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?
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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?
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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 x-axis 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?
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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 3-dimensional (x, y,z) space, the set of all points 5 units from the x-axis is:
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Solution
The set of all points 5 units from the x-axis in 3-dimensional space can be likened to the set of all points 5 units from the x-axis in 2-dimensional space, which are two lines. Revolving such lines about the x-axis creates a tube-like 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?
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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)?
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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 + 3x2.
What is the value of b in the solution to the system of equations below?
3a − b = 18
a + 3b = −4
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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?
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Solution
To solve this problem, recall that 102 = 100. Therefore since n = 10, n2 = 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 n2 = 100, 5(100) + 5(10) + 2 = 5n2 + 5n + 2.
If 0 < pr < 1, then which of the following CANNOT be true?
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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?
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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.