Use the distance formula, d=rt, to write an expression for the time it takes to run each half of the race.
About 7.54 mi/h
Practice makes perfect
Let's begin by breaking down the 10-miles race into two halves to work the information given. During the first half of the race, our speed is 2 mi/h greater than the speed in the second half. Let r be the speed during the second half. Let's write a table with this information!
Distance (mi)
Speed (mi/h)
First Half
5
r+2
Second Half
5
r
We can use the distance formula to find the time taken in each half of the race.
d=rt ⇒ t=d/r
We will include this information in our table.
Distance (mi)
Speed (mi/h)
Time (h)
First Half
5
r+2
5/r+2
Second Half
5
r
5/r
We are told that we take 94 minutes to complete the whole race. Let's convert this to hours using a conversion factor.
We can now add the time taken in each half of the race to write an equation for r.
5/r+2+5/r=47/30
Let's highlight the different factors in the denominators. This will help us find the least common denominator (LCD).
5/r+2+5/r=47/30
Now we will solve this equation by multiplying by the LCD to clear the denominators.
Finally, we will solve this equation using the Quadratic Formula.
ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 a
In this case, the variable is r. We can identify the values of a, b, and c.
47r^2-206r-300=0 ⇓
47r^2+( -206)r+ -300=0
Let's substitute these values into the Quadratic Formula.
Since the speed cannot be negative, we take r=5.54. The average speed during the second half of the race is about 5.54 miles per hour. Therefore, the average speed during the first half of the race is about 7.54 miles per hour.
