Pete's Plymouth travels 200 miles at a certain speed. If the car had gone 10 mph faster, the trip would have taken 1 hour less. Find Pete's speed.
I am very confused about how to solve this problem! Please explain well. Thank you so much!!! :-)
let his original speed be x mph
then his original time is 200/x hours
let his second speed be x+10 mph
then his second time is 200/(x+10) hours
so 200/x - 200/(x+10) = 1
multiply both sides by x(x+10)
200x + 2000 - 200x = x^2 + 10x
x^2 + 10x - 2000 = 0
(x-40)(x+50) = 0
x = 40 or x= -50 (negative speed makes no sense)
so Pete's speed is 40 mph.
check: at 40 mph, it would take him 200/40 = 5 hours
at 50 mph it would have taken him 200/50 = 4 hours.
OK then!
Lat V be the slower speed and T be the time it takes to go 200 miles at that speed. You can write these two equations:
T = 200/V
T = 200/(V + 10) + 1
Now get rid of the T and create and solve an equation for v alone.
200 [1/V - 1/(V+10) = 1
200(V + 10 - V) = V(V+10)
V^2 + 10V -2000 = 0
(V-40)(V+50) = 0
V = 40 since only positive roots are allowed.
To solve this problem, let's break it down step by step.
Let's start by assigning variables to the given information:
Let Pete's speed be represented by s (in mph).
Let the time it takes for Pete to travel the 200 miles at the given speed s be represented by t (in hours).
According to the question, if Pete had gone 10 mph faster, the trip would have taken 1 hour less. This can be expressed as:
t - 1 = time taken at s + 10 mph
Now, let's use the formula for speed:
speed = distance / time
We can write two equations using this formula:
1) For the speed s:
s = 200 / t
2) For the speed s + 10:
s + 10 = 200 / (t - 1)
Now we have a system of two equations. We can solve them simultaneously to find the value of s.
To solve the system, we'll use a method called substitution:
From equation 1, we can express t in terms of s:
t = 200 / s
Substituting this value of t into equation 2, we get:
s + 10 = 200 / (200 / s - 1)
Now, we need to simplify this equation:
s + 10 = 200 / (200 - s)
Now, we'll cross-multiply to eliminate the fraction:
s(200 - s) = 200(200 - s) / (s + 10)
Expanding both sides of the equation:
200s - s^2 = 200(200 - s) / (s + 10)
Multiply both sides of the equation by (s + 10) to eliminate the fraction:
(s + 10)(200s - s^2) = 200(200 - s)
Expanding both sides:
200s^2 - s^3 + 2000s - 10s^2 = 40000 - 200s
Rearranging the terms:
s^3 - 190s^2 + 2000s - 40000 = 0
Now we have a cubic equation. We can solve it using numerical methods or a calculator.
Once you have the value of s, it will represent Pete's speed.