Sophia :: Welcome https://app.sophia.org/spcc/college-algebra-3/milestone_take_feedback
Score 23/25
23/25 that's 92%
Well done! If you keep getting Milestone scores like this you'll pass the course.
23 questions were answered correctly.
2 questions were answered incorrectly.
Score Improvement Tips
1. Remember that the Milestone is an open book test – use your notes and unit tutorials to help you evaluate the questions and choose possible answers.
2. When you think you have the right answer, read the question again to be sure.
3. Use as much of the time limit as you need – don’t rush through the Milestone.
4. Use the Practice Milestone as your key preparation tool.
5. Take the Milestone at a time and in a place where you can be focused and undisturbed for the entire Milestone time limit.
MILESTONE
OK
1
The Elster family drove 9.25 hours on the first day of their road trip.
How many minutes is this equivalent to?
•
154 minutes
•
555 minutes
•
9,256 minutes
•
33,300 minutes
RATIONALE
In general, we use conversion factors to convert from one unit to another. A conversion factor is a fraction with equal quantities in the numerator and denominator, but written
with different units. We want to convert hours to minutes. We know how many minutes are in 1 hour. We will use this fact to set up a conversion factor.
There are 60 minutes in 1 hour so to convert 9.25 hours into minutes, we will multiply by the fraction fraction numerator 60 space m i n u t e s over
denominator 1 space h o u r end fraction.
Notice how the fractions are set up. The units of hours will cancel, leaving only minutes. Finally we can evaluate the multiplication by multiplying across the numerator and
denominator.
In the numerator, 9.25 times 60 equals 555. 9.25 hours is equivalent to 555 minutes.
CONCEPT
Converting Units
2
Consider the function .
What are the domain and range of this function?
•
•
•
•
1 of 22 2/13/2022, 3:53
,Sophia :: Welcome https://app.sophia.org/spcc/college-algebra-3/milestone_take_feedback
RATIONALE
Square root functions have the domain restriction that the radicand (the value underneath the radical) cannot be negative. To find the specific domain, construct an inequality showing that the
radicand must be greater than or equal to zero.
The expression under the radical, , must be greater than or equal to zero. To solve this inequality, add to both sides to undo the subtraction of .
This is the domain of the function, which means all x values must be greater than or equal to . To find the range, consider the fact that it is not possible for the input of the function to be a
negative number.
For all x-values greater than or equal to , the function will have non-negative values for y that only get bigger and bigger as x increases. The range is all values greater than or equal to zero.
CONCEPT
Finding the Domain and Range of Functions
3
In 2000, the median size of a new single-family house was 2,057 square feet. In 2010, the median size was 2,169 square feet. (Source: www.census.gov.)
What is the average rate of change in the median home size over this time period?
•
11.2 square feet per year
•
112 square feet per year
•
89 square feet per year
•
8.9 square feet per year
RATIONALE
The average rate of change can be computed like slope. It is the change in median home size divided by the change in years. The median home
size changed from square feet in , to square feet in . Substitue these values into the formula.
The change in median home size is the difference between and square feet, or square feet. The change in years is the
difference between and , or . Next, evaluate the subtraction in the numerator and denominator.
equals ; equals . The average rate of change is , which can be simplified.
divided by equals . The average rate of change in the median house size between and is square feet per year.
CONCEPT
Slope in Context
4
Bert is driving along a country road at a constant speed of 30 miles per hour.
How long does it take him to travel 50 miles?
•
An hour and 30 minutes
•
An hour and 40 minutes
•
An hour and 15 minutes
•
An hour and 50 minutes
RATIONALE
To find how long it will take Bert to travel 50 miles, we can use the distance, rate, time formula and solve for time. Plug in 50 miles for the distance, and 30 miles per hour
for the rate, or speed.
Once we have plugged in the values, we need to write miles per hour as a fraction: 30 miles over 1 hour.
