Tuesday, May 21, 2019

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.
After a brief hiatus, the Algebra 2 Problems of the Day are back. Hopefully, daily.

More Algebra 2 problems.

January 2019, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


16. Savannah just got contact lenses. Her doctor said she can wear them 2 hours the first day, and can then increase the length of time by 30 minutes each day. If this pattern continues, which formula would not be appropriate to determine the length of time, in either minutes or hours, she could wear her contact lenses on the nth day?
(1) a1 = 120; an = an - 1 + 30
(2) an = 90 + 30n
(3) a1 = 120; an = an - 1 + 0.5
(4) an = 2.5 + 0.5n

Answer: (4) an = 2.5 + 0.5n
On the first day, a1 must be 2 hours or 120 minutes.
In choice (4), 2.5 + 0.5(1) = 3 hours. The initial amount was already too big before anything was added to it.
Choice (1) starts with 120 minutes. Then adds 30 minutes to the previous day's total.
Choice (2) starts with 90 minutes plus an increment of 30 for a total of 120 on the first day
Choice (3) is the same as (1) except converted to hours.





17. If f(x) = ax, where a > 1, then the inverse of the function is

(1) f-1(x) = logx a
(2) f-1(x) = a log x
(3) f-1(x) = loga x
(4) f-1(x) = x log a

Answer: (3) f-1(x) = loga x
In ax, a is the base. When you take the log, a is the base.
Ex: If a=2, the f(3) = 23 = 8
f-1 ( f(3) ) = f-1(8) = log 2 8 = 3





18. Kelly-Ann has $20,000 to invest. She puts half of the money into an account that grows at an annual rate of 0.9% compounded monthly. At the same time, she puts the other half of the money into an account that grows continuously at an annual rate of 0.8%.
Which function represents the value of Kelly-Ann’s investments after t years?

(1) f(t) = 10,000(1.9)t + 10,000e0.8t
(2) f(t) = 10,000(1.009)t + 10,000e0.008t
(3) f(t) = 10,000(1.075)12t + 10,000e0.8t
(4) f(t) = 10,000(1.00075)12t + 10,000e0.008t

Answer: (4) f(t) = 10,000(1.00075)12t + 10,000e0.008t
This is almost purely a notation problem.
The choices with .9 and .8 should be eliminated immediately.
Similarly, 0.75 is .9 / 12, so that's eliminated.
Because the first account is compounded monthly and there are 12 months in a year, the 0.009 is divided by 12, giving .00075, and the exponent gets multiplied by 12.



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Monday, May 20, 2019

(x, why?) Mini: Directrix

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(C)Copyright 2019, C. Burke.

Something ... something ... follow directions ... something.

Sometimes the jokes write themselves. Sometimes they don't.




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Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.
After a brief hiatus, the Algebra 2 Problems of the Day are back. Hopefully, daily.

More Algebra 2 problems.

January 2019, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


13. The function f(x) = a cos bx + c is plotted on the graph shown below.


What are the values of a, b, and c?
(1) a = 2, b = 6, c = 3
(2) a = 2, b = 3, c = 1
(3) a = 4, b = 6, c = 5
(4) a = 4, b = π / 3, c = 3

Answer: (1) a = 2, b = 6, c = 3
Because c is not in parentheses, it is a vertical shift, not horizontal, and it will be equal to the midline of the function, which is 3 (halfway between 1 and 5). This eliminates choices (2) and (3).
Next, a is the amplitude it is equal to the distance from the midline to the maximum point (or one-half the distance from the maximum to the minimum). So a = 2, not 4. Eliminate choice (4).
As for b, it tells us that there will be 6 cycles in a 2π interval, and 2π / 6 = π / 3, which is the period for one cycle.





