answer:We have set f(x) = (e^(kx) + e^(-kx))/2 and g(x) = (e^(kx) - e^(-kx))/2, which satisfies the functional equations but does not meet the initial conditions f(1) = 1 and g(1) = 1. Let's adjust the coefficients in the expressions for f and g while maintaining their forms. We set f(x) = a(e^(kx) + e^(-kx)) and g(x) = b(e^(kx) - e^(-kx)), where a and b are constants to be determined. The condition f(1) = 1 gives us a(e^k + e^(-k)) = 1, and the condition g(1) = 1 gives us b(e^k - e^(-k)) = 1. To simplify the expressions, we can let c = e^k and d = e^(-k). Then, the conditions become a(c + d) = 1 and b(c - d) = 1. Solving these equations for a and b, we get a = 1/(c + d) and b = 1/(c - d). Substituting these values back into the expressions for f and g, we get f(x) = (e^(kx) + e^(-kx))/(e^k + e^(-k)) and g(x) = (e^(kx) - e^(-kx))/(e^k - e^(-k)). However, these expressions are not yet in a form that directly matches the hyperbolic function analogy. To simplify them, we can rewrite the expressions using the fact that c = e^k and d = e^(-k). By substituting c and d back into the expressions for f and g, we get f(x) = (c^x + d^x)/(c + d) and g(x) = (c^x - d^x)/(c - d). Notice that c and d are reciprocals of each other, i.e., cd = 1. Using this fact, we can further simplify the expressions for f and g. After simplification, we can express f and g in terms of c, which satisfies the given functional equations and initial conditions. The constant c can be determined using the condition f(1) = 1, which leads to a specific value for c that satisfies the given conditions. Ultimately, the adjustment of coefficients in the expressions for f and g allows us to satisfy the given initial conditions while maintaining the forms of the functions, providing a direct solution to the problem that aligns with the hyperbolic function analogy.

answer:1. Let the two numbers be (x) and (y). According to the problem, (x - y) and (x + y) must satisfy: [ (x - y)^2 : (x + y)^3 = 4 : 27 ] Also, we have: [ x + y = 5(x - y) + 3 ] 2. Express the ratio condition: [ frac{(x - y)^2}{(x + y)^3} = frac{4}{27} ] Rearranging gives: [ 27(x - y)^2 = 4(x + y)^3 ] 3. Substitute (x + y = 5(x - y) + 3) into the equation and simplify: [ 27(x - y)^2 = 4(5(x - y) + 3)^3 ] Let (d = x - y), then: [ 27d^2 = 4(5d + 3)^3 ] Solving this equation for (d), we find: [ d = 1 quad text{(among possible solutions after simplifying the equation)} ] 4. Substitute (d = 1) back into (x + y = 5d + 3): [ x + y = 5(1) + 3 = 8 ] 5. Calculate the product (xy) using (x = frac{x+y + x-y}{2}) and (y = frac{x+y - x-y}{2}): [ x = frac{8+1}{2} = 4.5, quad y = frac{8-1}{2} = 3.5 ] [ xy = 4.5 times 3.5 = 15.75 ] Thus, the product of the two numbers is (15.75). The final answer is boxed{textbf{(B)} 15.75}

answer:1. **Calculate the rate per dozen based on the provided cost and quantity:** Given that a half dozen costs 4.80, a full dozen would cost: [ text{Cost per dozen} = frac{4.80}{0.5} = 9.60 ] 2. **Calculate the cost for four dozen apples:** Using the rate of 9.60 per dozen, the cost for four dozen apples would be: [ text{Cost for four dozen} = 4 times 9.60 = 38.40 ] 3. **Conclusion with boxed answer:** [ boxed{38.40} ]

answer:This problem beautifully illustrates the relationships between tangents, radii, and the properties of right triangles in Euclidean geometry. First, let's establish some notation for the points of tangency: we'll call them A and B. The tangents from P to the circle at points A and B are segments PA and PB. Since PA and PB are tangents to the circle, we know that the radii CA and CB are perpendicular to PA and PB, respectively. This means that right triangles PAC and PBC are formed. Moreover, because PA and PB are tangents from the same external point P, their lengths are equal. Let's denote this length as x. Applying the Pythagorean Theorem to either triangle PAC or PBC (since they are congruent), we have x^2 + 5^2 = 13^2. Solving for x, we get x^2 = 144, and thus x = 12. Now, to find the distance between the two points of tangency, we can observe that this distance is equal to the length of segment AB. This can be seen by noting that angle APB is bisected by the line passing through the center of the circle (due to symmetry), making segments ACB and APB similar, and hence their corresponding sides proportional. However, a simpler approach to find AB is by recognizing that the distance between the two points of tangency can also be viewed as the base of an isosceles trapezoid CAPB, whose diagonals are CP and AB. Alternatively, we can leverage the fact that triangle APB and the triangle connecting the center C with the midpoints of PA and PB, and the midpoint of AB, are similar. The distance from the midpoint of PA (or PB) to C is 5 (the radius), and from this midpoint to the midpoint of AB is (1/2)AB. Also, the distance from P to the midpoint of AB is x = 12. Applying similar triangles, the ratio between (1/2)AB, 5, and 12 (half of the distance between P and C) will give us AB. But the most straightforward way is through applying Pythagorean Theorem for the right triangle PAB height, PH = 12 (same as the length of PA or PB since it is an isosceles trapezoid and PH is its height), we are not directly using PH but we are using this right triangle in mind. Since PH is dividing CAPB to two right triangles PHB, PHA and as PHA or PHB has height 12, then AB is derived directly from Pythagorean as follows: AB^2 = (PC)^2 - (PH)^2 then substitute PC value which is 13 and PH = 12, that follows AB^2 = 169 - 144 = 25, and hence AB is sqrt(25) = 5. Interestingly, we see that the length of AB, which is the distance between the points of tangency, is equal to the radius of the circle. This is not a coincidence and holds in general for tangents from an external point to a circle - a property that reflects the intricate symmetry of Euclidean geometry. The length of each tangent (PA or PB) is 12 units, and the distance between the two points of tangency (AB) is 5 units, reflecting the geometric relationships between tangents, radii, and the properties of right triangles in this configuration.