answer：First, I need to determine how much Andy spent in total. He bought a can of soda for 1 and two hamburgers, each costing 2. So, the cost for the hamburgers is 2 multiplied by 2, which equals 4. Adding the cost of the soda, Andy's total expenditure is 1 plus 4, totaling 5.Since Andy and Bob spent the same amount, Bob also spent 5. Bob purchased a can of fruit drink for 2 and some sandwiches. To find out how much Bob spent on the sandwiches, I subtract the cost of the fruit drink from his total expenditure: 5 minus 2 equals 3. Therefore, the total cost of Bob's sandwiches is 3.

answer：Okay, so I have this problem here: "Two sides of a right triangle have the lengths 6 and 8. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth." Hmm, right triangle, two sides given, need to find the third side and then multiply those possibilities together. Alright, let me think through this step by step.First, I remember that in a right triangle, the sides relate to each other through the Pythagorean theorem, which is (a^2 + b^2 = c^2), where (a) and (b) are the legs, and (c) is the hypotenuse. So, depending on whether the given sides are both legs, or one is a leg and the other is the hypotenuse, the third side will be different.So, let's consider the two possibilities:1. **Both 6 and 8 are legs of the triangle.** In this case, the third side will be the hypotenuse. Using the Pythagorean theorem, I can calculate it as: [ c = sqrt{6^2 + 8^2} = sqrt{36 + 64} = sqrt{100} = 10 ] So, the hypotenuse is 10.2. **One of the sides is the hypotenuse.** Now, I need to figure out which one could be the hypotenuse. Since the hypotenuse is always the longest side in a right triangle, it has to be longer than either of the legs. Here, 8 is longer than 6, so 8 could potentially be the hypotenuse, and 6 would be one of the legs. The other leg can be found using the Pythagorean theorem as well: [ b = sqrt{8^2 - 6^2} = sqrt{64 - 36} = sqrt{28} ] Simplifying (sqrt{28}), I get (2sqrt{7}), which is approximately 5.2915.Wait, hold on, is 8 the only possible hypotenuse? What if 6 was the hypotenuse? But 6 is shorter than 8, and the hypotenuse has to be the longest side. So, 6 can't be the hypotenuse because 8 is longer. Therefore, the only other possibility is that 8 is the hypotenuse, and 6 is a leg, giving us the other leg as (2sqrt{7}).So, now I have two possible lengths for the third side: 10 and approximately 5.2915. The problem asks for the product of these possible lengths. So, I need to multiply them together.Calculating the product:[10 times 5.2915 = 52.915]Rounding this to the nearest tenth, I look at the hundredths place, which is 1. Since 1 is less than 5, I don't round up. So, 52.915 rounded to the nearest tenth is 52.9.Wait, let me double-check my calculations to make sure I didn't make a mistake. For the first scenario, 6 and 8 as legs, hypotenuse is definitely 10 because (6^2 + 8^2 = 36 + 64 = 100), and the square root of 100 is 10. That seems right.For the second scenario, if 8 is the hypotenuse and 6 is a leg, then the other leg is (sqrt{8^2 - 6^2} = sqrt{64 - 36} = sqrt{28}). (sqrt{28}) is approximately 5.2915, which is correct because (5.2915^2) is roughly 28.Multiplying 10 and 5.2915 gives 52.915, which rounds to 52.9. That seems accurate.I think I've covered all the possibilities. There are only two cases: either both given sides are legs, or one is a leg and the other is the hypotenuse. Since 6 can't be the hypotenuse because it's shorter than 8, the only other case is 8 being the hypotenuse. So, I think I've got all the possible third sides and their product.Just to recap:- If 6 and 8 are legs, third side (hypotenuse) is 10.- If 8 is the hypotenuse and 6 is a leg, third side (other leg) is approximately 5.2915.- Product of 10 and 5.2915 is approximately 52.915, which rounds to 52.9.Yes, that all checks out. I don't think I missed anything here.

answer：To determine the number of businessmen who drank neither coffee nor tea, I can use the principle of inclusion-exclusion.First, I'll add the number of businessmen who drank coffee and those who drank tea:15 (coffee) + 13 (tea) = 28.However, this count includes the 6 businessmen who drank both coffee and tea twice. To correct for this double-counting, I'll subtract the number of businessmen who drank both:28 - 6 = 22.This means that 22 businessmen drank either coffee or tea. To find out how many drank neither, I'll subtract this number from the total number of businessmen:30 - 22 = 8.Therefore, 8 businessmen drank neither coffee nor tea.

