answer:To find the percentage difference in the gravitational pull between the equator and the north pole, we need to calculate the gravitational pull at both locations and then compare them. The force of gravity is inversely proportional to the square of the distance from the center of mass. We are given the equatorial radius as the baseline, and its gravitational pull is set to 100%, or 1. This means we can use the equation F ∝ 1/r^2, and for the equator, we can set it as 1 = k / (6399 km)^2, where k is the constant of proportionality. To find the gravitational pull at the north pole, we can use the same equation with the polar radius: F_pole = k / (6357 km)^2. We can then divide the two equations to eliminate k and find the ratio of the gravitational pull at the pole to that at the equator: F_pole / F_equator = (6399 km)^2 / (6357 km)^2 = (6399 / 6357)^2 ≈ 1.0066. Since the gravitational pull at the equator is set to 1, or 100%, the gravitational pull at the pole is 1.0066 times that at the equator, or 100.66%. The percentage difference is then (1.0066 - 1) * 100 ≈ 0.66%. So, the gravitational pull at the north pole is approximately 0.66% stronger than that at the equator.

answer:We know that f(x) is inversely proportional to x^2, which means f(x) = k / x^2, where k is a constant of proportionality. We also know that f(6399 km) = 1, so 1 = k / (6399 km)^2. To find the value of k, we can multiply both sides by (6399 km)^2, which gives k = (6399 km)^2. Now, we can express f(x) as f(x) = (6399 km)^2 / x^2. We are also given g(x) = x * f(x), so g(x) = x * (6399 km)^2 / x^2 = (6399 km)^2 / x. To find the change in g(x) as x goes from 6399 km to 6357 km, we can calculate g(6399 km) and g(6357 km): g(6399 km) = (6399 km)^2 / 6399 km = 6399 km g(6357 km) = (6399 km)^2 / 6357 km ≈ 6410.9 km Since g(6357 km) > g(6399 km), g(x) increases as x goes from 6399 km to 6357 km. To find the percentage change, we can calculate: ((6410.9 km - 6399 km) / 6399 km) * 100% ≈ 0.19% So, g(x) increases by approximately 0.19% as x goes from 6399 km to 6357 km.

answer:1. **Necessary Condition**: - Let the lengths of the sides of triangle ABC be a, b, and c with a leqslant c. - Define A B = c, B C = a, and C A = b. - Consider the incircle of triangle ABC. Let M be the midpoint of AC. - Let BM intersect the incircle at points P and Q. - Let BS and CT be tangents to the incircle of the triangle from vertex B and C, respectively. - We have the points of intersection S on BC and T on CA. We want to show the segments BP, PQ, and QM are equal if and only if: [ frac{a}{5} = frac{b}{10} = frac{c}{13} ] 2. **Midline Properties**: - Using segment properties: [ BM^2 = frac{1}{2}BC^2 + frac{1}{2}BA^2 - frac{1}{4}CA^2 ] Substitute the known lengths: [ BM^2 = frac{1}{2} a^2 + frac{1}{2} c^2 - frac{1}{4} b^2 ] Since BP = PQ = QM = x, we get: [ 9x^2 = frac{1}{2} a^2 + frac{1}{2} c^2 - frac{1}{4} b^2 ] 3. **Use of Tangent Segment Relationship**: - From point B to S: [ BS = frac{a+c-b}{2} ] Since BS^2 = BP cdot BQ, we substitute: [ left(frac{a+c-b}{2}right)^2 = 2x^2 ] 4. **Another Pair of Tangents**: - The segments that join the tangents CT and the midline: [ MT = frac{b}{2} - frac{a+b-c}{2} = frac{c-a}{2} ] Using: [ MQ cdot MP = MT^2 implies 2x^2 = left(frac{c-a}{2}right)^2 ] 5. **Combining the Equations**: - From equations derived: [ 9x^2 = frac{1}{2} a^2 + frac{1}{2} c^2 - frac{1}{4} b^2 ] [ left(frac{a+c-b}{2}right)^2 = 2 x^2 ] [ 2 x^2 = left(frac{c-a}{2}right)^2 ] By solving these equations simultaneously, we find: [ a : b : c = 5 : 10 : 13 ] - This shows that the only way for BP = PQ = QM is if the sides follow the proportion above. 6. **Sufficient Condition:** - Let (a = 5), (b = 10), and (c = 13). - Calculate the required segments: [ BM = 6sqrt{2}, quad BS = 4, quad MT=4 ] - Verifying area relationships: [ MT^2 = MQ = 16, quad MP = MQ = 2sqrt{2} ] Thus, [ BP=PQ=QM = 2sqrt{2} ] - As all three segments are equal: BP = PQ = QM. This verifies the correctness. In conclusion, the necessary and sufficient condition for the internal tangents to be equal in triangle ABC is: [ boxed{frac{a}{5} = frac{b}{10} = frac{c}{13}} ]

answer:Based on the data, there appears to be a significant decrease in the number of hours spent on social media as age increases. The 18-24 age group spends the most time on social media, with a mean of 15 hours per week, followed by the 25-34 age group with a mean of 10 hours per week, and the 35-44 age group with a mean of 8 hours per week. The standard deviations for each age group suggest that the data is relatively consistent within each group, with the 18-24 age group showing the most variability (standard deviation of 2.5 hours). This could indicate that younger adults are more diverse in their social media usage, with some individuals spending significantly more time on social media than others. Comparing the means across the three age groups, we can see that there is a decrease of 5 hours per week between the 18-24 and 25-34 age groups, and a further decrease of 2 hours per week between the 25-34 and 35-44 age groups. This suggests that social media usage tends to decrease as individuals enter their mid-to-late 20s and continues to decline into their 30s and 40s. These findings could be attributed to various factors, such as changes in lifestyle, career demands, or shifting priorities. For example, younger adults (18-24) may have more free time and be more inclined to use social media for socializing and entertainment, while older adults (25-44) may have more responsibilities, such as work and family obligations, that reduce their available time for social media. Recommendations for further study: 1. Investigate the underlying reasons for the decline in social media usage with age. This could involve collecting additional data on participants' lifestyle, career, and priorities to better understand the factors influencing their social media habits. 2. Examine the types of social media platforms used by each age group. Different platforms may appeal to different age groups, and understanding these preferences could help explain the observed differences in social media usage. 3. Consider collecting longitudinal data to track changes in social media usage over time. This would allow researchers to examine whether individual social media habits change as they age, and whether the observed decline in social media usage is a cohort effect or a true age-related phenomenon. By exploring these questions, future studies can provide a more nuanced understanding of the relationship between age and social media usage, and shed light on the underlying factors driving these differences.