14. Which equation represents the equation of the parabola with focus (-3,3) and directrix y = 7?

(1) y = 1/8 (x + 3)2 - 5
(2) y = 1/8 (x - 3)2 + 5
(3) y = -1/8 (x + 3)2 + 5
(4) y = -1/8 (x - 3)2 + 5

Answer: (3) y = -1/8 (x + 3)2 + 5
The standard form of a parabola is y = a(x − h)2 + k, with the vertex at (h, k) and the focus at (h, k + 1 / (4a)).
Since the vertex is equidistant between the focus and the directrix, then vertex must be at (-3, 5) because 5 is halfway between 3 and 7. (3 + 7) / 2 = 5.
Flipping the sign, we know that (x + 3) must be in the equation, so eliminate (2) and (4).
Also, the equation ends with + 5, so we can eliminate choice (1) as well.

At this point, we know the answer is (3). We have one more piece of information: because the directrix is above the vertex, which is above the focus, we know that the parabola opens downward, so a must be negative:

1 / (4a) = -2
1 / ((4)(-2)) = a
1 / (-8) = a





15. What is the solution set of the equation

2 / (3x + 1) = 1 / x - 6x / (3x + 1)


(1) { -1/3, 1/2 }
(2) { -1/3 }
(3) { 1/2}
(4) { 1/3, -2 }

Answer: (3) { 1/2}
Strategy: find a common denominator by multiplying each fraction by either (x / x) or (3x + 1)/(3x + 1).
Then eliminate all the denominators by multiplying both sides of the equation by (x)(3x + 1).
However, we will have to check for extraneous solutions. Because of the denominators in the original equations, we know that x =/= 0 and 3x + 1 =/= 0.

2 / (3x + 1) = 1 / x - 6x / (3x + 1)
2x / ((3x + 1)(x)) = 1 / x - 6x(x) / ((3x + 1)(x))
2x / ((3x + 1)(x)) = (3x + 1) / ((3x + 1)(x)) - 6x(x) / ((3x + 1)(x))
2x = 3x + 1 - 6x2
6x2 - x - 1 = 0
(3x + 1)(2x - 1) = 0
3x + 1 = 0 or 2x - 1 = 0
3x = -1 or 2x = 1
x = -1/3 or x = 1/2

However, 3x + 1 =/= 0, and 3(-1/3) + 1 = 0, so we eliminate that answer. That leaves only 1/2, which is Choice (3).



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Friday, May 17, 2019

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.
After a brief hiatus, the Algebra 2 Problems of the Day are back. Hopefully, daily.

More Algebra 2 problems.

January 2019, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


10. A random sample of 100 people that would best estimate the computations. proportion of all registered voters in a district who support improvements to the high school football field should be drawn from registered voters in the district at a

(1) football game
(2) supermarket
(3) school fund-raiser
(4) high school band concert

Answer: (2) supermarket
At the supermarket, you will encounter a more random set of people.
If you want to improve a high school football field, then football games and school functions will give you biased results.





11. Which expression is equivalent to (2x - i)2 - (2x - i)(2x + 3i), where i is the imaginary unit and x is a real number?

(1) -4 - 8xi
(2) -4 - 4xi
(3) 2
(4) 8x - 4i

Answer: (1) -4 - 8xi
First method: multiply and combine like terms:

(2x - i)2 - (2x - i)(2x + 3i)
4x2 - 4xi + i2 - ( 4x2 + 6xi - 2xi - 3i2) )
4x2 - 4xi + i2 - 4x2 - 4xi + 3i2)
4x2 - 4xi - 1 - 4x2 - 4xi - 3
-8xi - 4
-4 - 8xi

Second method: Factor out (2x - i) from each term first.

