Car Air Conditioning Psi
can you please help with a calculus exponential growth and decay problem?
you run over a nail. as the air leaks out of your tire, the rate of change of air pressure inside the tire is directly proportional to that pressure.
a) write a differentiable equation that states this fact. evaluate the proportionality constant if the pressure was 35 psi and decreasing at 0.28 psi/min at time zero
b) solve the differentiable equation subject to the initial condition implied in step a
c)what will be the pressure at 10 min after the tire was punctured
d) the car is safe to drive as long as the tire pressure is 12 psi or greater. for how long after the puncture will the car be safe to drive?
thank you!!!!
A fairly routine first semester calculus question.
You ought to know that: “rate of change” = “derivative”
So y=”air pressure”
You should also know that “is directly proportional to”
means “equals a constant times…” or ” = k* ”
Part (a), the differential equation, is a direct translation of the sentence preceding it.
Note: What is x=? (The variable.)
Hint: They give a value of pressure (y) of 35 psi. So you know units for y.
They then give a value for the rate of change (dy/dx) as 0.28 psi/min at time zero.
That should tell you what the variable (x) ought to be.
Part (b) involves some familiarity with derivatives.
What is the only function whose derivative is a multiple of the original function?
Note:
When you see multiplication by a constant in a derivative, think CHAIN RULE:
(d/dx)F(kx) = kF’(kx)
So, what function is it’s own derivative?
Another Note:
Constant Rule: Multiply a function by any constant and the derivative is multiplied by the same constant. So, if (d/dx)[F(x)]=F(x), then (d/dx)[A*F(x)]=A*F(x) for any constant A. The value F(0)=35 in part (a) allows you to determine which constant “A” is needed.
Parts (c) [Where do you substitute that 10 minutes? (Which letter represents "minutes"?)] and (d) [Where do you substitute that 12 psi? (Which letter represents "psi"?)] are strictly high school prerequisites (basic equation evaluation/solving) to be mastered before the study of calculus is commenced.
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