a) Dalton created a mixture composed of 90% water and 10% carbon, weighing 20 pounds. After 2 days, the water content decreases from 90% to 50%. How much does the mixture weigh now?

b) You have two identical balls in terms of diameter and weight. One is solid, and the other is hollow. Without touching the balls, how can you identify the solid ball?

c) There are four gears with the following specifications: Gear A has 60 teeth, Gear B has 40 teeth, Gear C has 20 teeth, and Gear D has 60 teeth. Gear B makes 15 complete turns every minute. What is the relative speed of Gear A compared to Gear B?

d) A man is trapped on a frictionless frozen lake with no grip. He has a mobile phone with no reception and there is no wind. How can he escape the lake?

e) I cannot be seen, touched, or felt, but I can cook your lunch. What am I?

a) The new weight of the mixture is 36 pounds. b) By comparing the moments of inertia, we can identify the solid ball. c) The relative speed of Gear A to Gear B is 90 degrees per minute.

Explanation:

a) To find out the weight of the mixture after the water quantity was reduced, we can calculate the weight of water and carbon separately. Initially, the weight of water was 90% of 20 pounds which is (90/100) * 20 = 18 pounds. After the reduction, 50% of the mixture weight is due to water, so the weight of water is (50/100) * x = 0.5x pounds, where x is the new weight of the mixture. Since the weight of carbon remains constant, it contributes 10% of x pounds, which is (10/100) * x = 0.1x pounds. So, the equation becomes: 0.5x + 0.1x = 18 pounds. Solving this equation, we find that x = 36 pounds. Therefore, the new weight of the mixture is 36 pounds.

b) To identify the solid ball among two identical balls, we can compare their moments of inertia. The solid ball will have a higher moment of inertia compared to the hollow ball. By simply measuring the time it takes for each ball to roll down an inclined plane, we can determine which ball has a greater resistance to motion, indicating the solid ball.

c) To find the relative speed between Gear A and Gear B, we need to compare the number of teeth on each gear. Gear B makes 15 complete turns in a minute, which means it covers 15 * 360 = 5400 degrees in a minute. Gear A has 60 teeth, so it covers 5400 degrees / 60 teeth = 90 degrees per tooth rotation. Therefore, Gear A rotates 90 degrees every minute, while Gear B rotates 360 degrees every minute. This means the relative speed of Gear A to Gear B is 90 degrees per minute.

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