answer:To solve the problem, we need to determine the position of point ( K ) along segment ( AB ) such that the sum of the lengths of segments ( AK ) and ( KB ) is minimized. Since the total length of ( AB ) is given as 9 cm, let's denote the distance from point ( A ) to point ( K ) as ( x ). Consequently, the distance from point ( K ) to point ( B ) will be ( 9 - x ).Our goal is to minimize the sum of ( AK ) and ( KB ). Mathematically, this can be expressed as:[ f(x) = x + (9 - x) ]Let's break down the steps to solve this problem:1. Define the length of the segment ( AB ) as 9 cm.2. Let ( x ) be the distance from point ( A ) to point ( K ).3. Express the sum of the lengths ( AK ) and ( KB ) as a function ( f(x) = x + (9 - x) ).4. Find the value of ( x ) that minimizes this function.Let's implement this in Python using the sympy library:```pythonimport sympy as sp# Define variablesx = sp.symbols('x')# Length of segment ABlength_AB = 9# Define the function to minimize: f(x) = x + (9 - x)f = x + (length_AB - x)# Minimize the function f(x)min_value = sp.simplify(f)print(min_value)``````output9```The output (9) essentially reflects the total length of the segment (AB). From the problem statement, we need to find the specific distance from point (A) to point (K) such that the sum of the lengths (AK) and (KB) is minimized.Notice that this problem is somewhat trivial as the sum (AK + KB) is always equal to the total length of (AB), regardless of the position of (K). Therefore, any point (K) on the segment (AB) will result in the same sum, which is the length of (AB).Since the sum (AK + KB) will always equal (9) cm, (K) can be placed at any point along segment (AB), and the sum will still be minimized and equal to (9) cm. Thus, the distance from (A) to (K) can be any value from (0) to (9) cm, inclusive. In other words, the sum of (AK) and (KB) doesn't actually change regardless of where (K) is.Therefore, the final answer is:(boxed{K text{ can be placed at any point along segment } AB})

answer:To solve this problem, we need to determine the cutoff score that ranks a student among the top 9100 out of 400,000 students in a normal distribution (N(98, 100)).Here's how we can break down the problem:1. Understand the Distribution: The scores follow a normal distribution (N(98, 100)), which means the mean (mu = 98) and the standard deviation (sigma = 10).2. Determine the Required Percentile: We need to find the score that corresponds to the top (9100) students out of (400,000). This means we are looking for the score at the percentile (1 - frac{9100}{400000}).3. Use the Percentile to Find the Z-Score: Using the cumulative distribution function (CDF) of the normal distribution, we can find the z-score corresponding to the required percentile.4. Convert the Z-Score to the Actual Score: Using the mean and standard deviation, we can convert the z-score to the actual score.Let's implement this step-by-step using Python and Sympy:```pythonimport sympy as spimport scipy.stats as stats# Given datamean = 98std_dev = 10total_students = 400000top_students = 9100# Calculate the required percentilerequired_percentile = 1 - (top_students / total_students)# Find the z-score corresponding to the required percentilez_score = stats.norm.ppf(required_percentile)# Convert the z-score to the actual scorerequired_score = mean + z_score * std_devprint(required_score)``````outputTraceback (most recent call last): File " import scipy.stats as statsModuleNotFoundError: No module named 'scipy'```Since the `scipy` library is not available, we can use the `sympy` library to calculate the inverse of the cumulative distribution function (CDF) for the normal distribution. Sympy provides tools to work with probability distributions.Here's the modified implementation using `sympy`:```pythonimport sympy as spfrom sympy import stats# Given datamean = 98std_dev = 10total_students = 400000top_students = 9100# Calculate the required percentilerequired_percentile = 1 - (top_students / total_students)# Define the normal distributionX = stats.Normal('X', mean, std_dev)# Find the score corresponding to the required percentilerequired_score = sp.N(stats.quantile(X)(required_percentile))print(required_score)``````output118.000024438996```The student's math score will not be lower than (boxed{118}) points.

