MIT

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question:Compute cos 210^circ.

answer:Since the cosine function has a period of 360^circ, we can subtract 360^circ from 210^circ to find an equivalent angle in the interval [0^circ, 360^circ). 210^circ - 360^circ = -150^circ. Since the cosine function is an even function, cos (-150^circ) = cos 150^circ. Using the unit circle, we see that cos 150^circ = -frac{sqrt{3}}{2}. Therefore, cos 210^circ = boxed{-frac{sqrt{3}}{2}}.The answer is: -frac{sqrt{3}}{2}

question:There were x peaches in each basket. Five baskets of peaches were delivered to a market. The farmers have eaten 5 peaches. The remaining peaches are packed into smaller boxes of 15 each. How many boxes of peaches are there? If we know the answer to the above question is 8, what is the value of unknown variable x?

answer:There were x peaches in each basket. Five baskets of peaches were delivered to the market, so the total number of peaches is 5x. The farmers ate 5 peaches, so the remaining peaches are 5x - 5. These remaining peaches are packed into smaller boxes of 15 each, so the number of boxes is (5x - 5) / 15. We are given that the number of boxes is 8, so we can write: (5x - 5) / 15 = 8. Multiplying both sides by 15, we get: 5x - 5 = 120. Adding 5 to both sides, we get: 5x = 125. Dividing both sides by 5, we get: x = 25. The value of x is 25. 25 The answer is: 25

question:Paul goes fishing every Saturday. Last week he was able to catch 5 fish for every 2 hours he was fishing. He caught 30 fish when he was fishing for x hours. What is the value of unknown variable x?

answer:To solve this problem, we need to determine the value of x, which represents the number of hours Paul was fishing. We know that Paul caught 5 fish for every 2 hours he was fishing. Therefore, the rate of catching fish is 5 fish / 2 hours. Paul caught 30 fish in total. We can set up the equation as follows: Rate of catching fish * Number of hours fishing = Total number of fish caught (5 fish / 2 hours) * x hours = 30 fish Let's simplify and solve for x: (5/2)x = 30 To isolate x, we multiply both sides of the equation by 2/5: (5/2)x * (2/5) = 30 * (2/5) x = 12 The value of x is 12. 12 The answer is: 12

question:The number 2^{1993}+3^{1993} is a multiple of X What is the units digit of the quotient frac{2^{1993}+3^{1993}}{5}? The answer is 3. What is the value of unknown variable X?

answer:To solve this problem, we need to determine the value of x, which represents the number that the expression 2^{1993}+3^{1993} is a multiple of. We know that the units digit of the quotient frac{2^{1993}+3^{1993}}{5} is 3. This means that the expression 2^{1993}+3^{1993} ends in 3 when divided by 5. In order for the expression to end in 3 when divided by 5, x must be equal to 5. Therefore, the value of unknown variable x is 5. The answer is: 5