Geometry problem

In the following figure, an isosceles triangle ABC is perfectly enclosed in a semi-circle of radius r. If AB is a diameter of the semi-circle, which of the following expressions shows the combined area of the shaded regions in terms of r?

A. r²(π – 1) / 2

B. r²(π – 2)

C. r²(π – 2) / 2

D. r²(π – 1) 

The combined area of the shaded regions is determined from subtracting the area of the triangle from the area of the semi-circle. AB = 2r, and this segment must pass through the center of the circle. This means that the height of the triangle (to the vertex C) must be equal to the radius r. Thus the area of the triangle is 2r(r)/2 = r². Area of the semi-circle is πr²/2. Thus the combined area of the shaded regions is πr²/2 – r² = r²(π – 2) / 2. The correct answer choice is (C).

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Nonlinear function word problem #2

Perimeter of a rectangular playing field measures 1,500 feet. Find the maximum area of the field.

Let the length of the field be l and width w. Then, 1,500 = 2(l + w), meaning that 750 = l + w. This means that w = 750 - l, and since the area is given by A = lw, we have
A = l(750 - l)
= -l² + 750l
= -(l² – 750l)
= -[l² – 750l + (-750/2)² – (-750/2)²]
= -(l² – 750l + 140,625)  + 140,625
= -(l – 375)² + 140,625. 

This is a vertex form of a parabola. Since there is a negative coefficient in front of l, the parabola expands downward, and the vertex represents its maximum value. The vertex is given by (375, 140625), thus for the length 375 feet, the maximum area of the playing field is 140,625 square feet with the given perimeter of the field. Note that you could have expressed the area in terms of the width w, and it would not have changed the result.
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Graphical Analysis problem

If the curve shown on the graph is shifted 2 units to the left and then 3 units down, what will be the coordinates of its new x-intercept?

A. (0, 0)
B. (1, 0)
C. (-1, 0)
D. (2, 0)


The easiest method to solve this problem is to find a point on the graph that will land on the x-axis (where y = 0) when performing the horizontal and vertical shifts. The point (2, 3) on the curve is a good choice. The shifted curve will intercept the origin. Thus the correct choice is (A).

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Nonlinear function word problem

A drug company X estimates that a person will lose 50 pounds of their weight in N months when increasing their daily consumption of a product by r%.  The model below shows this result. 

N(r) = 42/r + 1

Which of the following is true?

A. A person will lose 42 pounds in two months when increasing their daily consumption of a product by 1%

B. A person will lose 50 pounds in 43 months when increasing their daily consumption of a product by 42%

C. A person will lose 50 pounds in three months when increasing their daily consumption of a product by 21%

D. A person will lose 42 pounds in 43 months when increasing their daily consumption of a product by 42%


You must understand that N(r) represents the time needed, in months, to drop 50 pounds. This time is dependent on the person's daily product consumption percent increase r. Simply plug in each answer choice and translate the result into words. Choices (A) and (D) can be crossed out, because they do not represent 50 pounds as the target number given in the problem. Choice (B) suggests that 42/42 + 1 = 1 + 1 = 2 to equal 43, which is false. Choice (C) is correct, because 42/21 + 1 = 2 + 1 = 3, which is consistent with three months.

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Statistics problem

The following histogram displays data taken from a random sample of instrumentalists from a music conservatory who revealed their average daily practice duration (in hours). Which of the following statements is the most accurate interpretation of the data?

A. the mean number of hours of practice is less than the median number of hours of practice 
B. the mean number of hours of practice exceeds the median number of hours of practice 
C. the median number of hours of practice is exactly 5 hours
D. the mean number of hours of practice is exactly 5 hours 

When the median equals the mean, the graph will look symmetric (it is a bell-shaped curve or a symmetric histogram with equal fall to the right and left of the mean). In this example, there is no symmetry, thus the median cannot equal the mean. The median point divides the area in half (the two areas on both sides of the median point must be equal). The graph appears to be skewed to the right (it gets thinner to the right and builds up to the left). Thus the median number of hours of practice should be less than the mean number of hours of practice. 

Let's check this result: there are 20 + 35 + 25 + 10 + 5 = 95 instrumentalists in total, so that the median number of hours of practice will fall on the 48th lowest number of hours of practice (the middle number of students is 48 out of 95, calculated as 95/2 = 47.5, which is rounded to 48). Since 20 students practice 0-2 hours daily, we see 2-4 hours of practice from the 21st to 55th student. This is where the median (48) will fall, so that the median number of hours of practice is in the 2-4 hours range. 

Since there are outliers (values that are extreme to the right of the median), the mean will exceed the median. Thus the correct answer choice is (B).

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Trigonometry problem

Find the measures of angles x at which the function f(x) = -(2/3)cos(x - π/2) + 3 is at its maximum value.

A. -π/2 ± nπ, n an integer

B. 3π/2 ± 2nπ, n any real number

C. -π/2 ± 2nπ, n an integer

D. 3π/2 ± nπ, n any real number

The maximum value for this function will be 3 + 2/3 = 11/3. We know this from the fact that the minimum for cos(x - π/2) is -1, and -1(-2/3) = 2/3. 

