answer:1. **Triangle inscribed in a semicircle**: Let ( t_1 ) be the side length of the equilateral triangle inscribed in the semicircle of radius ( r ). The height of the triangle from the base to the top vertex on the diameter is ( r ). Using the properties of equilateral triangles, the altitude ( h_1 ) splits the triangle into two 30-60-90 right triangles. The hypotenuse is ( t_1 ), and the side opposite the 30° angle is ( r ). Thus, ( frac{t_1}{2} = r sec(30^circ) = r frac{2}{sqrt{3}} ). Solving for ( t_1 ), we find ( t_1 = 2r sec(30^circ) = frac{4r}{sqrt{3}} ). The area ( A_1 ) of the triangle is ( frac{sqrt{3}}{4} t_1^2 = frac{sqrt{3}}{4} left( frac{4r}{sqrt{3}} right)^2 = frac{16r^2}{3} ). 2. **Triangle inscribed in a circle**: Let ( t_2 ) be the side length of the equilateral triangle inscribed in the circle of radius ( r ). Using the fact that the radius to the vertex forms a 60° angle with the side, ( t_2 ) is given by ( t_2 = 2r sin(60^circ) = 2r frac{sqrt{3}}{2} = rsqrt{3} ). The area ( A_2 ) of this triangle is ( frac{sqrt{3}}{4} t_2^2 = frac{sqrt{3}}{4} (rsqrt{3})^2 = frac{3sqrt{3}r^2}{4} ). 3. **Ratio of the areas**: [ text{Ratio} = frac{A_1}{A_2} = frac{frac{16r^2}{3}}{frac{3sqrt{3}r^2}{4}} = frac{64}{9sqrt{3}} = frac{64sqrt{3}}{27} ] Simplifying, we get (boxed{frac{64sqrt{3}}{27}}).

answer:To determine the range of values for x that makes the expression sqrt{frac{5}{2-3x}} meaningful within the real number range, we must satisfy two conditions: 1. The expression under the square root, frac{5}{2-3x}, must be non-negative. 2. The denominator of the fraction, 2-3x, must not be zero to avoid division by zero. Let's address these conditions step by step: **Condition 1: Non-negative Expression** For the square root to be meaningful (real), the expression inside it must be greater than or equal to zero. This gives us: [2-3x geq 0] **Condition 2: Non-zero Denominator** To ensure the fraction is meaningful (defined), the denominator must not be zero. This gives us: [2-3x neq 0] Combining these two conditions, we find that 2-3x must be strictly greater than zero to satisfy both (being non-negative and not zero): [2-3x > 0] Solving this inequality for x gives us: [ begin{align*} 2-3x &> 0 -3x &> -2 x &< frac{2}{3} end{align*} ] Therefore, the range of values for x that makes the expression meaningful within the real number range is x < frac{2}{3}. Hence, the correct answer is boxed{B}.

answer:To find out how many empty cans Alyssa has collected, we need to subtract the number of cans Abigail collected and the number of cans they still need to collect from the total number of cans they need for their project. Alyssa's cans + Abigail's cans + cans still needed = total cans needed Alyssa's cans = total cans needed - (Abigail's cans + cans still needed) We know that: Total cans needed = 100 Abigail's cans = 43 Cans still needed = 27 Now we can plug in the numbers: Alyssa's cans = 100 - (43 + 27) Alyssa's cans = 100 - 70 Alyssa's cans = 30 Alyssa has collected boxed{30} empty cans so far.

answer:**Analysis** This question examines the application of function properties. One should judge based on the domain, range, odd-even properties, and monotonicity of the function. **Solution** The function y=x+cos x, (1) The domain of f(x) is mathbb{R}, so it is incorrect; (2) Since y'=1-sin x geqslant 0, the function y=x+cos x is increasing, so its range being mathbb{R} is correct; (3) The function y=x+cos x is neither odd nor even; (4) The intersection of the graph of f(x) with the line y=x satisfies x+cos x=x, which gives cos x=0 Rightarrow x=kpi+ frac{pi}{2}, kin mathbb{Z}, Thus, there is a point where the x-coordinate is frac{pi}{2}, which is correct. Therefore, the correct choice is boxed{text{C}}.