Final answer:
Emmett's jogging rate is found by setting up an equation with the given information and then solving the quadratic equation to get the value of his jogging rate, which is 9 mph.
Step-by-step explanation:
To determine Emmett's jogging rate, we can set up an equation based on the information given. The distance to his friend's house is 27 miles. Let x represent Emmett's jogging rate in miles per hour (mph). The rate at which he got a ride back is x + 9 mph, since it's stated that his jogging rate is 9 mph slower than the riding rate.
The time it takes to jog to his friend's house is the distance (27 miles) divided by the jogging rate (x), which can be represented as 27/x hours. Similarly, the time it takes to get a ride back is the distance (27 miles) divided by the riding rate (x + 9), or 27/(x + 9) hours. It's given that the jogging took 4 hours longer, so the equation is 27/x - 27/(x + 9) = 4.
To solve for x, first find a common denominator and subtract:
(27(x + 9) - 27x) / x(x + 9) = 4
Then simplify the equation:
27x + 243 - 27x = 4x(x + 9)
243 = 4x^2 + 36x
Now set up a quadratic equation:
4x^2 + 36x - 243 = 0
Using the quadratic formula, the possible values for x are found (ignoring negative values since the rate cannot be negative):
x = (9 mph)