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Problem 54 Medium Difficulty

Car tires need to be inflated properly because overinflation or underinflation can cause premature tread wear. The data in tge tabe shoe tire life $ L $ (in thousands of miles) for a certain type of tire at various pressures $ P (in lb/in^2). $

(a) Use a calculator to model tire life with a quadratic function of the pressure.
(b) Use the model to estimate $ dL/dP $ when $ P = 30 $ and when $ P = 40. $ What is the meaning of the derivative? What are the units? What is the significance of the signs of the derivatives?

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Video Transcript

it's clear someone named Reed here. So we're finding the tire tire life and thousands so we'll get and I could have 275 0.4 to 77 three piece square plus 19,748 0.5 to 7. Five p minus 2.7 35 for part B. We're gonna use our equation for l that we call it and differentiate it. So I get d l over d p this sequel to 0.55 through 896 p plus 19 0.748 5 to 8 to differentiate. I didn't use the equation above. I used the equation known in thousands. So when P is equal to 30 yet approximately 3.22 and when P is equal to 40 get approximately negative 2.286 and the unit are inches cubes per pound, and the derivative is the rate of change of life, of tyre and respect to the pressure. And when the derivative is positive, it means that, um, there's a further increase in pressure will increase the life of the tire and for negative means that a further increase in pressure will decrease the life of the tire