 3.4.4.7