Math - Geometry: Consider a rectangle with a length of 12 units and a width of 8 units. If you draw the diagonal of the rectangle, what is the length of the diagonal, and why?
Physics - Free Fall: An object is dropped from a height of 80 meters. Ignoring air resistance, how long will it take for the object to reach the ground, given the acceleration due to gravity is approximately 10 m/s^2?
Math - Probability: A fair six-sided die is rolled. What is the probability of getting an even number or a number greater than 4?
Physics - Optics: When light passes through a glass prism, it bends and spreads into a spectrum of colors. This phenomenon is called dispersion. What causes dispersion, and why do different colors of light bend by different amounts?
Math - Sequences: Consider the arithmetic sequence 3, 7, 11, 15, … . What is the 10th term of the sequence, and how did you find it?
Physics - Static Electricity: When you rub a balloon on your hair and then hold the balloon close to small pieces of paper, the paper will be attracted to the balloon. What causes this attraction between the balloon and the paper?
Math - Pythagorean Triples: A Pythagorean triple consists of three positive integers a, b, and c, such that a^2 + b^2 = c^2. One example is (3, 4, 5). Can you find another Pythagorean triple?
Physics - Buoyancy: When you place an object in water, it may either float or sink. What determines whether the object will float or sink, and how does the shape of the object affect this outcome?
Math - Algebra: Solve the following system of linear equations for x and y: 2x + 3y = 7 x - y = 3
Physics - Sound: When a piano key is struck, a string inside the piano vibrates to produce a sound. What causes the string to produce sound, and how does the length and tension of the string affect the pitch of the sound?
Math - Geometry: The length of the diagonal of the rectangle can be found using the Pythagorean theorem: diagonal^2 = length^2 + width^2. So, diagonal = sqrt(12^2 + 8^2) = sqrt(144 + 64) = sqrt(208) = 4√13 units.
Physics - Free Fall: Using the formula t = sqrt(2h/g), where h is the height and g is the acceleration due to gravity, we get t = sqrt(2 × 80/10) = sqrt(16) = 4 seconds.
Math - Probability: The probability of getting an even number (2, 4, or 6) is 3/6 = 1/2. The probability of getting a number greater than 4 (5 or 6) is 2/6 = 1/3. The numbers 6 is counted twice, so we subtract its probability, which is 1/6. Thus, the probability of getting an even number or a number greater than 4 is 1/2 + 1/3 - 1/6 = (3 + 2 - 1)/6 = 4/6 = 2/3.
Physics - Optics: Dispersion occurs because different colors (wavelengths) of light travel at slightly different speeds through the glass. As a result, they refract (bend) by different amounts, causing the light to spread into a spectrum of colors.
Math - Sequences: The common difference of the arithmetic sequence is 4. To find the 10th term, use the formula: T_n = a + (n - 1)d, where T_n is the nth term, a is the first term, n is the term number, and d is the common difference. T_10 = 3 + (10 - 1)4 = 3 + 36 = 39.
Physics - Static Electricity: The attraction between the balloon and the paper is caused by static electricity. Rubbing the balloon on your hair transfers electrons from your hair to the balloon, giving the balloon a negative charge. This negatively charged balloon attracts the positively charged paper.
Math - Pythagorean Triples: One example of another Pythagorean triple is (5, 12, 13), because 5^2 + 12^2 = 25 + 144 = 169 = 13^2.
Physics - Buoyancy: Whether an object floats or sinks is determined by its density relative to the density of the fluid (in this case, water). If the object’s density is less than the fluid’s density, it will float; if it’s greater, it will sink. The shape of the object affects its volume and surface area, which can influence its buoyancy.
Math - Algebra: To solve the system of linear equations, we can use the substitution method. From the second equation, we get x = y + 3. Substituting this into the first equation, we get 2(y + 3) + 3y = 7. Solving for y, we get y = 1. Substituting this back into the equation for x, we get x = 1 + 3 = 4. So, x = 4 and y = 1.
Physics - Sound: When the piano string vibrates, it creates pressure waves in the air, which we perceive as sound. The length and tension of the string affect the frequency of the vibration. A shorter or tighter string will produce a higher frequency