answer:To solve the system of equations and find the value of (xy + yz + xz), let's proceed with a systematic approach.We have:[left{begin{array}{l}x^{2}+xy+y^{2}=75 y^{2}+yz+z^{2}=4 z^{2}+xz+x^{2}=79end{array}right.]Let's consider the equations:1. ( x^2 + xy + y^2 = 75 )2. ( y^2 + yz + z^2 = 4 )3. ( z^2 + xz + x^2 = 79 )Sum all three equations:[ (x^2 + xy + y^2) + (y^2 + yz + z^2) + (z^2 + xz + x^2) = 75 + 4 + 79 ][ 2x^2 + 2y^2 + 2z^2 + xy + yz + xz = 158 ]Factoring out the common term:[ 2(x^2 + y^2 + z^2) + xy + yz + xz = 158]Rearrange to isolate (xy + yz + xz):[ xy + yz + xz = 158 - 2(x^2 + y^2 + z^2)]We need to find (x^2 + y^2 + z^2). From the rewritten equations, we can extract:1. ( x^2 + y^2 = 75 - xy )2. ( y^2 + z^2 = 4 - yz )3. ( z^2 + x^2 = 79 - xz )Add these three together:[ (x^2 + y^2) + (y^2 + z^2) + (z^2 + x^2) = 158 - (xy + yz + xz) ][ 2(x^2 + y^2 + z^2) = 158 - (xy + yz + xz) ]We still need that one missing piece. And it looks like we can solve these equations iteratively. Let's use Python to calculate (x, y, z) that satisfy the system of equations and compute the desired value (xy + yz + xz).```pythonimport numpy as np from scipy.optimize import fsolve # Define the system of equations def equations(vars): x, y, z = vars eq1 = x2 + x*y + y2 - 75 eq2 = y2 + y*z + z2 - 4 eq3 = z2 + x*z + x2 - 79 return [eq1, eq2, eq3] # Initial guess initial_guess = [1, 1, 1] # Solve the system of equations x, y, z = fsolve(equations, initial_guess) # Calculate the desired expression result = x*y + y*z + x*z print(((x, y, z), result))``````output((8.141523424439642, 0.9578262852360377, 1.3409567993233407), 20.000000000403674)```The values of (x), (y), and (z) that satisfy the system of equations are approximately (x approx 8.142), (y approx 0.958), and (z approx 1.341).The value of the expression (xy + yz + xz) is approximately:[boxed{20}]

answer:To solve this problem, we'll start by calculating the number of certificates awarded to boys and girls separately. Then, we'll determine the total number of students who received certificates and express this number as a percentage of the total number of students who participated in the competition.Let's break down the steps:1. Calculate 10% of the boys who received certificates.2. Calculate 20% of the girls who received certificates.3. Sum the total number of students who received certificates.4. Calculate the total number of students who participated.5. Determine the percentage of students who received certificates.We'll implement these steps using Python code for accuracy.```python# Number of boys and girlsboys = 30girls = 20# Percentage of boys and girls who received certificatespercent_boys_certificates = 10 / 100percent_girls_certificates = 20 / 100# Calculate the number of boys and girls who received certificatesboys_with_certificates = boys * percent_boys_certificatesgirls_with_certificates = girls * percent_girls_certificates# Total number of students who received certificatestotal_with_certificates = boys_with_certificates + girls_with_certificates# Total number of students who participatedtotal_students = boys + girls# Calculate the percentage of students who received certificatespercentage_with_certificates = (total_with_certificates / total_students) * 100print(percentage_with_certificates)``````output14.000000000000002```Based on the calculations, the percentage of students who received certificates is approximately 14%.Thus, the final answer is (boxed{14}).

answer:To solve the problem, we need to evaluate the expression (sqrt{4} - 1).Let's break it down into manageable steps:1. Calculate the square root of 4.2. Subtract 1 from the result.Let's implement this in Python using the sympy library.```pythonimport sympy as sp# Calculate the square root of 4sqrt_4 = sp.sqrt(4)# Subtract 1 from the resultresult = sqrt_4 - 1# Print the resultprint(result)``````output1```The result of the expression (sqrt{4} - 1) is (boxed{1}).

answer:To solve (displaystyle frac{66,!666^4}{22,!222^4}), let's perform the calculation step-by-step.First, we'll start by rewriting the expression:[frac{66,!666^4}{22,!222^4} = left( frac{66,!666}{22,!222} right)^4]Next, calculate the value of the fraction inside the parenthesis:[frac{66,!666}{22,!222}]We can simplify this fraction to a simpler form and then raise the result to the fourth power. We'll use Python to ensure the calculation is accurate.```python# Define the numbersnumerator = 66666denominator = 22222# Calculate the fractionfraction = numerator / denominator# Raise the fraction to the power of 4result = fraction 4print(result)``````output81.0```The value of (frac{66,!666}{22,!222}) simplifies to 3, since:[frac{66,!666}{22,!222} = 3]Therefore, the expression (left( frac{66,!666}{22,!222} right)^4) simplifies to:[3^4 = 81]So the final answer is (boxed{81}).