Related Rates

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The fundamental process of solving a related rates problem

• To solve a related rates problem, you have to use a formula and relate that formula to a change, usually a change in time.

Example problem

If y=x³ find dy/dt at t=1 if x=2 and dx/dt=4.

First step: Take an implicit derivative with respect to t. It has to be implicit because y and x are in terms of t.

Work:

d/dt[y]=d/dt[x³]

dy/dt[1]=3x²dx/dt

Second step:Plug in the values for t=1 and the values that show what’s happening at t=1 for the values of x and dx/dt.

Work:

3(2)² multiplied by 4

dy/dt = 48 units- This means that y is changing at 48 units per unit of time. This is the final answer for this problem.

Other example problem

Problem: There’s an oil spill, and the radius is spreading at a rate of 3 ft/s. How fast is the area increasing if r=30 fr?

First step: Assign variables. We know based on the highlighted words that we need a variable for time, area, and a radius.

T= time

A= area.

R= radius

Second step: Identify the formula that you need for this problem. The formula should use the variables found, with the exception of time, as that’s what implicit differentiation will account for. Therefore, the formula will be A=pir² (the formula for a circle).

Third Step: Set up derivatives that will be used in implicit differentiation

dA/dt= rate of change in area

dR/dt= rate of change of radians -The problem states that the radius is spreading at a rate of 3ft/s.

Fourth step: Perform implicit differentiation by differentiating with respect to time and plug in values.

d/dt[A]=d/dt[pir²]

dA/dt=2pirdr/dt

*We only needed to use power rule to differentiate d/dt[pir²] because pi is a constant.

dA/dT=2pi multiplied by 30 multiplied by 3 ft/second

dA/dT=180pi ft²/second

*We had to use square feet to make sure our units were adequate for area.