(2x - i)2 - (2x - i)(2x + 3i)
(2x - i) ( 2x - i) - (2x + 3i) )
(2x - i) (-4i)
-8xi + 4i2
-8xi - 4
-4 - 8xi





12. Suppose events A and B are independent and P(A and B) is 0.2. Which statement could be true?

(1) P(A) = 0.4, P(B) = 0.3, P(A or B) = 0.5
(2) P(A) = 0.8, P(B) = 0.25
(3) P(A|B) = 0.2, P(B) = 0.2
(4) P(A) = 0.15, P(B) = 0.05

Answer: (2) P(A) = 0.8, P(B) = 0.25
If A and B are independent then P(A and B) = P(A) * P(B)
Choice (1): P(A) * P(B) = 0.4 * 0.3 = 0.12 =/= 0.2.
Choice (2): P(A) * P(B) = 0.8 * 0.25 = 0.2.
Choice (4): P(A) * P(B) = 0.15 * 0.05 = 0.0075 =/= 0.2
In Choice (3), if A and B are independent, then P(A|B) = P(A). So 0.2 * 0.2 = 0.04 =/= 0.2.



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Surface Area of a Cylinder

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(C)Copyright 2019, C. Burke.

Basically, to turn 4 circles into a cylinder, you have to square the circle. Or maybe rectangle two circles.




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Thursday, May 16, 2019

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.
After a brief hiatus, the Algebra 2 Problems of the Day are back. Hopefully, daily.

More Algebra 2 problems.

January 2019, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


7. Tides are a periodic rise and fall of ocean water. On a typical day at a seaport, to predict the time of the next high tide, the most important value to have would be the

(1) time between consecutive low tides
(2) time when the tide height is 20 feet
(3) average depth of water over a 24-hour period
(4) difference between the water heights at low and high tide

Answer: (1) time between consecutive low tides
If the tides are periodic, then knowing the consecutive low tides will tell you when the next high tide is because it will be at the midpoint of those two times.
The water height by itself will not let you predict the next high tide.





8. An estimate of the number of milligrams of a medication in the bloodstream t hours after 400 mg has been taken can be modeled by the function below.

I(t) = 0.5t4 + 3.45t3 - 96.65t2 + 347.7t, where 0 ≤ t ≤ 6
Over what time interval does the amount of medication in the bloodstream strictly increase?

(1) 0 to 2 hours
(2) 0 to 3 hours
(3) 2 to 6 hours
(4) 3 to 6 hours

Answer: (1) 0 to 2 hours
If you graph the function, you will see that it it's rising from 0 to 2. It reaches a local maximum point at approximately x = 2.15. The graph decreases between x = 2.15 and x = 6.





9. Which representation of a quadratic has imaginary roots?


Answer: (4) 2x2 + 32 = 0
If the quadratic intersects or crosses the x-axis then it does not have imaginary roots.
In other words, if there is some value of x which makes y equal to 0, it has a real root.
Choice (1) has (-2.0, 0) and Choice (3) has (3, 0), so they both can be eliminated.

The equations in (2) and (4) have imaginary roots, if there are no real solutions that make the equation true.
If 2(x + 3)2 = 64
then (x + 3)2 = 32
and x + 3 = SQRT(32)
So x = -3 + SQRT(32), which is a real, irrational value.

If 2x2 + 32 = 0
then 2x2 = -32
and x2 = -16.
This means x = SQRT(-16), which is not a real number.



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Wednesday, May 15, 2019

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.
After a brief hiatus, the Algebra 2 Problems of the Day are back. Hopefully, daily.

More Algebra 2 problems.

January 2019, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


4. When a ball bounces, the heights of consecutive bounces form a geometric sequence. The height of the first bounce is 121 centimeters and the height of the third bounce is 64 centimeters. To the nearest centimeter, what is the height of the fifth bounce?
(1) 25
(2) 34
(3) 36
(4) 42

Answer: (2) 34
We know a1 = 121 and a3 = 64.
The common ratio, r = 64 / a2 or a2 / 121
Then r2 = (64 / a2) (a2 / 121) = 64 / 121
And r = SQRT(64/121) = 8/11

To get from a3 to a5, you need to multiply by the common ratio two more times (or multiply by r2).
64 * (8/11)*(8/11) = 33.851... = 34





5. The solutions to the equation 5x2 - 2x + 13 = 9 are
(1) 1/5 + SQRT(21)/5
(2) 1/5 + SQRT(19)/5 i
(3) 1/5 + SQRT(66)/5 i
(4) 1/5 + SQRT(66)/5


