The volume, V, of a sphere is determined by the formula V =\(\frac{4\pi R^{3}}{3}\) , where r is the radius of the sphere. What is the volume, in cubic inches, of a sphere with a diameter 12 inches long?

Solution
If a sphere has a diameter of 12, then its radius is 6. A sphere with radius 6 has volume V =\(\frac{4\pi R^{3}}{3}=\frac{4\pi (6)^{3}}{3}=\frac{4\pi (216)}{3}=\frac{864\pi }{3}\), or 288π.
The number 0.002 is 100 times as large as which of the following numbers?

Solution
To solve this problem, express the number 0.002 in scientific notation, which is 2×10^{−3}, because the decimal point moved 3 places to the right. If the number 10^{−3} is 100 times larger than some number x, then:
10^{−3} = 100x, or x = \(\frac{10^{−3}}{100}\)
\(\frac{10^{−3}}{10^{2}}\) = 10^{−3−2}
10^{−3−2} = 10^{−5}
10^{−5}
Therefore, 2 × 10^{−3} is 100 times larger than 2 × 10^{−5} which equals 0.00002 because you must move the decimal point 5 places to the right.
An overlay of an accessibility ramp of a building is placed on the standard (x, y) coordinate plane so that the xaxis aligns with the horizontal. The line segment representing the side view of the ramp goes through the points (−2,−1) and (16,2). What is the slope of the accessibility ramp?

Solution
To solve this problem,calculate the slope of the ramp in the same manner as calculating the slope of a line through the points (−2,−1) and (16,2). Slope is defined as \(\frac{rise}{run}\), or \(\frac{change\: in\: y}{change\: in\: x}\), therefore (−2,−1) and (16,2) the equation is:
\(\frac{(2 + 1)}{(16 + 2)}=\frac{3}{18}\)
\(\frac{3}{18}=\frac{1}{6}\)
In a 3dimensional (x, y,z) space, the set of all points 5 units from the xaxis is:

Solution
The set of all points 5 units from the xaxis in 3dimensional space can be likened to the set of all points 5 units from the xaxis in 2dimensional space, which are two lines. Revolving such lines about the xaxis creates a tubelike shape that is a cylinder.
Due to inflation, a refrigerator that formerly sold for $450 now sells for 7% more. Which of the following calculations gives the current cost, in dollars, of the refrigerator?

Solution
Since the refrigerator that sold for $450 now sells for 7% more, the current price is $450 + (7% of $450), or 450+450(0.07). Remember that 0.07 is the decimal equivalent of 7%.
Which of the following is an equivalent form of x + x(x + x + x)?

Solution
To solve this problem, first combine like terms. The expression x + x(x + x + x) becomes x + x(3x). Multiplying the second term yields x + 3x^{2}.
What is the value of b in the solution to the system of equations below?
3a − b = 18
a + 3b = −4

Solution
To solve this problem, add the two equations, or multiples of them, so that cancellation occurs and you can solve for the b variable. If the equation a + 3b = −4 is multiplied by −3, the result is −3a − 9b = 12. The addition of −3a − 9b = 12 to the second equation is shown below:
3a − b = 18
−3a − 9b = 12
−10b = 30
By canceling out the a terms, you can solve for b:b = 30
−10 = −3.
If n = 10, then which of the following represents 552?

Solution
To solve this problem, recall that 102 = 100. Therefore since n = 10, n^{2} = 100. The number 552 can be broken up into 500 + 50 + 2, which can be written as 5(100) + 5(10) + 2. Again, since n = 10 and n^{2} = 100, 5(100) + 5(10) + 2 = 5n^{2} + 5n + 2.
If 0 < pr < 1, then which of the following CANNOT be true?

Solution
At first glance, you might think that you don’t have enough information to solve this problem. However, if you recognize that pr must be a positive fraction since it lies between 0 and 1, you can work your way through the answer choices and eliminate those that could be true:
Answer choice F: If both p and r were less than 0, their product would be positive. It’s possible for pr to be a positive fraction, because both p and r could be negative fractions, so eliminate answer choice F.
Answer choice G: If p were −1 and r were also a negative number, their product would be positive. It’s possible for pr to be a positive fraction, because r could be a negative fraction, so eliminate answer choice G.
Answer choice H: If both p and r were less −1, then pr would be greater than 1, so this statement cannot be true, and answer choice H is correct. Answer choice J: If both p and r were less than 1, their product could be positive. It’s possible for pr to be a positive fraction, because both p and r could be negative fractions, so eliminate answer choice J.
Answer choice K: If p were less than 1, p could be a positive fraction. If r were greater than 0,it would be a positive number, and it’s possible for pr to be a positive fraction; eliminate answer choice K.
Which of the following is NOT a solution of (x − 5)(x − 3)(x + 3)(x + 9) = 0?

Solution
The solutions of (x − 5)(x − 3)(x + 3)(x + 9) = 0 are the values of x that make the product (x − 5) (x − 3)(x + 3)(x + 9) equal to zero.
Such values of x will be the values that make any one of the four parts of (x − 5)(x − 3)(x + 3)(x + 9) equal to zero, which would be 5, 3, −3, and −9.
Of the answer choices, only −5 is not a value of x such that (x − 5)(x − 3)(x + 3)(x + 9) = 0.