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Problem:In the equation r = 4/(2 + k), k represents a positive integer. As k gets larger without bound, the value of r: F. gets closer and closer to 4. G. gets closer and closer to 2. H. gets closer and closer to 0. J. remains constant. K. gets larger and larger Answer:Answer is H.
As k gets larger and larger without bound, the expression 4/(2+k) becomes 4 divided by an increasingly large number. For example, think about the trend between the following fractions: 4/100, 4/10,000, 4/1,000,000, ... Looking at it this way, you can see that the expression for r gets closer and closer to zero.
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QuestionWhich of the following is a factored form of 3x^3y^3 + 3xy? A. 3xy(x^2y^2 + 1) B. 3(3x^2y^2) C. (3x + 3y)(3x + 3y) D. 3x^2y^2(xy) E. 3x(x^2y^2 + 3) AnswerThe correct answer is A. This problem requires you to find the Greatest Common Factor. The Greatest Common Factor is 3xy, because each term has at least 1 factor of 3, 1 factor of x, and 1 factor of y. When you factor 3xy out of 3x^3y^3 you are left with x^2 y^2 , and when you factor 3xy out of 3xy, you are left with 1. Therefore, when factored, 3x^3 y^3 + 3xy = 3xy(x^2 y^2 + 1).
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QuestionA waffle ice cream cone is pictured above. If the volume of the cone is 14.4pi cubic inches, what is the diameter of the cone (in inches)? AnswerAnswer is 6.
Solution: The volume of a cone is found by using the formula V= 1/3 x pi x r^2 x h, where r is the radius of the circular base and h is the height. Input the known data and solve for r. 14.4pi = 1/3 pi r^2 (4.*) 14.4 = 1.6 r^2 9 = r^2 3 = r or, d, the diameter is 6.
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Question:A gymnast has a routine in which he sways back and forth on a high bar, making an arc that measures 135 deg. As he swings, the bottom of his shoes create an arc that measures 9 feet. At the conclusion of his routine, he swings completely around for one full circle around the bar. What is the circumference of that circle (answer in feet)? Answer:Answer is 24.
Solution: A circle measures 360 deg. Find what portion a 135 deg arc is of a circle and then use that information to create a proportion. We get: 135/260 which can be reduced to 3/8. So, 3/8 = 9 feet/x feet, or 3/8 = 9/x, or, 3x = 72 or, x = 24
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The volume of a 3 dimensional solid is the amount of space it occupies.
Since the end (base) of a cylinder is a circle, the area of that circle is given by the formula:
\[area = π r^2\]
The volume of a cylinder is found my multiplying the area of one end of the cylinder by its height. So the formula of the volume of a cylinder is:
\[volume = π r^2 h\]
where:
π is Pi, approximately 3.142 r is the radius of the circular end of the cylinder h height of the cylinder
Our problem involves calculating the volume of Amazon Echo. The height of the Echo is 235mm while the diameter of the base is 84 mm or 8.4cm . Let's first find the area of the base.
The area of the base of Amazon Echo is:
\[area = π r^2\]
\[= 3.142 x 8.4^2\]
= 221.56 cm^2
To calculate the volume, we multiply the area with the height of the Echo, which is 235mm or 23.5cm
So, volume of Alexa Echo = 221.56 * 23.5 = 5,206.6224 cm^3
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N.B: SAT Subject Test Math Level 2 Question
Question
What is the sum of the roots of the equation?
\[(x  \sqrt{2})(x^2  \sqrt{3x} + \pi) = 0\]
(A) 0.315
(B) 0.318 (C) 1.414 (D) 3.15 (E) 4.56
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Question:If there are known to be 4 broken transistors in a box of 12, and 3 transistors are drawn at random, what is the probability that none of the 3 is broken?
(A) 0.250 (B) 0.255 (C) 0.375 (D) 0.556 (E) 0.750
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N.B. This is for SAT Subject Test Math Level 2 Problem:What is the sum of the infinite geometric series:
6 + 4 + 8/3 + 16/9 + ... ? (A) 18 (B) 36 (C) 45 (D) 60 (E) There is no sum.