When dividing by a fraction, we can change this into a multiplication problem and multiply by the reciprocal of 30 miles per 1 hour, which would be 1 hour over 30 miles.
Rewrite 50 miles as a fraction over 1, and multiply this by the reciprocal of 30 mph. Next, multiply the numerators and denominators of the fractions.
2 of 22 2/13/2022, 3:53
,Sophia :: Welcome https://app.sophia.org/spcc/college-algebra-3/milestone_take_feedback
Multiplying across the numerator and denominator, produces 50 over 30. The units of miles cancel, so we are left with hours. Finally, divide 50 by 30.
It will take Bert about 1.67 hours. However, because we must express our answer in hours and minutes, we must convert 0.67 hours to minutes.
Using the conversion factor, 60 minutes to 1 hour, evaluate the fractions by multiplying the numerators and denominators.
Multiplying across the numerator and denominator produces 0.67 times 60, or approximately 40. Units of hours cancel, leaving only minutes.
It will take Bert about 1 hour and 40 minutes to travel 50 miles at a constant speed of 30 miles per hour.
CONCEPT
Distance, Rate, and Time
5
Write the following as a single rational expression.
•
•
•
•
RATIONALE
Just as with numeric fractions, we can re-write division of algebraic fractions as multiplication and multiply across numerators and denominators. To re-write fraction division as multiplication, re-
write the second fraction as its reciprocal (flipping the numerator and denominator).
changes to and division changes to multiplication. We can now multiply across the numerators and denominators.
times x squared is equal to and x cubed times is equal to . Next, find any common factors in the numerator and denominator.
Both the numerator and denominator have two factors of . We can cancel out these factors and simplify.
Once all common factors have been canceled out in the numerator and denominator, write the fraction in simplest form.
This is the the simplified fraction written as a single rational expression.
CONCEPT
Multiplying and Dividing Rational Expressions
6
Write the following expression as a single complex number.
•
•
•
•
RATIONALE
When multiplying two complex numbers, we can use FOIL.
3 of 22 2/13/2022, 3:53
Score 23/25
23/25 that's 92%
Well done! If you keep getting Milestone scores like this you'll pass the course.
23 questions were answered correctly.
2 questions were answered incorrectly.
Score Improvement Tips
1. Remember that the Milestone is an open book test – use your notes and unit tutorials to help you evaluate the questions and choose possible answers.
2. When you think you have the right answer, read the question again to be sure.
3. Use as much of the time limit as you need – don’t rush through the Milestone.
4. Use the Practice Milestone as your key preparation tool.
5. Take the Milestone at a time and in a place where you can be focused and undisturbed for the entire Milestone time limit.
MILESTONE
OK
1
The Elster family drove 9.25 hours on the first day of their road trip.
How many minutes is this equivalent to?
•
154 minutes
•
555 minutes
•
9,256 minutes
•
33,300 minutes
RATIONALE
In general, we use conversion factors to convert from one unit to another. A conversion factor is a fraction with equal quantities in the numerator and denominator, but written
with different units. We want to convert hours to minutes. We know how many minutes are in 1 hour. We will use this fact to set up a conversion factor.
There are 60 minutes in 1 hour so to convert 9.25 hours into minutes, we will multiply by the fraction fraction numerator 60 space m i n u t e s over
denominator 1 space h o u r end fraction.
Notice how the fractions are set up. The units of hours will cancel, leaving only minutes. Finally we can evaluate the multiplication by multiplying across the numerator and
denominator.
In the numerator, 9.25 times 60 equals 555. 9.25 hours is equivalent to 555 minutes.
CONCEPT
Converting Units
2
Consider the function .
What are the domain and range of this function?
•
•
•
•
1 of 22 2/13/2022, 3:53
,Sophia :: Welcome https://app.sophia.org/spcc/college-algebra-3/milestone_take_feedback
RATIONALE
Square root functions have the domain restriction that the radicand (the value underneath the radical) cannot be negative. To find the specific domain, construct an inequality showing that the
radicand must be greater than or equal to zero.