Answer: (2) 1/5 + SQRT(19)/5 i
If 5x2 - 2x + 13 = 9
then 5x2 - 2x + 4 = 0
If you graph this, you will see that there are no real roots, and you can eliminate (1) and (4).
Calculate the discriminate, b2 - 4ac = (-2)2 - 4(5)(4) = 4 - 80 = -76
SQRT(-76) = SQRT(-1 * 4 * 19) = 2i * SQRT(19), which elminates choice (3).

x = (-b + SQRT (b2 - 4ac) ) / (2a)
x = ( -(-2) + SQRT (-76) ) / (2*5)
x = ( 2 + 2i SQRT (19)) / (10)
Split the fraction
x = 2/10 + 2i SQRT (19) / 10
x = 1/5 + SQRT(19)/5 i





6. Julia deposits $2000 into a savings account that earns 4% interest per year. The exponential function that models this savings account is y = 2000(1.04)t, where t is the time in years. Which equation correctly represents the amount of money in her savings account in terms of the monthly growth rate?
(1) y = 166.67(1.04)0.12t
(2) y = 2000(1.01)t
(3) y = 2000(1.0032737)12t
(4) y = 166.67(1.0032737)t

Answer: (3) y = 2000((1.0032737)12t
If you take the 12th root (1/12 power) of 1.04, you get 1.00327373978...
Conversely, if you raise 1.0032737 to the 12th power, you will get 1.039999... or 1.04.



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Tuesday, May 14, 2019

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.
After a brief hiatus, the Algebra 2 Problems of the Day are back. Hopefully, daily.

More Algebra 2 problems.

January 2019, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


1. Suppose two sets of test scores have the same mean, but different standard deviations, σ1 and σ2, with σ2 > σ1. Which statement best describes the variability of these data sets?
(1) Data set one has the greater variability.
(2) Data set two has the greater variability.
(3) The variability will be the same for each data set.
(4) No conclusion can be made regarding the variability of either set.

Answer: (2) Data set two has the greater variability.
A lower standard deviation means that the data is closer to the mean, and a higher standard means that the data is more spread out.





2. If f(x) = log3 x and g(x) is the image of f(x) after a translation five units to the left, which equation represents g(x)?
(1) g(x) = log3(x + 5)
(2) g(x) = log3x + 5
(3) g(x) = log3(x - 5)
(4) g(x) = log3x - 5

Answer: (1) g(x) = log3(x + 5)
Adding 5 inside the parentheses shifts the function 5 units to the left. Subtracting 5 moves it to the right.
Adding 5 outside the parentheses shifts the function 5 units up. Subtracting 5 outside (or without) the parentheses moves it down.





3. When factoring to reveal the roots of the equation x3 + 2x2 - 9x - 18 = 0, which equations can be used?

I. x2(x + 2) - 9(x + 2) = 0
II. x(x2 - 9) + 2(x2 - 9) = 0
III. (x - 2)(x2 - 9) = 0

(1) I and II, only
(2) I and III, only
(3) II and III, only
(4) I, II, and III

Answer: (1) I and II, only
If you distribute I, you will get the original four terms.
Likewise, if you distribute II, you will get the four terms, which can be rearranged back into standard form.
If distribute, FOIL, box, or whatever III, you will get -2x2 and +18, which are incorrect.



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Sunday, May 12, 2019

Happy Mother's Day 2019

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(C)Copyright 2019, C. Burke.

Happy Mother's Day!




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Thursday, May 09, 2019

August 2018 Common Core Geometry Regents, Part I (multiple choice)

The following are some of the multiple questions from the recent August 2018 New York State Common Core Geometry Regents exam.
The answers to Part II can be found here
The answers to Parts III and IV can be found here

August 2018 Geometry, Part I

Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.


1. In the diagram below, AEFB || CGD, and GE and GF drawn.

If m∠EFG = 32° and m∠AEG 137°, what is m∠EGF?

