answer:Sam originally had 9 dimes. His dad gave him 7 more dimes. To find out how many dimes Sam has now, we add the two amounts together: 9 dimes + 7 dimes = 16 dimes Sam now has boxed{16} dimes.

answer:To solve this problem, we need to determine the speed of the train. We can use the information given about the train passing the platform to find the speed. When the train passes the platform, it covers the length of the train plus the length of the platform. So the total distance covered by the train when passing the platform is: Length of the train + Length of the platform = 250 m + 1250 m = 1500 m We are given that it takes the train 60 seconds to pass the platform. Therefore, we can calculate the speed of the train (Speed = Distance / Time) as follows: Speed = 1500 m / 60 s = 25 m/s Now that we have the speed of the train, we can calculate the time it takes to pass the pole. When passing the pole, the train only needs to cover its own length, which is 250 m. Using the speed we calculated, we can find the time it takes to pass the pole (Time = Distance / Speed): Time = 250 m / 25 m/s = 10 s So, it takes the train boxed{10} seconds to pass the pole.

answer:The sum of the first three terms of the arithmetic sequence is S_{3} = a_1 + a_2 + a_3 = 3a_1 + 3d because a_2 = a_1 + d and a_3 = a_1 + 2d. Similarly, the sum of the first two terms is S_{2} = a_1 + a_2 = 2a_1 + d. Given that frac{S_{3}}{3} -frac{S_{2}}{2} = 1, we substitute the expressions for S_3 and S_2 to find: frac{3a_1 + 3d}{3} - frac{2a_1 + d}{2} = frac{a_1 + d}{3} - frac{a_1 + frac{d}{2}}{2} = 1 Simplifying, we have: a_1 + d - (a_1 + frac{d}{2}) = 2d - frac{d}{2} = frac{3d}{2} = 1 Solving for d, we find: d = frac{2}{3} cdot 1 = frac{2}{3} cdot frac{2}{2} = frac{4}{3} However, this result is not one of the options provided. Notice that there seems to be an error in proceeding with the calculation. The correct approach is: frac{3a_1 + 3d}{3} - frac{2a_1 + d}{2} = a_1 + d - (a_1 + frac{d}{2}) = d - frac{d}{2} = frac{d}{2} = 1 Solving for d gives us: d = 2 Thus, the correct option is boxed{C}.

answer:To solve for a+b+c, we start by expressing w as a complex number, w = x + yi, where x and y are real numbers. The three roots of P(z) given are w+3i, w+9i, and 2w-4. We can express the sum of these roots as follows: begin{align*} (w + 3i) + (w + 9i) + (2w - 4) &= 4w - 4 + 12i &= 4(x + yi) - 4 + 12i &= (4x - 4) + (4y + 12)i. end{align*} By Vieta's formulas, the sum of the roots of a cubic polynomial x^3+ax^2+bx+c is -a, which must be a real number. Therefore, the imaginary part of the sum of the roots, (4y + 12)i, must equal 0 for the sum to be real. This implies that 4y + 12 = 0, leading to y = -3. Substituting y = -3 back into the roots, we get: - The first root: w + 3i = x - 3i + 3i = x, - The second root: w + 9i = x - 3i + 9i = x + 6i, - The third root: 2w - 4 = 2(x - 3i) - 4 = 2x - 4 - 6i. Given that the coefficients of P(z) are real, any nonreal roots must occur in conjugate pairs. Thus, x + 6i and 2x - 4 - 6i must be conjugates. For these to be conjugates, their real parts must be equal, which gives x = 2x - 4. Solving this equation yields x = 4. With x = 4 and y = -3, we can express P(z) as: [P(z) = (z - 4)(z - 4 - 6i)(z - 4 + 6i).] To find a+b+c, we evaluate P(1): begin{align*} P(1) &= (1 - 4)(1 - 4 - 6i)(1 - 4 + 6i) &= (-3)(-3 - 6i)(-3 + 6i) &= -3 cdot 9 cdot (-1) &= -27 cdot (-1) &= -135. end{align*} Since P(1) = 1 + a + b + c, we have 1 + a + b + c = -135. Solving for a + b + c gives a + b + c = -136. Therefore, the sum of the coefficients a+b+c is boxed{-136}.