The expression under the radical, , must be greater than or equal to zero. To solve this inequality, add to both sides to undo the subtraction of .
This is the domain of the function, which means all x values must be greater than or equal to . To find the range, consider the fact that it is not possible for the input of the function to be a
negative number.
For all x-values greater than or equal to , the function will have non-negative values for y that only get bigger and bigger as x increases. The range is all values greater than or equal to zero.
CONCEPT
Finding the Domain and Range of Functions
3
In 2000, the median size of a new single-family house was 2,057 square feet. In 2010, the median size was 2,169 square feet. (Source: www.census.gov.)
What is the average rate of change in the median home size over this time period?
•
11.2 square feet per year
•
112 square feet per year
•
89 square feet per year
•
8.9 square feet per year
RATIONALE
The average rate of change can be computed like slope. It is the change in median home size divided by the change in years. The median home
size changed from square feet in , to square feet in . Substitue these values into the formula.
The change in median home size is the difference between and square feet, or square feet. The change in years is the
difference between and , or . Next, evaluate the subtraction in the numerator and denominator.
equals ; equals . The average rate of change is , which can be simplified.
divided by equals . The average rate of change in the median house size between and is square feet per year.
CONCEPT
Slope in Context
4
Bert is driving along a country road at a constant speed of 30 miles per hour.
How long does it take him to travel 50 miles?
•
An hour and 30 minutes
•
An hour and 40 minutes
•
An hour and 15 minutes
•
An hour and 50 minutes
RATIONALE
To find how long it will take Bert to travel 50 miles, we can use the distance, rate, time formula and solve for time. Plug in 50 miles for the distance, and 30 miles per hour
for the rate, or speed.
Once we have plugged in the values, we need to write miles per hour as a fraction: 30 miles over 1 hour.
When dividing by a fraction, we can change this into a multiplication problem and multiply by the reciprocal of 30 miles per 1 hour, which would be 1 hour over 30 miles.
Rewrite 50 miles as a fraction over 1, and multiply this by the reciprocal of 30 mph. Next, multiply the numerators and denominators of the fractions.
2 of 22 2/13/2022, 3:53
,Sophia :: Welcome https://app.sophia.org/spcc/college-algebra-3/milestone_take_feedback
Multiplying across the numerator and denominator, produces 50 over 30. The units of miles cancel, so we are left with hours. Finally, divide 50 by 30.
It will take Bert about 1.67 hours. However, because we must express our answer in hours and minutes, we must convert 0.67 hours to minutes.
Using the conversion factor, 60 minutes to 1 hour, evaluate the fractions by multiplying the numerators and denominators.
Multiplying across the numerator and denominator produces 0.67 times 60, or approximately 40. Units of hours cancel, leaving only minutes.
It will take Bert about 1 hour and 40 minutes to travel 50 miles at a constant speed of 30 miles per hour.
CONCEPT
Distance, Rate, and Time
5
Write the following as a single rational expression.
•
•
•
•
RATIONALE
Just as with numeric fractions, we can re-write division of algebraic fractions as multiplication and multiply across numerators and denominators. To re-write fraction division as multiplication, re-
write the second fraction as its reciprocal (flipping the numerator and denominator).
changes to and division changes to multiplication. We can now multiply across the numerators and denominators.
times x squared is equal to and x cubed times is equal to . Next, find any common factors in the numerator and denominator.
Both the numerator and denominator have two factors of . We can cancel out these factors and simplify.
Once all common factors have been canceled out in the numerator and denominator, write the fraction in simplest form.
This is the the simplified fraction written as a single rational expression.
CONCEPT
Multiplying and Dividing Rational Expressions
6
Write the following expression as a single complex number.
•
•
•
•
RATIONALE
When multiplying two complex numbers, we can use FOIL.
3 of 22 2/13/2022, 3:53
