Chapter 1.7, Problem 114E

Bonfire Temperature In the vicinity of a bonfire the temperature T in °C at a distance of x meters from the center of the fire was given by

$T=\frac{600,000}{x^2+300}$

At what range of distances from the fire’s center was the temperature less than 500°C?

To determine

The range of the distances from the fire’s center where the temperature is less than $500°\text{C}$.

Answer to Problem 114E

The temperature is less than $500°\text{C}$ if the distance from the fire’s center is more than $\underset{\_}{30\text{m}}$.

Explanation of Solution

Given:

The temperature T in $°\text{C}$ at a distance of x meters from the center of the fire was given by $T=\frac{600,000}{x^2+300}$.

Calculation:

To obtain the range of the distances from the fire’s center where the temperature is less than $500°\text{C}$, set $T<500$ and solve this inequality.

That is, solve the inequality $\frac{600,000}{x^2+300}<500$.

Simplify the above inequality as follows.

$\begin{array}{l}600,000<500\left(x^2+300\right)\left[\text{Multiply both sides by }\left(x^2+300\right)\right]\\ 1200<x^2+300\left[\text{Divide both sides by }500\right]\\ 900<x^2\left[\text{Subtract 300 from both sides}\right]\\ x^2>900\end{array}$

Simplify further as follows.

$\begin{array}{l}{x}^2>900\\ \left|x\right|>\sqrt{900}\\ \left|x\right|>30\\ x>30\text{or }x<-30\left[\because \left|x\right|>a\Rightarrow x>a\text{or }x<-a\right]\end{array}$

Since x represents the distance, x will be positive.

Hence, the temperature is less than $500°\text{C}$ if the distance from the fire’s center is more than $\underset{\_}{30\text{m}}$.