answer:First, we evaluate ( g(4) ): [ g(4) = 3 cdot 4^2 - 3 cdot 4 - 4 = 3 cdot 16 - 12 - 4 = 48 - 12 - 4 = 32 ] Next, we calculate ( f(g(4)) = f(32) ): [ f(32) = 3sqrt{32} + frac{18}{sqrt{32}} ] We simplify ( sqrt{32} ) as ( 4sqrt{2} ), thus: [ f(32) = 3 cdot 4sqrt{2} + frac{18}{4sqrt{2}} ] To rationalize the denominator of the second term: [ frac{18}{4sqrt{2}} = frac{18 cdot sqrt{2}}{4 cdot 2} = frac{18sqrt{2}}{8} = frac{9sqrt{2}}{4} ] Thus: [ f(32) = 12sqrt{2} + frac{9sqrt{2}}{4} = 12sqrt{2} + 2.25sqrt{2} = (12 + 2.25)sqrt{2} = 14.25sqrt{2} ] In exact form, this is: [ boxed{14.25sqrt{2}} ]

answer:1. **Define the coordinates of the points:** - Let C be at the origin (0, 0). - Let A be at (18, 0) since AC = 18. - Let E be at (0, 18) since CE = 18 and AC perp CE. 2. **Find the midpoints B and D:** - B is the midpoint of AC, so B = left(frac{18+0}{2}, frac{0+0}{2}right) = (9, 0). - D is the midpoint of CE, so D = left(frac{0+0}{2}, frac{0+18}{2}right) = (0, 9). 3. **Find the equations of lines EB and AD:** - Line EB passes through E(0, 18) and B(9, 0). The slope of EB is frac{0-18}{9-0} = -2. Thus, the equation of EB is y = -2x + 18. - Line AD passes through A(18, 0) and D(0, 9). The slope of AD is frac{9-0}{0-18} = -frac{1}{2}. Thus, the equation of AD is y = -frac{1}{2}x + 9. 4. **Find the intersection point F of EB and AD:** - Set the equations equal to each other: -2x + 18 = -frac{1}{2}x + 9. - Solve for x: [ -2x + 18 = -frac{1}{2}x + 9 implies -2x + frac{1}{2}x = 9 - 18 implies -frac{3}{2}x = -9 implies x = 6 ] - Substitute x = 6 back into y = -2x + 18 to find y: [ y = -2(6) + 18 = -12 + 18 = 6 ] - Thus, F is at (6, 6). 5. **Calculate the area of triangle BDF:** - Use the vertices B(9, 0), D(0, 9), and F(6, 6). - The area of a triangle with vertices (x_1, y_1), (x_2, y_2), (x_3, y_3) is given by: [ text{Area} = frac{1}{2} left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) right| ] - Substituting the coordinates: [ text{Area} = frac{1}{2} left| 9(9 - 6) + 0(6 - 0) + 6(0 - 9) right| = frac{1}{2} left| 9 cdot 3 + 0 + 6 cdot (-9) right| = frac{1}{2} left| 27 - 54 right| = frac{1}{2} left| -27 right| = frac{27}{2} = 13.5 ] The final answer is boxed{13.5}

answer:Firstly, let's rewrite the function f(x) in terms of the completed square to make it simple for comparison purposes: f(x) = x^2 - 2x + c = (x - 1)^2 + c - 1 The expression (x - 1)^2 is always non-negative because it’s the square of a real number. Also, (x - 1)^2 represents a parabola that opens upwards and has its vertex at (1, c-1). Since the parabola opens upwards and (x - 1)^2 reaches its minimum value of 0 when x = 1, it is clear that the value of f(x) at x = 0 will be less than the value at x = 4 or x = -4 due to the symmetry of the parabola about the line x = 1. Therefore, we can assert that f(0) is the smallest among the three. Next, let's evaluate f(x) for x = 4 and x = -4: For x = 4: f(4) = (4 - 1)^2 + c - 1 = 3^2 + c - 1 = 9 + c - 1 = 8 + c For x = -4: f(-4) = (-4 - 1)^2 + c - 1 = (-5)^2 + c - 1 = 25 + c - 1 = 24 + c Now, we have to compare f(4) and f(-4). Since 8+c is always less than 24+c for any real number c, we have: f(4) < f(-4) Combining the two comparisons, we come to the conclusion that: f(0) < f(4) < f(-4) Therefore, the correct inequality is option C. boxed{text{C: } f(0) < f(4) < f(-4)}

answer:Let's denote the number as N. According to the first part of the problem, we have: (1/4) * (1/3) * (2/5) * N = 17 To find N, we can solve for it by multiplying both sides of the equation by the reciprocal of the fraction on the left side: N = 17 * (4/1) * (3/1) * (5/2) N = 17 * 4 * 3 * (5/2) N = 17 * 2 * 3 * 5 N = 34 * 3 * 5 N = 102 * 5 N = 510 Now, according to the second part of the problem, we have: √(0.6 * N) = (N^(1/3)) / 2 Let's substitute the value of N we found into this equation: √(0.6 * 510) = (510^(1/3)) / 2 First, let's calculate 0.6 * 510: 0.6 * 510 = 306 Now, let's take the square root of 306: √306 ≈ 17.49 (rounded to two decimal places) Now, let's calculate the cube root of 510 and then divide it by 2: (510^(1/3)) / 2 ≈ (8.02) / 2 (rounded to two decimal places) (510^(1/3)) / 2 ≈ 4.01 (rounded to two decimal places) Since √(0.6 * N) is supposed to be equal to (N^(1/3)) / 2, we can see that the values we've calculated are approximately equal (17.49 ≈ 4.01), which means our value for N is consistent with the given conditions. Finally, we need to find 40% of N: 0.4 * N = 0.4 * 510 0.4 * N = 204 Therefore, 40% of the number is boxed{204} .