<li> A circle with equation (x 3)2 + (y 3)2 = 16 is rotated around the line y = x. What is the volume of the solid that is created?<br>
(A) 256.29<br>
(B) 268.08<br>
(C) 301.97<br>
(D) 312.75<br>
(E) 316.33<br></li>
</ol>
<p>Even I’m doing AP Calculus, I don’t really know how to find the volume. It’s so hard to find the radius and also the intergral.</p>
<p>Well, it’s from the reputable Sparknotes.</p>
<p>that is pretty harsh...so you just skip it. If you have time later, figure out if you can do something with your calculator</p>
<p>are you sure? that wouldnt be on a Math II, I thought math 2 only went to pre-calc..i doubt an integral would appear...that's a bad question..probably wont ecounter that</p>
<p>but that is from sparknotes... ... interesting...</p>
<p>hmm i may be completetly wrong but:</p>
<p>This problem does not involve around a line it creates a sphere... so if a circle with a radius of 4 is rotated around a line y= x it would create a sphere with radius of 4. After plugging 4 into the formula for the volume for the sphere (4/3pi-r-cubed) i got 268.08 or B. Please correct me if i'm wrong im taking this test on saturday as well.</p>
<p>I'm not sure... it sounds right... but I was thinking that the volume produced was like a washer...</p>
<p>if you have PR turn to page 127. A sphere is produced when a circle is rotated around its diameter. A cylinder is formed by the rotation of a rectangle around a central line or one edge. A cone is formed by the rotation of a right triangle around one of its legs, or by the rotation of an isosceles triangle around its line of symmetry. This question, if I understand it correctly, is a 44 not because it is that difficult.. its just that many people arnt familiar with using the concept of 3D rotation to go from coordinate geometry to solid geometry. Peace good luck.</p>
<p>Since the center is at (3,3), that means the line y=x IS a diameter. So the shape formed would be a sphere, with radius 4, right? And then it's easy from there.</p>
<p>yea ballz053 and thecomisar you guys have it right</p>
<p>That hasn't shown up on the Math 2. The one thing they test repeatedly is if you have a rectangle of given dimensions that gets rotated around one of it's sides. So, it becomes a cylinder, where the radius is the bottom edge, and the height is...well, the height. And that question has shown up only near the end of the test (hard). So there's no way they'd give you some really weird solid of revolution.</p>
<p>^ The practice tests I've been taking also frequently have a triangle with two sides on the axes that gets rotated, forming a cone. So that's not too bad, either.</p>
<p>Why is this question weird?</p>
<p>I solved it in about 20 seconds and got B as an answer using the same method as the people above. It's a perfectly doable question that does not involve calculus.</p>
<p>However, if the center does not lie on y=x, it probably would take some extra work.</p>
<p>I agree that there is nothing hard about it. It just has not shown up on the math 2. The reason is pretty clear: it is too involved a question....those do not appear on the Math 2.</p>
<p>Also this one, </p>
<p>An indirect proof of the statement "If x=2, then sqrt x is NOT a rational number" could begin with the assumption that
A, x=sqrt x
B, x^2=2
C, sqrt x is rational
D, sqrt is not rational
E, x is nonnegative</p>
<p>How do you actually do it? I don't even have a clue about this question.</p>
<p>This is a logic question.</p>
<p>If you know anything about proofs, you know that there are two major kinds of proofs: indirect and direct. Direct proof is proving what you are trying to prove (q) directly through mathematical steps (meaning: the last step to your proof is the thing you are trying to prove). Indirect proof is also known as proof by contradiction, which often sets the OPPOSITE (~q) of what you are trying to prove as the first step. Thus, if you can show that the OPPOSITE of what you are trying to prove is false through contradiction, it follows that what you are trying to prove (q) is true.</p>
<p>The proof that sqrt(2) is irrational is a classical indirect proof.</p>
<p>A) This is a choice that has nothing to do with the question. x is not mentioned as being equal to sqrt(x) at all in the question.</p>
<p>B) This statement is true and follows from sqrt(2) = x, but we are trying to show that sqrt(2) is irrational and thus this statement is irrelevant.</p>
<p>C) A possible answer. sqrt(x) is rational is actually the OPPOSITE of what you are trying to prove (~q) and is the logical beginning to an indirect proof. If you can prove that the statement "sqrt(x) is rational" (~q) is false, it must follow logically that the statment "sqrt(x) is irrational" (q) is true. Thus this is the right answer.</p>
<p>D) A possible answer. However, note that sqrt(x) is not rational is (q), or what you are trying to prove! It is possible to prove a statement by setting what you are trying to prove as the first step and showing that the statement must be true, but this is called a constructive proof and not the indirect proof that the question asks for.</p>
<p>E) Again, this statement has nothing to do with the question; we're not discussing about positive/negatives here.</p>
<p>In short, remember that indirect proofs have the opposite of what you are trying to prove (~q) always as the first step.</p>
<p>Ohh, Thanks oasis!~</p>
<p>Great explanation. More info:
This kind of question shows up once in a while (a similar type shows up on Math 1 and the SAT Reasoning Test). Because of the multiple-choice format, you can expect that the answer will always be related to proof by counterexample (i.e. indirect proof).
In a few instances, the ETS gives you a statement in quotes or a little box, and then asks "Which of the following values of x could be used to prove or disprove the statement above?" Answers are numerical values. The part that says "prove or" is merely a smokescreen. You see, you cannot prove a general principle (a "must be true" statement on these tests) with a single value. However, you can certainly disprove a statement with a single value (one that shows that the proposition is not true in all cases).</p>
<p>Rotating triangle (forming a cone) showed up on the 06.03 SAT II math 2.</p>