answer：To find the coefficient of the ( x^2 ) term in the expansion of ( left(x - frac{2}{x}right)^4 ), we can use the binomial theorem. The binomial theorem states that:[(a + b)^n = sum_{k=0}^{n} binom{n}{k} a^{n-k} b^k]In this case, ( a = x ), ( b = -frac{2}{x} ), and ( n = 4 ).The general term in the expansion is:[binom{4}{k} x^{4 - k} left(-frac{2}{x}right)^k]Simplifying the term:[binom{4}{k} (-2)^k x^{4 - k} x^{-k} = binom{4}{k} (-2)^k x^{4 - 2k}]We need the exponent of ( x ) to be 2:[4 - 2k = 2 implies k = 1]Now, calculate the coefficient for ( k = 1 ):[binom{4}{1} (-2)^1 = 4 times (-2) = -8]Therefore, the coefficient of the ( x^2 ) term is (-8).

answer：Okay, so I have this equation here: x plus 2y minus 3 equals zero. I need to find the value of 2 to the power of x multiplied by 4 to the power of y. Hmm, let me think about how to approach this.First, the equation is x + 2y - 3 = 0. Maybe I can rearrange this to make it simpler. If I add 3 to both sides, I get x + 2y = 3. That seems a bit cleaner. So, x + 2y equals 3. Got that.Now, I need to find 2^x times 4^y. I remember that 4 is the same as 2 squared, right? So, 4^y is the same as (2^2)^y. When you raise a power to another power, you multiply the exponents. So, (2^2)^y becomes 2^(2y). Okay, so 4^y is 2^(2y).So, now, the expression 2^x times 4^y becomes 2^x times 2^(2y). Since both terms have the same base, which is 2, I can add the exponents when multiplying. That means 2^x times 2^(2y) is equal to 2^(x + 2y).Wait a minute, from the equation I had earlier, x + 2y equals 3. So, substituting that into the exponent, 2^(x + 2y) becomes 2^3. And 2^3 is 8. So, the value of 2^x times 4^y is 8.Let me double-check that to make sure I didn't make any mistakes. Starting from the beginning, x + 2y = 3. Then, 2^x times 4^y is 2^x times (2^2)^y, which is 2^x times 2^(2y), which equals 2^(x + 2y). Since x + 2y is 3, that's 2^3, which is 8. Yeah, that seems right.I think I got it. The key was recognizing that 4 is 2 squared, so I could rewrite the expression with the same base and then combine the exponents. That made it much simpler to substitute the known value from the equation.