If cos(x - π/2) = -1, then x - π/2 = π, so that x = π + π/2 = 3π/2. Now, since cos(x) shows the same value with angle increments of 2π, cos(x - π/2) function will also show the same certain value with angle increments of 2π. This means that our x can be 3π/2, 3π/2 - 2π, 3π/2 + 2π, 3π/2 - 4π, 3π/2 + 4π, etc. This means that angles x produce the same function values for f(x) = -(2/3)cos(x - π/2) + 3 at 3π/2 + 2nπ, 3π/2 - 2nπ, where n is an integer. Thus the angles with the maximum values are 3π/2 ± 2nπ, n an integer, and this is the same thing as the answer (C) shows, since 3π/2 - (-π/2) = 2π. Choice (B) is not correct because n must be an integer, otherwise the angles will not produce the same value, and choices (A) and (D) also fail for these reasons as well as adding and subtracting the wrong angle measures. You may plug in each answer choice to make an educated guess, but you may get confused with angle additions if you are not familiar with this topic.

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Function problem

If f(x) = -|-(x – 2)² + 1| - 1, and g(x) = 1/x, what is the value of (2g + f)(5)? 

A. -43/5
B. 37/5
C. -43/9
D. -53/5

First, notice that we need to combine the two functions. In other words, we must find the value of the function, that consists of the sum of twice the function of g and function f, at x = 5.

The best way is to denote a third function as a sum of the two functions g and f. If we let h(x) = 2g(x) + f(x), then we need to simply find h(5).

Now, h(x) = 2/x - |-(x – 2)² + 1| - 1. Setting x = 5, we have h(5) = 2/5 - |-(5 - 2)² + 1| - 1 = 2/5 - |-9 + 1| - 1 = 2/5 - |-8| - 1 = 2/5 - 8 - 1 = 2/5 - 9 = (2 - 45) / 5 = -43/5.

Thus the correct answer choice is (A). ________________________________________________

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Algebra problem

For -2k / (k + 1) = 3/4, find the value of k.

We need to cross-multiply to solve for k. We then have -8k = 3k + 3. Subtracting 3k from both sides, we get -11k = 3, which gives k = -3/11.

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Data Analysis problem

*CALCULATOR REQUIRED 

 The following table shows points (x, y) on the same line. Which of the following choices shows variable y in terms of variable x for the line? 

 A. y = -5x/2 - 12.15 

B. y = -2x/5 - 2.28 

C. y = -2x/5 - 1.48 

D. y = -5x/2 - 11.35

Since every point on the table (x, y) shares the same line, we can determine the slope of the line using any pair of three available points. We will use the first two for convenience. The slope is found by the formula (change in y) / (change in x) = [0 - 0.4] / (-3.7 - (-4.7)] = -0.4 / 1 = -2/5. This rules out choices A and D. We now have the equation in the form y = -2x/5 + b, where b is the y-intercept or point on the line where x = 0, that is, (0, b). Using the point-slope formula (and the first point on the table) we have the equation as y - 0.4 = -2/5[x - (-4.7)], which simplifies to y = -2x/5 - 1.88 + 0.4, or y = -2x/5 - 1.48. Be sure to check this equation with all of the given points x above to see if you are getting the corresponding y values. Thus the correct answer choice is (C). You may also plug in each equation in the answer choices if you forget how to set up your equation, but it may take extra time. You may, however, take a shortcut after you find the correct slope -2/5 to check answers (B) and (C) with the table values to find the correct y-intercept.

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A hybrid Algebra-Geometry Problem

A parabola, whose equation is ax² + bx + 3 = y, has a vertex at (-1, -2). Find the value of a + b.

We know that any parabola can also be expressed in the form of a(x – c)² + d = y, where (c, d) is the point of the vertex of the parabola. In this example c = -1, and d = -2. Thus we have the equation in the vertex form as a(x –  (-1))² + (-2) = y. Setting this expression for y and the given expression for y equal, we have a(x + 1)² – 2 = ax² + bx + 3. Simplifying the left side, we have ax² + 2ax + a – 2 = ax² + bx + 3. Since the two sides are equal, we can compare their coefficients. The second and third terms provide with information we need, so we have 2a = b, and a – 2 = 3, which gives a = 5, and b = 2(5) = 10. Thus a + b = 5 + 10 = 15.

The second way of solving this problem is to remember that the x-coordinate of the vertex of a parabola is given by -b/2a, so that -1 = -b/2a, so that 2a = b. Now, we can plug in the x-coordinate of the given vertex point into the given equation to get a second expression for constants a and b: a(-1)² + b(-1) + 3 = -2, which gives a – b = -5. Substituting 2a = b into this equation, we have a – 2a = -5, so that a = 5, and b = 10 as before. 

Always check the constants you found with the vertex coordinate in the original equation to see if you are correct. If you are not sure how to use either of the two methods, skip this question on the exam and come back to it.  

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The Redesigned Math SAT: should I be concerned?

You do NOT have to be afraid. There are plenty of great books out there. However, being a mathematician myself, I tend to sort out books before I buy them. Now I write my own study guides.

 The new SAT Math exam coming up in 2016 will require the knowledge of trigonometry, geometry, sequences, logic, algebra, word problems, graphical reasoning, data analysis, among other topics.

 Typically, word problems must be solved in the following way: you are given information in words, and you are required to put it in mathematical form.

 Example word problem: Rita is 8 years older than John was 3 years ago. Tom will be twice the current age of John in 4 years time. How old is Tom now if Rita is 25?

 Solution: let r be Rita's age now, j John's age now, and t Tom's age now. Then we have r = j - 3 + 8 (we got this from sentence "Rita is 8 years older than John was 3 years ago."), t + 4 = 2j (from "Tom will be twice the current age of John in 4 years time."). Since r = 25, we have 25 = j - 3 + 8, which means that j = 25 - 5 = 20, and t = 2(20) - 4 = 40 - 4 = 36. Thus Tom is 36 years old.

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