4.8 mi
Explanation
[tex]\text{time}=\text{ }\frac{\text{distance}}{\text{rate}}[/tex]
Step 1
Set the equations
a) uphill
let
rate1= 1.5 miles per hour
time= unknow= t1
distance = x
b) down hille
rate=4 miles per hour
time=time2=one hour less than the time to climb = t1-1
distance = x
so
replacing
[tex]\begin{gathered} t_1=\frac{x}{1.5\frac{mi}{\text{hour}}}\rightarrow t_1=\frac{x}{1.5}\rightarrow equation(1) \\ t_2=\frac{x}{4\frac{mi}{\text{hour}}} \\ \text{replace t}_2=t_1-1 \\ t_1-1=\frac{x}{4} \\ \text{add 1 in both sides} \\ t_1-1+1=\frac{x}{4}+1 \\ t_1=\frac{x}{4}+1\rightarrow equation(2) \end{gathered}[/tex]
Step 2
solve the equations
[tex]\begin{gathered} t_1=\frac{x}{1.5}\rightarrow equation(1) \\ t_1=\frac{x}{4}+1\rightarrow equation(2) \end{gathered}[/tex]
set t1= t1
[tex]\begin{gathered} t_1=t_1 \\ \frac{x}{1.5}=\frac{x}{4}+1 \\ \frac{x}{1.5}=\frac{x+4}{4} \\ 4x=(x+4)1.5 \\ 4x=1.5x+6 \\ subtract\text{ 1.5 x in both sides} \\ 4x-1.5x=1.5x+6-1.5x \\ 2.5x=6 \\ \text{divide both sides by 2.5} \\ \frac{2.5x}{2.5}=\frac{6}{2.5} \\ x=2.4 \end{gathered}[/tex]
it means the distance to the top of the mountain is 2.4 miles, so the entire hike is twice that amount
total distance=2.4 mi*2
total distance=4.8 miles
Step 3
now, the times
[tex]\begin{gathered} t_1=\frac{x}{1.5} \\ t_1=\frac{2.4}{1.5} \\ t_1=1.6\text{ hours} \\ t_2=t_1-1 \\ t_2=1.6-1=\text{ 0.6 hours} \end{gathered}[/tex]
table
I hope this helps you
