Question

application: the local police need a math expert to help them find an appropriate formula for speed (km/min) which will help establish who will be getting fines over that section of the highway. fines are given if the speed exceeds 60 km/h. you are given the following information: cars over a given stretch of highway follow the formulas for distance and time given: $d = \frac{2x^2 + 5x + 2}{4x^2 - 4}$ in kilometers, and time $t = \frac{x^2 + x - 2}{2x^2 + x - 1}$ in minutes. a) determine the rational expression that represents the speed. (decimals are ok) hint: speed $= \frac{distance}{time}$. dont forget restrictions. b) use your formula to decide if a fine should be given if $x = 20$.

Explanation:

Part (a)

Step1: Recall speed formula

Speed \( s=\frac{d}{t} \), where \( d = \frac{2x^{2}+5x + 2}{4x^{2}-4} \) and \( t=\frac{x^{2}+x - 2}{2x^{2}+x - 1} \). So \( s=\frac{\frac{2x^{2}+5x + 2}{4x^{2}-4}}{\frac{x^{2}+x - 2}{2x^{2}+x - 1}}=\frac{2x^{2}+5x + 2}{4x^{2}-4}\times\frac{2x^{2}+x - 1}{x^{2}+x - 2} \)

Step2: Factor numerators and denominators

Factor \( 2x^{2}+5x + 2=(2x + 1)(x + 2) \), \( 4x^{2}-4 = 4(x^{2}-1)=4(x - 1)(x + 1) \), \( 2x^{2}+x - 1=(2x - 1)(x + 1) \), \( x^{2}+x - 2=(x + 2)(x - 1) \)

Step3: Substitute factored forms

\( s=\frac{(2x + 1)(x + 2)}{4(x - 1)(x + 1)}\times\frac{(2x - 1)(x + 1)}{(x + 2)(x - 1)} \)

Step4: Cancel common factors

Cancel \( (x + 2) \), \( (x + 1) \), then we get \( s=\frac{(2x + 1)(2x - 1)}{4(x - 1)^{2}}=\frac{4x^{2}-1}{4(x - 1)^{2}} \)
Restrictions: Denominators cannot be zero. So \( 4x^{2}-4
eq0\Rightarrow x
eq\pm1 \), \( 2x^{2}+x - 1
eq0\Rightarrow x
eq - 1,\frac{1}{2} \), \( x^{2}+x - 2
eq0\Rightarrow x
eq - 2,1 \). So \( x
eq\pm1,-2,\frac{1}{2} \)

Step1: Substitute \( x = 20 \) into speed formula

\( s=\frac{4(20)^{2}-1}{4(20 - 1)^{2}}=\frac{4\times400-1}{4\times361}=\frac{1600 - 1}{1444}=\frac{1599}{1444}\approx1.107 \) km/min

Step2: Convert speed to km/h

Since 1 hour = 60 minutes, speed in km/h is \( 1.107\times60\approx66.42 \) km/h

Step3: Compare with 60 km/h

Since \( 66.42>60 \), a fine should be given

Answer:

The speed formula is \( \boldsymbol{s=\frac{4x^{2}-1}{4(x - 1)^{2}}} \) with restrictions \( x
eq\pm1,-2,\frac{1}{2} \)

Part (b)