answer:1. Simplify the given expression: [ frac{x^4}{x^2} = x^2 ] This reduces the problem to solving the inequality x^2 < 18. 2. Determine the square roots to find the maximum integer x: Since 4^2 = 16 < 18 and 5^2 = 25 > 18, the largest integer x for which x^2 < 18 is 4. 3. Conclude with the final answer: [ boxed{4} ]

answer:: 1. Start with the given equation: [ x^{2} + frac{36}{x^{2}} = 13 ] 2. Multiply through by ( x^2 ) to clear the fraction: [ x^2 cdot x^2 + x^2 cdot frac{36}{x^2} = 13 cdot x^2 ] [ x^4 + 36 = 13x^2 ] 3. Rearrange the equation to bring all terms to one side: [ x^4 - 13x^2 + 36 = 0 ] 4. Recognize that this is a quadratic equation in terms of ( y = x^2 ): [ y^2 - 13y + 36 = 0 ] 5. Solve the quadratic equation using the factorization method. We need two numbers that multiply to 36 and add to -13: [ y^2 - 13y + 36 = (y - 4)(y - 9) = 0 ] 6. Thus, the roots of the quadratic equation are: [ y - 4 = 0 quad text{or} quad y - 9 = 0 ] [ y = 4 quad text{or} quad y = 9 ] 7. Recall that ( y = x^2 ), so: [ x^2 = 4 quad text{or} quad x^2 = 9 ] 8. Taking the square root of both sides, we get: [ x = pm 2 quad text{or} quad x = pm 3 ] Therefore, the solutions are ( x = pm 2 ) and ( x = pm 3 ). [ boxed{x = pm 2 text{ and } x = pm 3} ]

answer:Let's break down the problem step by step: 1. There are 10 neighborhoods. 2. Each neighborhood has 4 roads passing through it. 3. Therefore, the total number of roads in the town is 10 neighborhoods * 4 roads/neighborhood = 40 roads. 4. The total number of street lights in the town is 20,000. 5. We need to find out how many street lights are on each opposite side of a road. Since the street lights are on each opposite side of the road, we can assume that the total number of street lights is evenly divided between the two sides. First, let's find out how many street lights there are per road: 20,000 street lights / 40 roads = 500 street lights per road. Now, since there are street lights on each opposite side of the road, we divide the number of street lights per road by 2 to find out how many are on each side: 500 street lights per road / 2 sides = 250 street lights on each opposite side of the road. So, there are boxed{250} street lights on each opposite side of each road.

answer:Consider a right triangle where the adjacent side (to the angle theta) is 5 units and the opposite side is 12 units. [asy] unitsize (0.5 cm); draw((0,0)--(5,0)--(5,12)--cycle); label("5", (2.5,0), S); label("12", (5,6), E); label("theta", (0.8,0.5)); [/asy] Given the triangle setup, [ cot theta = frac{text{adjacent}}{text{opposite}} = frac{5}{12}. ] Thus, [ theta = operatorname{arccot} frac{5}{12}. ] The tangent of this angle theta, which is the multiplication inverse of the cotangent (since tan theta = frac{1}{cot theta}), results in: [ tan (operatorname{arccot} frac{5}{12}) = frac{1}{cot theta} = frac{1}{frac{5}{12}} = frac{12}{5}. ] Thus the final answer is [ boxed{frac{12}{5}}. ]