Answer: (4) 105°
If ∠AEG = 137, then ∠FEG = 180 – 137 = 43.
The sum of the angles of triangle EFG is 180.
So 180 – (43 + 32) = 105. Angle EGF is 105°.

Alternatively, Angle AEG is an exterior angle to triangle EFG. The Remote Angle Theorem says that it must be the sum of the two remote angles EFG and and EGF.
So EGF = 137 – 32 = 105 degrees.


2. If triangle ABC is mapped onto triangle DEF after a line reflection and triangle DEF is mapped onto triangle XYZ after a translation, the relationship between triangle ABC and triangle XYZ is that they are always

Answer: (1) congruent and similar
Reflections and translations are rigid motions which preserve the shape of the object. Also, choice (2) congruent but not similar is not possible. If two triangles are congruent, they are automatically similar.


3. An isosceles right triangle whose legs measure 6 is continuously rotated about one of its legs to form a three-dimensional object. The three-dimensional object is a

Answer: (4) cone with a diameter of 12
When a right triangle is rotated about a leg, the resulting 3-D shape will be a cone. Because the base of the triangle is 6, the radius will be 6, so the diameter is twice that, or 12.


4. In regular hexagon ABCDEF shown below, AD, BE, and CF all intersect at G.

When triangle ABG is reflected over BG and then rotated 180° about point G, triangle ABG is mapped onto

Answer: (1) Triangle FEG
When it is reflected, it is mapped onto triangle BGC, the top center of the hexagon. When it is rotated 180 degrees, it maps onto FEG, the bottom center of the hexagon.


5. A right cylinder is cut perpendicular to its base. The shape of the cross section is a

Answer: (3) rectangle
A horizontal (parallel to base) cut would give a circle. A vertical (perpendicular to base) cut gives a rectangle. If you look at a can on a shelf, it appears to be a rectangle.


6. Yolanda is making a springboard to use for gymnastics. She has 8-inch-tall springs and wants to form a 16.5° angle with the base, as modeled in the diagram below.

To the nearest tenth of an inch, what will be the length of the springboard, x?

Answer: (4) 28.2
Opposite and hypotenuse mean you need to use sine.
sin 16.5 = 8 / x
x = 8 / sin 16.5 = 28.167…


7. In the diagram below of right triangle ABC, altitude BD is drawn to hypotenuse AC.

If BD = 4, AD = x – 6, and CD = x what is the length of CD?

Answer: (3) 8
According to the Right Triangle Altitude Theorem, (BD)2 = (AD)(CD)
So 42 = (x – 6)(x)
16 = x2 - 6x. At this point, you could substitute the choices, or solve.
x2 - 6x – 16 = 0
(x – 8)(x + 2) = 0
x = 8 or x = -2. Discard the negative length.


8. Rhombus STAR has vertices S(–1,2), T(2,3), A(3,0), and R(0,–1). What is the perimeter of rhombus STAR?

Answer: (4) 4 * SQRT(10)
Each side of the rhombus has the same length, so calculate the length of one side using the distance formula, and multiply the result by 4.
SQRT( (2 - -1)2 + (3-2)2 ) = SQRT( (3)2 + (1)2 )
= SQRT(9 + 1) = SQRT(10) per side.
Perimeter is 4 * SQRT(10).


9. In the diagram below of triangle HAR and triangle NTY, angles H and N are right angles, and HAR ~ NTY.

If AR = 13 and HR = 12, what is the measure of angle Y, to the nearest degree?

Answer: (1) 23°
Angle Y corresponds to angle R. AR is the hypotenuse, and HR is adjacent to R. This means use cosine to find the size of angle R, which will also be the size of angle Y.
cos R = 12/13
r = cos-1(12/13) = 22.61 = 23 degrees. Note that choice (4) is the angle if you either used sine, or if you solved for angle A. You would get choices (2) and (3) if you used tangent, and if you used tangent and the wrong angle.


