## Tuesday, June 23, 2015

### 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 rThus 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).

___________________________________________________

## Friday, June 19, 2015

### 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)2140,625

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.

___________________________________________________

## Tuesday, June 16, 2015

### 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).

_____________________________________________________________

## Saturday, June 13, 2015

### 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.

_________________________________________________________________

## Tuesday, June 9, 2015

### 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).

_________________________________________________________________

## Friday, June 5, 2015

### 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 ± , n an integer
B. 3π/2 ± 2, n any real number
C. -π/2 ± 2, n an integer
D. 3π/2 ± , 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 is an integer. Thus the angles with the maximum values are 3π/2 ± 2n 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.

_______________________________________________________

## Tuesday, June 2, 2015

### 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). _________________________________________________________________________________