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ProblemIn April of 2004, d dogs and c cats lived in an animal shelter. If 4 cats arrived at the shelter in May of 2004 and the ratio of dogs to cats remained unchanged, in terms of c and d, how many dogs arrived at the shelter in May of 2004?
(A) 4 (B) 4d/c (C) d/c (D) d^2  4d (E) (2cd + 4d)/c
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In the complex plane, the horizontal axis is called the real axis and the vertical axis is called the imaginary axis. The complex number a + bi graphed in the complex plane is comparable to the point (a, b) graphed in the standard (x, y) coordinate plane. The modulus of the complex number a + bi is given by:
\[\sqrt{a^2 + b^2}\]
Question:
Which of the complex numbers z1, z2, z3, z4 and z5 below has the greatest modulus?
Answer:
Okay, so what is this question really asking?
All it is asking is for you to find the largest value of the square root of the sum of the squares of our coordinate points, or find:
\[\sqrt{x^2 + y^2}\]
The challenge is that we are not given the real x and y values of the coordinate points. So we have to estimate the coordinate points of our z points.
Because we are working with squares, negatives are not a factor, so we can eliminate the smaller numbers as we are just taking a number, positive or negative and taking the square of it. Let us estimate our coordinates: z1 = (4, 5) z2 = (2, 1) z3 = (2, 3) z4 = (2, 2) z5 = (4, 2) We are looking for whichever point has the largest combination of its coordinate points. At a glance, the two points with the largest coordinates are z1 and z5. Let's find the modulus of z5, and smaller of the two.
\[\sqrt{4^2 + (2)^2}\]
\[\sqrt{16 + 4}\]
\[\sqrt{20}\]
\[4.5\]
And the modulus of z1:
\[\sqrt{(4)^2 + 5^2}\]
\[\sqrt{16 + 25}\]
\[\sqrt{41}\]
\[6.4\]
We can see that the modulus of z1, 6.4, is higher than that of z5.
Final answer: F, z1.
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Question:Consider the functions f(x) = sqrt(x) and g(x) = 7x + b. In the standard (x,y) coordinate plane, y = f(g(x)) passes through (4,6). What is the value of b?
A. 8 B. 8 C. 25 D. 26 E. 4  7. sqrt(6)
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Question:
N.B: This question is for SAT Subject Test Math Level 2
\[if \ log_a 5 = x \ and \ log_a 7 = y, \ then \ log_a \sqrt{1.4} = \]
(A) (xy)/2
(B) x/2  y (C) (x + y)/2 (D) (y  x)/2 (E) y/(2x) Answer:
The answer is D.
\[log_a\sqrt{1.4} = log_a\sqrt{\frac{7}{5}} = \frac{1}{2} (log_a 7  log_a 5) = \frac{1}{2} (y  x) \]
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Two whole numbers have a greatest common factor of 8 and a least common multiple of 48. Which of the following pairs of whole numbers will satisfy the given conditions?
F. 4 and 9 G. 5 and 10 H. 10 and 16 J. 14 and 20 K. 16 and 24
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ProblemA total of f men went on a fishing trip. Each of the r boats that were used to carry the fishermen could accommodate a maximum number of m passengers. If one boat had 5 open spots and the remaining boats were filled to capacity, which of the following expresses the relationship among f , r, and m?
