Math 22 Related Rates

Related Variables

We will study problems involving variables that are changing with respect to time. If two or more such variables are related to each other, then their rates of change with respect to time are also related.

Example 1: Given Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y^{3}=x^{2}y+3} Find ${\frac {dy}{dt}}$ when $x=3$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=1} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {dx}{dt}}=1}

Solution: Differentiate both sides of the equation with respect to $t$:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {d}{dt}}[y^{3}]={\frac {d}{dt}}[x^{2}y+3]}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =3y^{2}{\frac {dy}{dt}}=2x{\frac {dx}{dt}}y+x^{2}{\frac {dy}{dt}}}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =3y^{2}{\frac {dy}{dt}}-x^{2}{\frac {dy}{dt}}=2xy{\frac {dx}{dt}}}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =(3y^{2}-x^{2}){\frac {dy}{dt}}=2xy{\frac {dx}{dt}}}

Hence, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {dy}{dt}}={\frac {2xy{\frac {dx}{dt}}}{3y^{2}-x^{2}}}} .

Substitute, we get Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {dy}{dt}}={\frac {2(3)(1)(1)}{3(1)^{2}-(3)^{2}}}=-1}

Example 2: The revenue $R$ from selling $x$ units of a product is given by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R=1200x-x^{2}} . The sales are increasing at a rate of $30$ units per day. Find the rate of change of the revenue when Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=400}

Solution: The sales are increasing at a rate of $30$ units per day. So, ${\frac {dx}{dt}}=30$. we want to find the rate of change of the revenue, this means we want to find Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {dR}{dt}}} . So, differentiate both sides of the equation with respect to $t$ to get:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {d}{dt}}[R]={\frac {d}{dt}}[1200x-x^{2}]}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ={\frac {dR}{dt}}=1200{\frac {dx}{dt}}-2x{\frac {dx}{dt}}}

Substitute to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dR}{dt}=1200(30)-2(400)(30)=12000}

Therefore, the rate of change of the revenue is $12000 per day.

Guidelines for Solving a Related-Rate Problem

 1. Identify all given quantities and all quantities to be determined. If possible, make a sketch 
 and label the quantities.
 
 2. Identify all given quantities and all quantities to be determined. If possible, make a sketch 
 and label the quantities.
 
 3. Use the Chain Rule to implicitly differentiate both sides of the equation with respect to 
 time.
 
 4. After completing Step 3, substitute into the resulting equation all known values for the 
 variables and their rates of change. Then solve for the required rate of change.

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