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question:A chemical model of a pyramid with a square base is to be made, where each vertex of the pyramid is connected to a small ball of a different color, and the two ends of each edge must be of different colors. If only four different colors of balls are available, then the number of different ways to make this model is A: 24 B: 48 C: 60 D: 72
answer:**Answer** D Therefore, the number of different ways to make this model is boxed{72}.
question:Let M be the number of ways to write 2023 in the form 2023 = b_3 cdot 10^3 + b_2 cdot 10^2 + b_1 cdot 10 + b_0, where the b_is are integers, and 0 le b_i le 999. Find M.
answer:If we choose b_3 and b_1 such that (10^3)(b_3) + (10)(b_1) leq 2023, there is a unique choice of b_2 and b_0 that makes the equality hold. So M is simply the number of combinations of b_3 and b_1 we can pick. We analyze possible values of b_3: - If b_3 = 0, b_1 can range from 0 to 202 (as 10 times 202 = 2020 leq 2023). - If b_3 = 1, then b_1 can range from 0 to 99 (as 1000 + 990 leq 2023). - If b_3 = 2, then b_1 must be 0 or 1 or 2 (as 2000 + 20 leq 2023). Thus, counting: - 203 options (0 to 202) when b_3 = 0. - 100 options (0 to 99) when b_3 = 1. - 3 options (0, 1, 2) when b_3 = 2. Total M = 203 + 100 + 3 = boxed{306}.
question:Charles bought a tshirt for some amount, a backpack for 10, and a blue cap for 5. The store gave him a discount, and in total, he only spent 43. They gave him a 2 discount. How much did the tshirt cost before the discount?
answer:Let's call the original price of the t-shirt "T". The total amount Charles spent after the discount is 43. The store gave him a 2 discount, so the total cost of his items before the discount was 43 + 2 = 45. We know the backpack cost 10 and the blue cap cost 5. So the combined cost of the backpack and cap is 10 + 5 = 15. Now, we subtract the combined cost of the backpack and cap from the total cost before the discount to find the original price of the t-shirt: 45 (total before discount) - 15 (backpack and cap) = 30. So, the t-shirt cost boxed{30} before the discount.
question:Find the vertex of the function x^2 + 2px + 3q where p and q are positive constants. **A)** 2p **B)** p **C)** 3p **D)** 3q **E)** p^2
answer:1. **Identify the form of the function**: The given function is x^2 + 2px + 3q. This is a quadratic function in the form ax^2 + bx + c, where a = 1, b = 2p, and c = 3q. 2. **Determine the vertex**: The formula for the x-coordinate of the vertex of a parabola represented by the quadratic function ax^2 + bx + c is x = -frac{b}{2a}. Plugging in the values: [ x = -frac{2p}{2 cdot 1} = -frac{2p}{2} = -p ] 3. **Conclusion**: Since the coefficient of x^2 is 1 (positive), the parabola opens upwards, making the vertex the minimum point of the function. Therefore, the minimum value of the function x^2 + 2px + 3q occurs at x = -p. [ -p ] The final answer is **B)** boxed{-p}