answer:1. **Determine individual rates**: - Cagney's rate of frosting cupcakes is 1 cupcake per 20 seconds. - Lacey's rate of frosting cupcakes is 1 cupcake per 30 seconds. 2. **Calculate combined rate**: - The combined rate of frosting cupcakes when Cagney and Lacey work together can be calculated using the formula for the combined work rate of two people working together: [ text{Combined rate} = frac{1}{text{Time taken by Cagney}} + frac{1}{text{Time taken by Lacey}} = frac{1}{20} + frac{1}{30} ] - To add these fractions, find a common denominator (which is 60 in this case): [ frac{1}{20} = frac{3}{60}, quad frac{1}{30} = frac{2}{60} ] - Therefore, the combined rate is: [ frac{3}{60} + frac{2}{60} = frac{5}{60} = frac{1}{12} text{ cupcakes per second} ] 3. **Convert the combined rate to a single cupcake time**: - The time taken to frost one cupcake together is the reciprocal of the combined rate: [ text{Time for one cupcake} = frac{1}{frac{1}{12}} = 12 text{ seconds} ] 4. **Calculate total cupcakes in 5 minutes**: - Convert 5 minutes to seconds: [ 5 text{ minutes} = 5 times 60 = 300 text{ seconds} ] - The number of cupcakes frosted in 300 seconds is: [ frac{300 text{ seconds}}{12 text{ seconds per cupcake}} = 25 text{ cupcakes} ] 5. **Conclusion**: - Cagney and Lacey can frost 25 cupcakes in 5 minutes. Thus, the answer is boxed{textbf{(D)} 25}.

answer:Writing the right-hand side with 2 as the base, we have 8^{x-5} = (2^3)^{x-5} = 2^{3(x-5)} = 2^{3x-15}. Our equation becomes: 2^{x^2 - 5x - 4} = 2^{3x - 15}. Thus, by setting the exponents equal to each other, we obtain: x^2 - 5x - 4 = 3x - 15. Solving this equation, we rearrange terms to form a quadratic: x^2 - 8x + 11 = 0. Factoring this quadratic equation: (x-1)(x-11) = 0. This gives roots x = 1 and x = 11. The sum of these solutions is boxed{12}.

answer:From sqrt {2}rho= frac {1}{sin ( frac {pi}{4}+theta)}, we get rho(sin theta+ cos theta)=1, which simplifies to the Cartesian equation x+y=1. The Cartesian equation of the line theta= frac {pi}{3} is y= sqrt {3}x. Solving the simultaneous equations, we get x= frac { sqrt {3}-1}{2} and y= frac { sqrt {3}( sqrt {3}-1)}{2}. Therefore, x^{2}+y^{2}=( sqrt {3}-1)^{2}. Hence, rho= sqrt {x^{2}+y^{2}}= sqrt {3}-1. Substituting theta= frac {pi}{3}, we get the coordinates of the intersection point as boxed{( sqrt {3}-1, frac {pi}{3})}. To solve this problem, we converted the polar coordinate equations to Cartesian equations, solved the simultaneous equations, found the value of rho, and then substituted the value of theta to get the coordinates of the intersection point. The crucial part of the solution is to find x^{2}+y^{2}=( sqrt {3}-1)^{2}=rho^{2}.

answer:First, we find the inverse function of g. By definition, g(g^{-1}(x)) = x, so frac{g^{-1}(x) - 2}{g^{-1}(x) - 5} = x. We solve this equation for g^{-1}(x). Multiplying both sides by g^{-1}(x) - 5: g^{-1}(x) - 2 = x(g^{-1}(x) - 5). Expanding and rearranging to isolate terms involving g^{-1}(x): g^{-1}(x) - xg^{-1}(x) = 2 - 5x. Factoring the left side: g^{-1}(x)(1 - x) = 2 - 5x. Dividing both sides by 1 - x (assuming x neq 1): g^{-1}(x) = frac{2 - 5x}{1 - x}. This function is undefined when the denominator is zero, which occurs when 1 - x = 0, or x = 1. Hence, g^{-1}(x) is undefined for boxed{1}.