answer:We set y = 35 to find: [ 35 = -4.9t^2 + 30t ] Rearranging this equation gives: [ 0 = -4.9t^2 + 30t - 35 ] Multiplying all terms by 10 to clear the decimal, we have: [ 0 = -49t^2 + 300t - 350 ] We factor the quadratic equation: [ 0 = (7t - 50)(t - 2) ] The possible values for t are t = frac{50}{7} approx 7.143 or t = 2. Since the question asks when the projectile first reaches 35 meters, we select the smaller t: boxed{2}.

answer:1. **Initial Board and Cut:** - The initial board is an 8 times 8 grid consisting of 64 squares. - A corner cell has been removed from this board, leaving us with 64 - 1 = 63 squares. 2. **Area Consideration:** - We seek to determine if the remaining 63 squares can be divided into 17 equal-area triangles. 3. **Equilateral Triangle's Side Analysis:** - Each triangle must have equal areas. Since the board's area is quantified by units squared, our target equilateral triangles should closely match this metric. - Calculate the area per triangle needed: [ text{Area per triangle} = frac{text{Total remaining area}}{text{Number of triangles}} = frac{63 text{ squares}}{17} ] 4. **Evaluating Area per Triangle:** - This calculation gives: [ text{Area per triangle} = frac{63}{17} approx 3.70588 ] 5. **Side Length Bound Analysis:** - Focus on the possible configurations of these triangles. If each triangle is to fit well within the grid: - Each triangle side length, considering it must fit within the confines of the partially cut grid (not more than 8 times 8), must assess feasibility. - Each triangle side shares grid boundaries; target maximum height (perpendicular to the longest side) within the current constraint of the grid size (bounded by side 1). The simplification steps highlight getting the length feasibility. 6. **Height Maximization:** - For any triangle to fit effectively: - Its area calculation should fit well. - Considering it's not possible due to smaller fitting units will be less than exact. 7. **Infeasibility of Required Division:** - Given one side length bound not exceeding 1, and each height must fit within the confines: - The height directly not more exceeding half the overall side configurations leaves each fitting less than half; a crucial measure is ≤ 7, implying: - Triangle sides fitting leave height bounds less than frac{7}{2} fits into any left configuration suffices less than 3.70588, proving infeasibility. 8. **Conclusion:** [ boxed{text{No, it's impossible to divide remaining board into 17 equal-area triangles}} ] Hence using these logical deductions ensures the impossibility of the designated configuration for equilateral triangles within constrained parts. (blacksquare)

answer:First, use the Pythagorean theorem to find the length of the missing leg b: [ c^2 = a^2 + b^2 ] [ 30^2 = 18^2 + b^2 ] [ 900 = 324 + b^2 ] [ b^2 = 900 - 324 ] [ b^2 = 576 ] [ b = sqrt{576} = 24 , text{inches} ] Now, calculate the area using the area formula for a right triangle: [ text{Area} = frac{1}{2} times text{base} times text{height} ] [ text{Area} = frac{1}{2} times 18 times 24 ] [ text{Area} = 9 times 24 ] [ text{Area} = boxed{216} , text{square inches} ]

answer:**Analysis** This problem examines the definition and standard equation of a hyperbola. The key to solving the problem is to understand that the distance from point P to the right focus equals 2a plus the distance from point P to the left focus. **Solution** From the equation of the hyperbola frac{x^2}{16} - frac{y^2}{9} = 1, we can determine that a=4. According to the definition of a hyperbola, the distance from point P to the right focus equals the distance from point P to the left focus minus 2a. Therefore, the distance from point P to the right focus is 12 - 8 = 4. Hence, the answer is boxed{4}.