The length L of a metal rod is given in terms of the temperature T(^∘ C) by L=39.6+0.025 T cm .

step 1 Here we are given that the length follows the expression L is equal to 39 .6 plus 0 .025T, where

The length L of a metal rod is given in terms of the...

The length $L$ of a metal rod is given in terms of the temperature $T\left(^{\circ} \mathrm{C}\right)$ by $L=39.6+0.025 T \mathrm{cm} .$ Within what range of temperature must the rod be kept if its length must be maintained within $\pm 1 \mathrm{mm}$ of $40 \mathrm{cm} ?$

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Key Concepts

Solving Inequalities

This involves transforming and simplifying inequalities to isolate the variable of interest. Here, it means translating the tolerance condition into an inequality involving the temperature, and then solving that inequality to determine the acceptable range of temperatures.

Unit Conversion

Unit conversion is the process of converting a measurement from one unit to another to ensure consistency in calculations. In this problem, conversion between millimeters and centimeters is crucial since the tolerance and the rod's length are given in different units.

Linear Relationships

This concept involves modeling the relationship between two quantities with a straight-line equation, typically in the form L = a + bT. It shows how one variable changes in a constant rate with respect to the other, which is key to understanding how temperature affects the length of the rod.

Absolute Value Inequalities

Absolute value inequalities express how far a variable is allowed to deviate from a central value. In this context, they are used to represent the tolerance limits—maintaining the rod’s length within a specified range (± a given value) of a target measurement.

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