Sat practice Test #8 Answer Explanations

The correct answer is 25.4. The average speed is the total distance 
divided by the total time. The total distance is 11 miles and the total 
time is 26 minutes. Thus, the average speed is 11 
_ 
26 miles per minute. 
The question asks for the average speed in miles per hour, and there 
are 60 minutes in an hour; converting miles per minute to miles per 
hour gives the following:
Average speed = 11 miles 
—
26 minutes ×
60 minutes
__ 
1 hour
= 660 
_ 
26 miles per hour 
≈ 25.38 miles per hour
Therefore, to the nearest tenth of a mile per hour, the average speed of 
Paul Revere’s ride would have been 25.4 miles per hour.

---

QUESTION 35
The correct answers are 2 and 8. Substituting x = a in the definitions 
for f and g gives f (a) = − 1 
_ 
2 (a − 4)
2
+ 10 and g(a) = −a + 10, respectively. 
If f (a) = g(a), then − 1 
_ 
2 (a − 4)
2
+ 10 = −a + 10. Subtracting 10 from both 
sides of this equation gives − 1 
_ 
2 (a − 4)
2
= −a. Multiplying both sides 
by −2 gives (a − 4)
2
= 2a. Expanding (a − 4)
2
gives a
2
− 8a + 16 = 2a. 
Combining the like terms on one side of the equation gives
a 
2
− 10a + 16 = 0. One way to solve this equation is to factor
a 
2
− 10a + 16 by identifying two numbers with a sum of −10 and
a product of 16. These numbers are −2 and −8, so the quadratic 
equation can be factored as (a − 2)(a − 8) = 0. Therefore, the 
possible values of a are either 2 or 8. Either 2 or 8 will be scored as 
a correct answer.
Alternate approach: Graphically, the condition f (a) = g(a) implies the 
graphs of the functions y = f (x) and y = g(x) intersect at x = a. The 
graph y = f (x) is given, and the graph of y = g(x) may be sketched as 
a line with y-intercept 10 and a slope of −1 (taking care to note the 
different scales on each axis). These two graphs intersect at x = 2
and x = 8.
QUESTION 36