Sunday, December 09, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

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June 2017, Part III

All Questions in Part III are worth up to 4 credits. Partial credit is possible.


33. Solve the following system of equations algebraically for all values of x, y, and z:

x + y + z = 1
2x + 4y + 6z = 2
-x + 3y - 5z = 11

Answer:
Any system of three equations with three variables can be solved by eliminating one of the variables and then solving the remaining system of equations with two variables.

I was away from the PC for a while the other day, so I worked the solution out long hand, two different ways. First by using elimination, then by using substitution, to remove the x. After that, the work would be the same.

Note that the 2x + 2y + 2z = 2 came from multiplying the first equation by 2.
At this point, by either method, you should be left with the two equations:

2y + 4z = 0 and
4y - 4z = 12
Add these together, and the z term is eliminated, allowing you to solve for y.
Once you have y, you can find z.
Once you have y and z, you can find x.



34. Jim is looking to buy a vacation home for $172,600 near his favorite southern beach. The formula to compute a mortgage payment, M, is M = P • ( r (1+r)N) / ( (1 + r)N - 1) where P is the principal amount of the loan, r is the monthly interest rate, and N is the number of monthly payments. Jim’s bank offers a monthly interest rate of 0.305% for a 15-year mortgage.

With no down payment, determine Jim’s mortgage payment, rounded to the nearest dollar.

Algebraically determine and state the down payment, rounded to the nearest dollar, that Jim needs to make in order for his mortgage payment to be $1100.

Answer:
For the first part of the question, use the given formula with the following substitutions:
P = 172,600, r = 0.00305, N = 12 * 15 = 180 payments
M = P • ( r (1+r)N) / ( (1 + r)N - 1)
M = 172600 • ( 0.00305 (1+0.00305)180) / ( (1 + 0.00305)180 - 1)
M = 1247.4933... = 1247 to the nearest dollar. (Round 49 cents down.)

The down payment reduces the Principal that needs to be borrowed. To answer the second part, we need to solve for P when M = 1100. Once we know that, we need to subtract 172600 - P to find the amount of the down payment.
M = P • ( r (1+r)N) / ( (1 + r)N - 1)
1100 = P • ( 0.00305 (1+0.00305)180) / ( (1 + 0.00305)180 - 1)
P = 1100 / • ( 0.00305 (1+0.00305)180) / ( (1 + 0.00305)180 - 1)
P = 152193.19...
The down payment is 172600 - 152193.19 = 20406.81 = 20407.



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Friday, December 07, 2018

(x, why?) Mini: Revolution

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(C)Copyright 2018, C. Burke.

Why, yes, I have been covering science classes again. Why do you ask?




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Thursday, December 06, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

June 2017, Part II

All Questions in Part II are worth up tp 2 credits. Partial credit is possible.


31. Write (x)(1/3) * (x)(1/2) as a single term with a rational exponent.


Answer:
First of all, please look at the image. This is how the question was asked. I typed it the best way that I could for this medium.
That said, you're first step is to rewrite the equation as:

(x)(1/3) * (x)(1/2)

Now use the rules of exponents:
(x)(1/3) * (x)(1/2) = (x)(5/6)

Leave your answer in this form with the rational (fractional) exponent. Do NOT rewrite it with a radical symbol.

Note: (1/3) + (1/2) = (2/6) + (3/6) = (5/6)





32. Data collected about jogging from students with two older siblings in the table below.

Neither Sibling Jogs One Sibling Jogs Both Sibling Jog
Student Does Not Jog 1168 1823 1380
Student Jogs 188 416 400

Using these data, determine whether a student with two older siblings is more likely to jog if one sibling jogs or if both siblings job. Justify your answer.

Answer:
Sum up the second and the third columns. The number of students with one sibling who jogs is 1823 + 416 = 2239. The number of students with two siblings who jog is 1380 + 400 = 1780.

The probability that a student jogs if one sibling jogs is
P(student | one sibling) = 416/2239 = .186

The probability that a student jogs if both siblings jog is
P(student | both sibling) = 400/1780 = .225

It is more likely that a student with two older siblings will jog if both of their older siblings jog.



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Tuesday, December 04, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

June 2017, Part II

All Questions in Part II are worth up tp 2 credits. Partial credit is possible.


29. Graph y = 400(.85)2x - 6 on the set of axes below. (image omitted, see below)

Answer:
See the graph below.
The x-axis has a scale of 0.5, but using integer values for x is sufficient for full credit. The y-axis has a scale of 20.
I included a table of values for reference. It isn't needed, but it's not a bad idea to include it, or to label the points.
This was a very straightforward probably, using the graphing calculator. A little time-consuming, but simple.





30.Solve algebraically for all values of x:

sqrt(x - 4) + x = 6


Answer:
Follow along on the image below. Subtract x from both sides of the equation to isolate the radical. Next, square both sides of the equation. Subtract (x - 4) from both sides to set the quadratic equation equal to 0. Factor the trinomial.





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Monday, December 03, 2018

Elemeno P

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(C)Copyright 2018, C. Burke.

''Elemental P'' might've been a rockier pun.

This is the dude from the Alphabet Song.




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Saturday, December 01, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

June 2017, Part II

All Questions in Part II are worth up tp 2 credits. Partial credit is possible.


27.Over the set of integers, factor the expression 4x3 - x2 + 16x - 4 completely.

Answer:
Group the terms into the pairs. You will notice that in the first two terms, you can factor out x2, as follows: x2(4x - 1).
In the last two terms, you can factor out a 4, as follows: (4)(4x - 1).
This gives you x2(4x - 1) + (4)(4x - 1).
Factor out the (4x - 1) from the two terms, and you get (x2 + 4)(4x - 1).
(Distributive Property in reverse.)
You can check your answer by multiplying (FOIL, box method) and getting your original polynomial back.





28. The graph below represents the height above the ground, h, in inches, of a point on a triathlete's bike wheel during a training ride in terms of time, t, in seconds.


Identify the period of the graph and describe what the period represents in this context.

Answer:
First of all, you don't not need to know what a "triathlete" is, other than that it's someone who is riding a bike. Okay? Don't freak out over the word if you've never seen it before.
The period of the graph is 2/3, which you can see because the graph starts at a height of 26 and returns to 26 after 2/3 seconds.
The period represents the time it takes for a point at the top of the wheel to make one complete revolution and return back to the top of the wheel.



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