answer：Alright, so I've got this geometry problem here, and I need to figure out the measure of angle S. Let me try to visualize the problem first. There are two segments, PQ and RS, intersecting at point T. The problem states that PT equals QT, which equals TR, which equals RS. So, PT = QT = TR = RS. That means all these segments are of equal length. Interesting.Also, it says that angle R is three times angle P. So, if I let angle P be some measure, then angle R would be three times that. I need to find the measure of angle S. The options given are 36°, 45°, 72°, and 90°. Hmm, okay.Let me start by drawing a rough sketch to understand the problem better. I imagine two lines intersecting at point T. PQ is one line, and RS is the other. Since PT = QT, triangle PQT is isosceles with PT and QT as the equal sides. Similarly, since TR = RS, triangle TRS is also isosceles with TR and RS as the equal sides.Given that PT = QT = TR = RS, all these segments are equal. So, PT = QT = TR = RS. That means the lengths from T to each of these points are equal. Interesting. So, T is a common point where these segments intersect, and all the segments from T to the other points are equal in length.Now, the problem mentions angle R and angle P. I need to figure out what these angles are. Let me assume angle P is at point P, and angle R is at point R. Since angle R is three times angle P, I can denote angle P as x degrees, which would make angle R equal to 3x degrees.Since triangles PQT and TRS are isosceles, their base angles are equal. In triangle PQT, PT = QT, so the angles opposite these sides, which would be angles at P and Q, are equal. Similarly, in triangle TRS, TR = RS, so the angles opposite these sides, which would be angles at T and S, are equal.Wait, but I need to be careful here. The angles at P and Q in triangle PQT are not necessarily the same as angles at T and S in triangle TRS. Let me clarify.In triangle PQT, since PT = QT, the base angles at P and Q are equal. Let's call each of these angles x. Then, the angle at T in triangle PQT would be 180° - 2x, since the sum of angles in a triangle is 180°.Similarly, in triangle TRS, since TR = RS, the base angles at T and S are equal. Let's call each of these angles y. Then, the angle at R in triangle TRS would be 180° - 2y.But the problem states that angle R is three times angle P. Angle P is x, so angle R is 3x. But angle R is also part of triangle TRS, which we've denoted as 180° - 2y. Therefore, 180° - 2y = 3x.Now, I need another equation to relate x and y. Let's look at the angles around point T where the two segments intersect. The sum of the angles around a point is 360°, so the angles at T from both triangles should add up to 360°.In triangle PQT, the angle at T is 180° - 2x, and in triangle TRS, the angle at T is y (since the base angles are y). But wait, actually, in triangle TRS, the angle at T is y, but since it's an isosceles triangle, the angle at S is also y, and the angle at R is 180° - 2y.But around point T, we have two angles: one from triangle PQT and one from triangle TRS. So, the angle from triangle PQT at T is 180° - 2x, and the angle from triangle TRS at T is y. Since these two angles are adjacent and form a straight line, their sum should be 180°. Wait, no, actually, they are vertical angles, so they should be equal.Wait, I'm getting confused here. Let me think again. When two lines intersect, the opposite angles are equal. So, the angle at T in triangle PQT is equal to the angle at T in triangle TRS. Therefore, 180° - 2x = y.So, now I have two equations:1. 180° - 2y = 3x2. 180° - 2x = yLet me substitute equation 2 into equation 1.From equation 2: y = 180° - 2xSubstitute into equation 1:180° - 2(180° - 2x) = 3x180° - 360° + 4x = 3x-180° + 4x = 3x-180° + x = 0x = 180°Wait, that can't be right. An angle of 180° would mean a straight line, which doesn't make sense in a triangle. I must have made a mistake somewhere.Let me go back and check my assumptions. I assumed that the angle at T in triangle PQT is 180° - 2x, and the angle at T in triangle TRS is y. But if these angles are vertical angles, they should be equal. So, 180° - 2x = y.But then, in triangle TRS, the angle at R is 180° - 2y, which is equal to 3x.So, substituting y = 180° - 2x into 180° - 2y = 3x:180° - 2(180° - 2x) = 3x180° - 360° + 4x = 3x-180° + 4x = 3x-180° + x = 0x = 180°Again, I get x = 180°, which is impossible. Clearly, there's an error in my approach.Maybe I misapplied the angle relationships. Let me try a different approach. Since PT = QT = TR = RS, all these segments are equal, so PT = QT = TR = RS.Let me consider triangles PQT and TRS. Both are isosceles with equal sides PT = QT and TR = RS, respectively.In triangle PQT, PT = QT, so angles at P and Q are equal. Let's call each of these angles x. Then, the angle at T is 180° - 2x.In triangle TRS, TR = RS, so angles at T and S are equal. Let's call each of these angles y. Then, the angle at R is 180° - 2y.Given that angle R = 3 angle P, so 180° - 2y = 3x.Now, looking at the angles around point T. The angle from triangle PQT at T is 180° - 2x, and the angle from triangle TRS at T is y. Since these are vertical angles, they should be equal. Therefore, 180° - 2x = y.Now, we have two equations:1. 180° - 2y = 3x2. 180° - 2x = yLet me solve these equations simultaneously.From equation 2: y = 180° - 2xSubstitute into equation 1:180° - 2(180° - 2x) = 3x180° - 360° + 4x = 3x-180° + 4x = 3x-180° + x = 0x = 180°Again, I'm getting x = 180°, which is impossible. This suggests that my initial assumption about the angles might be incorrect.Perhaps I need to consider that the angles at T are not vertical angles but rather adjacent angles forming a straight line. Let me think about that.If segments PQ and RS intersect at T, then the angles formed at T are vertical angles. So, the angle between PT and RT is equal to the angle between QT and ST, and the angle between PT and ST is equal to the angle between QT and RT.Wait, maybe I need to consider the angles in a different way. Let me denote the angle at T in triangle PQT as angle PTQ, and the angle at T in triangle TRS as angle RT S.Since PT = QT, triangle PQT is isosceles with base PQ. Therefore, angles at P and Q are equal, denoted as x each. Then, angle PTQ = 180° - 2x.Similarly, in triangle TRS, since TR = RS, it's isosceles with base RS. Therefore, angles at T and S are equal, denoted as y each. Then, angle RT S = 180° - 2y.Given that angle R = 3 angle P, so angle R = 3x.But angle R is also part of triangle TRS, which has angles y, y, and 180° - 2y. Therefore, angle R = 180° - 2y = 3x.Now, looking at the angles around point T. The angles formed by the intersection of PQ and RS are vertical angles. So, angle PTQ (which is 180° - 2x) is equal to angle RT S (which is 180° - 2y). Therefore:180° - 2x = 180° - 2ySimplifying:-2x = -2yx = ySo, x equals y.Now, from the earlier equation:180° - 2y = 3xBut since x = y, substitute:180° - 2x = 3x180° = 5xx = 36°So, angle P = x = 36°, and angle R = 3x = 108°.Now, in triangle TRS, angles at T and S are equal to y, which is equal to x, so y = 36°. Therefore, angle S = y = 36°.Wait, but let me check this. In triangle TRS, angles at T and S are 36° each, and angle R is 108°. Sum of angles: 36 + 36 + 108 = 180°, which checks out.Similarly, in triangle PQT, angles at P and Q are 36° each, and angle PTQ is 180 - 2*36 = 108°. Sum of angles: 36 + 36 + 108 = 180°, which also checks out.Therefore, angle S is 36°, which corresponds to option A.I think this makes sense now. Initially, I was confused about the angle relationships, but by carefully considering the vertical angles and setting up the equations correctly, I arrived at the solution.