Solve the problem. A man rode a bicycle for 12 miles and then hiked an additional 8 miles. The total time for the trip was 5 hours. If his rate when he was riding a bicycle was 10 miles per hour faster than his rate walking, what was each rate?
let walking rate be x mph
let biking rate be x+10
time for walking = 12/x
time biking = 8/(x+10)
isn't 12/x + 8(x+10) = 5 ?
multiply each term by x(x+10)
you will get quadratic, solve for x
Let the walking speed be v and the bike speed V = v + 10.
12/(v+10) + 8/v = 5 is the sum of the times spent bidking and walking.
Solve that equation for v. You will have to start with a common denominator.
[12v + 8(v+10)]/[v(v+10)] = 5
5 v^2 + 50 v = 20 v + 80
v^2 + 6v - 16 = 0
(v + 8)(v -2) = 0
Choose the positive root, v = 2 miles/hr
The bike speed was then v + 10 = 12 mph.
go with drwls solution, I mixed up the two rates.
Solve. V^2-6v-16=0
To solve this problem, we will use the concept of rates and time. Let's assume that the man's rate walking is "x" miles per hour.
Since we know that his rate when riding a bicycle was 10 miles per hour faster than his rate walking, his rate when riding a bicycle would be "x+10" miles per hour.
The time it takes for the man to ride a bicycle for 12 miles can be calculated using the formula: time = distance / rate.
So, the time taken for the bicycle ride is 12 / (x+10).
Similarly, the time taken for the hike of 8 miles can be calculated as 8 / x.
We are given that the total time for the trip was 5 hours. So, we can set up the equation:
12 / (x+10) + 8 / x = 5
To solve this equation, we can start by clearing the fractions by multiplying throughout by the common denominator, which is x(x+10):
12x + 8(x+10) = 5x(x+10)
Now, simplify the equation:
12x + 8x + 80 = 5x^2 + 50x
Combine like terms:
20x + 80 = 5x^2 + 50x
Rearrange the equation to set it equal to zero:
5x^2 + 50x - 20x - 80 = 0
Combine like terms:
5x^2 + 30x - 80 = 0
This is now a quadratic equation in the form ax^2 + bx + c = 0, where a = 5, b = 30, and c = -80.
To solve the quadratic equation, we can either factor or use the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (-30 ± √(30^2 - 4 * 5 * -80)) / (2 * 5)
Calculating the discriminant:
x = (-30 ± √(900 + 1600)) / 10
x = (-30 ± √2500) / 10
Now, let's simplify further:
x = (-30 ± 50) / 10
We have two solutions:
x1 = (-30 + 50) / 10 = 20 / 10 = 2
x2 = (-30 - 50) / 10 = -80 / 10 = -8
Since rates cannot be negative, the solution x2 = -8 can be disregarded.
Therefore, the man's rate walking (x) is 2 miles per hour, and his rate riding a bicycle (x + 10) is 2 + 10 = 12 miles per hour.