Math

How can you determine the volume of an irregular polyhedron using the decomposition method?To determine the volume of an irregular polyhedron using the decomposition method, decompose it into simpler polyhedra like tetrahedra or prisms. Calculate the volume of each simpler shape using known formulas, then sum these volumes to obtain the total volume of the irregular polyhedron.

How do you determine the area of a regular hexagon, given only the length of one side?To determine the area of a regular hexagon given the side length ‘a’, use the formula: Area = (3√3/2) * a². This formula is derived from dividing the hexagon into six equilateral triangles, calculating the area of one triangle, and then multiplying by six.

What is the least common multiple (LCM) of 4 and 6?The least common multiple (LCM) of 4 and 6 is 12. The LCM is the smallest number that is a multiple of both 4 and 6. This can be found by determining the multiples of each number and identifying the smallest common multiple.

What is the value of sin(30 degrees)?The value of sin(30 degrees) is 0.5. This is derived from the unit circle or trigonometric functions, where the sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. For 30 degrees, this ratio is 1/2.

How do you apply the Central Limit Theorem to demonstrate that the sampling distribution of the sample mean approximates a normal distribution, even when the sample data is not normally distributed?The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean will approximate a normal distribution as the sample size becomes large, regardless of the population’s distribution. This approximation improves with larger sample sizes, typically n > 30. Therefore, even if the sample data is not normally distributed, the sample mean will tend to follow a normal distribution due to the CLT.

How do you calculate the probability of flipping a coin 10 times and getting exactly 6 heads and 4 tails using the binomial distribution formula?To calculate the probability of getting exactly 6 heads and 4 tails in 10 coin flips, use the binomial distribution formula: P(X=k) = (n choose k) * p^k * (1-p)^(n-k). Here, n=10, k=6, and p=0.5. Therefore, P(X=6) = 210 * (0.5)^6 * (0.5)^4 = 0.2051.

How does the Mean Value Theorem connect the derivative of a function to its average rate of change over an interval?The Mean Value Theorem states that for a continuous function f(x) that is differentiable on the interval (a, b), there exists at least one point c in (a, b) where the instantaneous rate of change (the derivative f'(c)) equals the average rate of change over [a, b], i.e., f'(c) = (f(b) – f(a)) / (b – a).

What is the probability of rolling a 2 on a standard six-sided die?The probability of rolling a 2 on a standard six-sided die is 1/6. This is because a standard die has six faces, each with an equal chance of landing face up. Therefore, the probability of any one specific outcome, such as rolling a 2, is 1 out of 6.

What are the maximum number of solutions to the trigonometric equation sin(2x) + cos(x) = 1 on the interval [0, 2π] and how can they be calculated?To find the maximum number of solutions to the equation sin(2x) + cos(x) = 1 on the interval [0, 2π], we first rewrite sin(2x) as 2sin(x)cos(x). Thus, the equation becomes 2sin(x)cos(x) + cos(x) = 1. Factoring out cos(x), we get cos(x)(2sin(x) + 1) = 1. Solving cos(x) = 0 and 2sin(x) + 1 = 1, we find that there are a maximum of 4 solutions in the given interval.

How can one determine whether two seemingly different polyhedra are actually topologically homeomorphic?To determine if two polyhedra are topologically homeomorphic, one must check if there exists a continuous bijection with a continuous inverse between them. This involves verifying that the polyhedra have the same Euler characteristic and can be deformed into each other without tearing or gluing.