10. In the diagram below, AKS, NKC, AN, and SC are drawn such that AN = SC.

Which additional statement is sufficient to prove triangle KAN = triangle KSC by AAS?

Answer: (4) AN || SC
If AN is parallel to SC, then you would get alternate interior angles, along with the vertical angles. That is enough to prove congruency by either ASA or AAS. Choice (1) would prove congruency by SSS, but wouldn’t give any more information about the angles. Choice (2) would yield SSA, which is not sufficient to prove congruency. Choice (3) would establish that the vertical angles are also right angles, but we already know that those two angles are congruent.


11. Which equation represents a line that is perpendicular to the line represented by y = (2/3)x + 1 ?

Answer: (1) 3x + 2y = 12
The negative reciprocal of 2/3 is -3/2, so choices (3) and (4) are both incorrect.
y = -(3/2)x + b
3/2x + y = b, multiply by 2 to get rid of the fraction
3x + 2y = 2b, which gives us the first equation, if 2b is replaced by 12.


12. In the diagram of ABC below, points D and E are on sides AB and CB respectively, such that DE || AC.

If EB is 3 more than DB, AB = 14, and CD = 21, what is the length of AD?

Answer: (2) 8
Be careful because you need to find the length of DB first, but they are asking for the length of AD, so you have to subtract DB from AB.
The sides are proportional, so you can set up an equation:
x / 14 = (x + 3) / 21. You can substitute the choices from this point if you want.
21x = 14x + 42
7x = 42
x = 6
AD = 14 – 6 = 8.
Notice that the four choices represent the lengths of DB, AD, BE, and CE. You might have reasoned this one out from the diagram and then checked your work to see if you were correct.


13. Quadrilateral MATH has both pairs of opposite sides congruent and parallel. Which statement about quadrilateral MATH is always true?

Answer: (4) ∠MAT = ∠MHT
Check Part IV of this exam for a coordinate geometry problem involving a quadrilateral with vertices M, A, T, and H.
Opposite sides congruent and parallel mean that this is a parallelogram. That means that the opposite angles are also congruent.
Choice (1) says that the diagonals must be congruent, which is only true in rectangles. (You could use this in Part IV)
Choice (2) says that the consecutive angles are right angles, which is only true in rectangles. (Again, Part IV)
Choice (3) would occur in rectangles as well because the two diagonals would create four isosceles triangles.


14. In the figure shown below, quadrilateral TAEO is circumscribed around circle D. The midpoint of TA is R, and HO = PE .

If AP = 10 and EO = 12, what is the perimeter of quadrilateral TAEO?

Answer: (2) 64

AP = 10 so AR =10. R is the midpoint of AT, so RT = 10, which means TH = 10.
HO = PE means that OZ = ZE, and since OE = 12, then HO = PE = OZ = PE = 6.
4 * 10 + 4 * 6 = 64


15. The coordinates of the endpoints of directed line segment ABC are A(-8,7) and C(7,-13). If AB:BC = 3:2, the coordinates of B are

Answer: (1) (1, -5)
B is 3/5 of the distance from A to C. The x-coordinates of A and C are 7 – (-8) = 15 units apart.
3/5(15) = 9, and -8 + 9 = 1, so the x-coordinate of B is 1, which is choice (1).
To check, -13 – 7 = -20, and (3/5)(-20) = -12, and 7 – 12 = -5, which is the y-coordinate of B.


16. In triangle ABC, points D and E are on sides AB and BC, respectively, such that DE || AC, and AD:DB 3:5.

If DB = 6.3 and AC = 9.4, what is the length of DE, to the nearest tenth?

Answer: (3) 5.9

If AD:DB = 3:5, then AB:DB = 8:5, which is the ratio of the larger triangle to the smaller triangle.
So 8/5 = 9.4/x
8x = (5)(9.4)
x = (5)(9.4)/8 = 5.875.


17. In the diagram below, rectangle ABCD has vertices whose coordinates are A(7,1), B(9,3), C(3,9), and D(1,7).

Which transformation will not carry the rectangle onto itself?