F. rm+5=f G. rm−5=f H. r+m+5=f J. rf = m + 5 K. rf = m − 5
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ProblemSamantha is making gluten free brownies for the family picnic. If the recipe calls for 2 ½ cups of cocoa to serve 4 people, how many cups will he need if there will be 60 people at the picnic? SolutionSamantha's recipe calls for 2 ½ cups of cocoa to serve 4 people. Or, in other words, > 4 people require 2 ½ cups of cocoa. > 1 person would then require (2 ½) / 4cups of cocoa, which is 0.625 cup of cocoa. > So 60 people would require 0.625 x 60 cups of cocoa = 37.5 cups of cocoa. Answer: 37.5 cups of cocoa. Ingredients
Instructions
Source: http://www.kingarthurflour.com/recipes/glutenfreebrowniesrecipe
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Problem:
An interior designer is creating a custom coffee table for a client. The top of the table is a glass triangle that needs to balance on a single support. If the coordinates of the vertices of the triangle are at (3, 6), (5, 2), and (7, 10), at what point should the support be placed?
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A proportion is a set of 2 fractions that equal each other. This article focuses on how to use proportions to solve cooking problems, especially when the numbers are not friendly. But first, let's start with an easy problem.
Let's say you are cooking rice to serve exactly 3 people. The recipe calls for 2 cups of water and 1 cup of dry rice. However, you found out that there are 12 guests coming. How would the recipe change? If you’ve ever made rice, you know that this ratio — 1 part dry rice and 2 parts water — is important. Mess it up, and your rice will become soggy or something else. Because you are quadrupling your guest list (3 people * 4 = 12 people), you must quadruple your recipe. Cook 8 cups of water and 4 cups of dry rice. This demonstrates how you can use ration to solve proportion problems in real life. What happens when the numbers are not so friendly? Let's say you are throwing a party for 25 people. How much water do you need?
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The following problem appeared on the SAT.
A cubic box with edge of length x inches is tied with a string 106 inches long. The string crosses itself at right angles on the top and bottom of the box. If the bow required 10 inches of string, what is the maximum number of inches x could be? 1. Write an equation based on this information. 2. Solve the equation for x.
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Are we born with the ability to tell which things are close and which are far away? Or, do we learn it? Two psychologists at Cornell University, Eleanor Gibson and Richard Walk, did an interesting experiment to answer this question.
The kitten in the photograph above is sitting on a strip in the middle of a sheet of glass. Behind the kitten is a floor directly underneath the glass; in front of it is another floor several feet below the glass. Even when the kitten is just old enough to move about, it is afraid to move off the "cliff" to the "deep" side. Chicks and baby goats just 1 day old and human infants just old enough to crawl behave exactly the same way. What explains this phenomenon? Surprisingly, it all about inequalities!
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Randall is scheduling his classes for next term. He has a choice of 3 different science classes, 4 different math classes, and 5 different humanities classes. How many different class schedules can Randall create if he must take 1 science class, 1 math class, and 1 humanities class?
F. 14 G. 23 H. 30 J. 45 K. 60
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The figure above appeared in a problem on an SAT exam. All of the triangles are congruent, the area of the shaded region is 84, and the area of square ABCD is 100.
1. What is the area of one of the triangles? 2. What is the total area of the entire figure?
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QuestionA prime number is a whole number greater than 1, whose only two whole number factors are 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
What’s the big deal about prime numbers? Encryption. Large prime numbers are used for encryption. RSA encryption is based on prime numbers — two prime numbers multiplied together. The original two prime numbers are known as your ‘private key’. When you multiply them together, the product (a number that’s only divisible by one, itself and those two prime numbers) is called the ‘public key’. Question: One prime number divided by another prime number is always A) a fraction between 0 and 1 B) an integer C) a positive noninteger D) a prime number
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QuestionCarter was on a diet. When he weighed himself as the start of his diet, he weighed 220 pounds. At the end of 6 months, Carter weighed 180 pounds. What fraction of his original weight did he lose?
(A) 1/8 (B) 4/13 (C) 3/16 (D) 2/11
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Question
What is the solution set for the above equation?
A) {5} B) {20} C) {−5, 20} D) {5, 20}
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Question
\[ (x^2 + bx  2)(x + 3) = x^3 + 6x^2 + 7x 6 \]
In the equation above, b is a constant. If the equation is true for all values of x, what is the value of b?
A) 2 B) 3 C) 7 D) 9 