college algebra final milestone questions with 100
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1
Angela is an electrical engineer who is testing the voltage of a circuit given a certain current and resistance. She uses the following formula to calculate voltage:
The circuit she tests has a current of amps and a resistance of ohms.
What is the voltage of the circuit?
MILESTONE
volts
volts
volts
volts
RATIONALE
The voltage of the circuit is the product of the current and resistance. Recall that we can write as and
as .
Once we have expressed voltage in terms of i, we need to multiply these two complex numbers by using FOIL.
Multiply the first terms , the outside terms , the inside terms , and the last terms . Next,
evaluate each multiplication.
The expression multiplies to Next, combine like terms.
and can combine to . Now we can simplify the last term, , which contains the imaginary unit
squared. Recall that the i squared is equivalent to short dash 1, which can be substituted in our expression.
i squared is replaced by short dash 1. Next, evaluate .
times is equal to . Finally, combine like terms and .
The voltage is equivalent to , or .
CONCEPT
Complex Numbers in Electrical Engineering
2
Simplify the following expression.
‐3 + 4(a – 3b + 2) – 5(2 – a + 3b)
-a + 3b – 5
, 3a – 9b – 5
9a – 27b – 5
3a – b – 11
RATIONALE
To simplify this expression, we must use the Distributive Property to distribute the number outside parentheses by multiplying each
term inside. There are two instances here. First, we will distribute 4 into (a – 3b + 2).
4(a – 3b + 2) is equivalent to 4a – 12b + 8. Next we have to distribute -5 into (2 – a + 3b). Notice the negative sign in front of .
-5(2 – a + 3b) is equivalent to -10 + 5a – 15b. Now we have to combine like terms.
The terms 4a and 5a combine to 9a, the terms-12b and -15b combine to -27b, and the constant terms -3, 8, and -10 combine to -5.
The expression ‐3 + 4(a – 3b + 2) – 5(2 – a + 3b) is equivalent to 9a – 27b – 5.
CONCEPT
Terms and Factors in Algebraic Expressions
3
Consider the function .
What are the domain and range of this function?
RATIONALE
A Square root function has 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 .
When we add to both sides, we now have on the left side. To isolate x, divide both sides by .
divided by is equivalent to . 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
4
The number of employees who will be able to perform a specific task is given by the equation , where N is the number of people who are able to perform the
task, and t is time (in days).
How long will it take until 100 employees are able to perform the task?
, 8 days
4 days
20 days
16 days
RATIONALE
The number of employees who can perform the task is modeled by this equation. To find how long it will take until 100 employees can perform the task, we
will substitute this value for N and solve for t.
Once N is replaced, we can solve for t. First, divide both sides by 25 to undo 25 multiplied by the square root of t.
On the left, 100 divided by 25 is 4. To undo the square root of t on the right, square both sides.
On the left, 4 squared is 16. It will take 16 days for 100 employees to be able to perform the task.
CONCEPT
Solving Multi-step Equations
5
Briana monitors the number of E. coli infections reported in a certain neighborhood in a given week. The recent numbers are shown in this table:
Week Number of People
0 40
1 60
2 90
3 135
According to her reports, the reported infections are growing at a rate of 50%.
If the number of infections continues to grow exponentially, what will the number of infections be in week 6?
304 people
203 people
297 people
456 people
RATIONALE
In general, exponential growth is modeled using this equation. We will use information from the problem to find values to plug into this equation.
The initial number of infections is , so this is the value for a. The infection rate is , so this is our value for b (remember to write it as a
decimal). We want to know how many infections there will be in week 6, so we will use for the value for x. We will need to solve for y. Start by
simplifying what's inside the parentheses.
1 plus is . Next, take this value to the power of .
to the power of is . Finally, multiply this by 40.
There will be 456 people infected in week 6.
CONCEPT
Exponential Growth
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