Answer
(a) $L=−0.2754~P^2+19.7485~P−273.5523$
(b) $\frac{dL}{dP} = −0.5508~P+19.7485$
When $P=30~~$ then $~~\frac{dL}{dP} = 3.2245$
When $P=40~~$ then $~~\frac{dL}{dP} = -2.2835$
The derivative $\frac{dL}{dP}$ is the rate of change of the tire life for each unit of increase in pressure.
The units of $\frac{dL}{dP}$ are $\frac{thousands~of~miles}{lb/in^2}$
When the sign of $\frac{dL}{dP}$ is positive, it means that a one unit increase in pressure will result in an increase in the tire life.
When the sign of $\frac{dL}{dP}$ is negative, it means that a one unit increase in pressure will result in a decrease in the tire life.
Work Step by Step
(a) With the quadratic regression function on a calculator, we can use the data in the table to find the following function:
$L=−0.2754~P^2+19.7485~P−273.5523$
(b) $\frac{dL}{dP} = −0.5508~P+19.7485$
When $P=30$:
$\frac{dL}{dP} = −0.5508~(30)+19.7485 = 3.2245$
When $P=40$:
$\frac{dL}{dP} = −0.5508~(40)+19.7485 = -2.2835$
The derivative $\frac{dL}{dP}$ is the rate of change of the tire life for each unit of increase in pressure.
The units of $\frac{dL}{dP}$ are $\frac{thousands~of~miles}{lb/in^2}$
When the sign of $\frac{dL}{dP}$ is positive, it means that a one unit increase in pressure will result in an increase in the tire life.
When the sign of $\frac{dL}{dP}$ is negative, it means that a one unit increase in pressure will result in a decrease in the tire life.