Answer: (3) a rotation of 180° about the point (6,6)
The rectangle will carry onto itself in a rotation about the rectangle's center. (5, 5) is the center of the rectangle. (6, 6) is not. If you rotate about (6, 6), the image will be adjacent to the original rectangle.


18. A circle with a diameter of 10 cm and a central angle of 30° is drawn below.


What is the area, to the nearest tenth of a square centimeter, of the sector formed by the 30° angle?

Answer: (2) 6.5
The radius is 5 cm. The central angle is 30 degrees, which is 1/12 of the total circle. (30/360 = 1/12)
V = (1/12)(pi)(5)2 = 6.54..


19. A child’s tent can be modeled as a pyramid with a square base whose sides measure 60 inches and whose height measures 84 inches. What is the volume of the tent, to the nearest cubic foot?

Answer: (2) 58
60 inches = 5 feet, 84 inches = 7 feet
Volume = (1/3) (Area of the base) (height)
V = (1/3) (5) (5) (7) = 58.333...


20. In the accompanying diagram of right triangle ABC, altitude BD is drawn to hypotenuse AC.

Which statement must always be true?

Answer: (2) AD/AB = AB/AC
Short leg is to hypotenuse as short leg is to hypotenuse.
The other choices do not use corresponding sides in the correct order.


21. An equation of circle O is x2 + y2 + 4x - 8y = -16. The statement that best describes circle O is the

Answer: (4) center is (-2,4) and is tangent to the y-axis
Rewrite the equation in standard form by grouping the variables and completing the square.
x2 + y2 + 4x - 8y = -16
x2 + 4x + y2 - 8y = -16
x2 + 4x + 4 + y2 - 8y + 16 = -16 + 4 + 16
(x + 2)2 + (y - 4) = 4 = 22
The center of the circle is (-2, 4) and the radius is 2, which would make it tangent to the y-axis.


22. In triangle ABC, BD is the perpendicular bisector of ADC. Based upon this information, which statements below can be proven?


I. is a median.
II. bisects ∠ABC.
III. ABC is isosceles.

Answer: (4) I, II, and III
If BD is a bisector of ADC, it's also a median.
If BD is a perpendicular bisector, than AB and AC must be congruent. (This comes in handy to know in Part II!)
IF AB = AC, the triangle is isosceles. In an isosceles triangle, the median bisects the vertex angle.


23. Triangle RJM has an area of 6 and a perimeter of 12. If the triangle is dilated by a scale factor of 3 centered at the origin, what are the area and perimeter of its image, triangle R'J'M'?

Answer: (3) area of 54 and perimeter of 36
If the image is dilated by a scale factor of 3, then the perimeter is increased by a factor of 3, but the area is increased by a factor of 32 = 9.
12 * 3 = 36, and 6 * 9 = 54.


24. If sin (2x + 7)° = cos (4x - 7)°, what is the value of x?

Answer: (2) 15
The Sine of any angle is equal to the Cosine of the complementary angle. And the sum of the two complementary angles is 90 degrees
2x + 7 + 4x - 7 = 90
6x = 90
x = 15

End of Part I

How did you do?

Questions, comments and corrections welcome.

Wednesday, May 08, 2019

Rationalize the Denominator, Part 2

(Click on the comic if you can't see the full image.)

(C)Copyright 2019, C. Burke.

At least he gave them a common denominator to start with.

Long-time readers will recognize the brown-haired girl as Bibi, from previous comics. I've been referring to the other two (in a non-canonically way) as Freedom and Serenity. Why I'm doing that is left as an exercise for the reader.




Come back often for more funny math and geeky comics.




Tuesday, May 07, 2019

Rationalize the Denominator

(Click on the comic if you can't see the full image.)

(C)Copyright 2019, C. Burke.

Who says the denominator has to be rational anyway? Other than the curriculum? And my old teachers? And the text books?




Come back often for more funny math and geeky comics.