A man drove 48 more miles an hour than the number of hours he drove. if he drove 208 miles in all, how many hours did he drive? What was his rate?

if he drove x hours, his speed was x+48

distance = speed * time, so

(x+48)(x) = 208
x^2 + 48x - 208 = 0
(x-4)(x+52)
x=4

so, he drove 52 mph for 4 hours

Let's assume the number of hours the man drove is represented by x.

According to the given information, the man drove 48 more miles per hour than the number of hours he drove. This means his rate of driving can be represented as x + 48 miles per hour.

As per the question, the man drove a total distance of 208 miles. Therefore, we can set up the equation:

Distance = Rate × Time
208 = (x + 48) × x

Next, we can simplify the equation by expanding:

208 = x^2 + 48x

Now, rearranging the equation into standard form, we have:

x^2 + 48x - 208 = 0

To solve this quadratic equation, we can either factorize it or use the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = 48, and c = -208. Substituting these values into the quadratic formula, we get:

x = (-48 ± √(48^2 - 4(1)(-208))) / (2(1))

Simplifying further:

x = (-48 ± √(2304 + 832)) / 2
x = (-48 ± √3136) / 2
x = (-48 ± 56) / 2

Now, calculate both values of x:

x1 = (-48 + 56) / 2 = 8 / 2 = 4
x2 = (-48 - 56) / 2 = -104 / 2 = -52

Since time cannot be negative, we disregard the negative value. Hence, the man drove for 4 hours.

To find his rate, we substitute the value of x into the expression x + 48:

Rate = x + 48 = 4 + 48 = 52 miles per hour

Therefore, the man drove for 4 hours and his rate was 52 miles per hour.

To solve this problem, let's break it down step by step.

Let's assume that the number of hours the man drove is "x".

According to the given information, the man drove 48 more miles per hour than the number of hours he drove. So, his rate of driving can be calculated as x + 48 miles per hour.

The total distance he drove is given as 208 miles. We can set up an equation as follows:

Distance = Rate × Time
208 = (x + 48) × x

To solve this quadratic equation, we need to rearrange it and set it equal to zero:

0 = x² + 48x - 208

Next, we can solve this quadratic equation either by factoring, completing the square, or by using the quadratic formula. For simplicity, we'll use factoring.

Factoring 0 = x² + 48x - 208:

0 = (x - 8) (x + 26)

Setting each factor equal to zero:

x - 8 = 0 or x + 26 = 0

Solving for x in each equation:

x = 8 or x = -26

Since the number of hours can't be negative, we discard x = -26 as an extraneous solution.

Hence, the man drove for 8 hours.

To calculate his rate of driving, we substitute this value back into the equation x + 48:

Rate = x + 48
Rate = 8 + 48
Rate = 56 miles per hour

Therefore, the man drove for 8 hours and his rate of driving was 56 miles per hour.