This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Question 3: Step 1: Identify the given dimensions of the right rectangular prism. The height h = 9 cm. The length of the base l = x cm. The length is 7 cm more than the width, so l = w + 7. Step 2: Express the width in terms of x. Since l = w + 7 and l = x, we have x = w + 7. Solving for w: w = x - 7 Step 3: Write the formula for the volume of a rectangular prism. The volume V is given by the formula V = length × width × height. V = l × w × h Step 4: Substitute the expressions for length, width, and height into the volume formula. V(x) = x × (x - 7) × 9 V(x) = 9x(x - 7) Comparing this with the given options, the correct function is D. The final answer is D. Question 4: Step 1: Split the compound inequality into two separate inequalities. The given inequality is 2x - 3 ≤ 4x + 5 < 3x + 7. This can be written as two inequalities: 1) 2x - 3 ≤ 4x + 5 2) 4x + 5 < 3x + 7 Step 2: Solve the first inequality. 2x - 3 ≤ 4x + 5 Subtract 2x from both sides: -3 ≤ 2x + 5 Subtract 5 from both sides: -3 - 5 ≤ 2x -8 ≤ 2x Divide by 2: (-8)/(2) ≤ (2x)/(2) -4 ≤ x Step 3: Solve the second inequality. 4x + 5 < 3x + 7 Subtract 3x from both sides: x + 5 < 7 Subtract 5 from both sides: x < 7 - 5 x < 2 Step 4: Combine the solutions from both inequalities. We have x ≥ -4 and x < 2. Combining these, the solution is: -4 ≤ x < 2 Comparing this with the given options, the correct solution is A. The final answer is A. Got more? Send 'em.