Respuesta :
Given Information:
Distance = 160 miles
Required Information:
Rate = r = ?
Answer:
Rate = r = 20 mph
Step-by-step explanation:
Recall that the rate is given by
Rate = distance/time
[tex]r = \frac{d}{t} \\\\t = \frac{d}{r} \\\\t = \frac{160}{r} \\\\[/tex]
It is given that the same trip would have taken 2 hours longer if he had decreased his speed by 4 mph.
Mathematically,
[tex]\frac{160}{(r-4)} = \frac{160}{r} + 2 \\\\[/tex]
Simplify the equation
[tex]\frac{160}{(r-4)} - \frac{160}{r} = 2 \\\\\frac{160r - 160(r-4)}{r(r-4)} = 2 \\\\\frac{160r - 160r+640)}{r(r-4)} = 2 \\\\160r - 160r+640 = r(r-4) (2) \\\\640 = r(r-4) (2) \\\\640 = (r^2 - 4r) (2) \\\\640 = 2r^2 - 8r \\\\2r^2 - 8r-640 =0 \\\\2(r^2 - 4r-320) =0 \\\\r^2 - 4r-320 =0 \\\\[/tex]
Now we are left with a quadratic equation.
We may solve the quadratic equation using the factorization method
[tex]r^2 - 4r-320 =0 \\\\r^2-20r+16r-320=0 \\\\r(r-20)+16(r-20)=0 \\\\(r-20) (r+16)=0 \\\\[/tex]
So,
[tex](r-20) = 0 \\\\r = 20 \\\\[/tex]
OR
[tex](r+16)=0 \\\\r = -16 \\\\[/tex]
Since rate cannot be negative, discard the negative value of r
Therefore, the rate is
[tex]r = 20 \: mph[/tex]
