answer:1. **Calculate the total amount invested at known rates:** The woman invests 5000 at 3% and 4000 at 4.5%. We need to calculate the income from these investments. [ text{Income from } 5000 = 5000 times frac{3}{100} = 150 ] [ text{Income from } 4000 = 4000 times frac{4.5}{100} = 180 ] 2. **Calculate the total income from these investments:** [ text{Total income} = 150 + 180 = 330 ] 3. **Determine the remaining amount to be invested:** The total amount available for investment is 12,000. The amount already invested is 5000 + 4000 = 9000. Therefore, the remainder to be invested is: [ 12,000 - 9000 = 3000 ] 4. **Calculate the required income from the remaining investment:** The woman desires a total yearly income of 600. She already has 330 from the first two investments, so the additional income needed from the remaining investment is: [ 600 - 330 = 270 ] 5. **Set up the equation to find the required rate of return on the remaining investment:** Let x be the rate of return (in percent) needed on the remaining 3000 to achieve 270 income. [ 3000 times frac{x}{100} = 270 ] 6. **Solve for x:** [ frac{x}{100} = frac{270}{3000} ] [ x = frac{270}{3000} times 100 = frac{270 times 100}{3000} = frac{27000}{3000} = 9 ] 7. **Conclusion:** The woman must invest the remaining 3000 at a rate of 9% to achieve her desired total yearly income of 600. [ 9% ] The final answer is boxed{C) 9%}

answer:To solve for the height h of the frustum, we use the formula for the volume of a frustum, which is given by V = frac{1}{3}pi h(r_1^2 + r_2^2 + r_1r_2), where r_1 and r_2 are the radii of the top and bottom bases, respectively, and h is the height of the frustum. Given that r_1 = 2, r_2 = 6, and V = 104pi, we can substitute these values into the formula to find h: [ 104pi = frac{1}{3}pi h(2^2 + 6^2 + 2 times 6) ] Simplifying the equation: [ 104pi = frac{1}{3}pi h(4 + 36 + 12) = frac{1}{3}pi h(52) ] Solving for h: [ 104pi = frac{52}{3}pi h Rightarrow h = frac{104pi}{frac{52}{3}pi} = 6 ] Now that we have the height h = 6, we can find the slant height l of the frustum using the Pythagorean theorem. The slant height forms a right triangle with the height of the frustum and the difference in radii of the bases. Thus, [ l = sqrt{h^2 + (r_2 - r_1)^2} = sqrt{6^2 + (6 - 2)^2} = sqrt{36 + 16} = sqrt{52} = 2sqrt{13} ] Therefore, the length of the slant height of the frustum is boxed{2sqrt{13}}.

answer:We need to find all values of (n) such that the sum of the first (n) terms of the arithmetic sequence (22, 19, 16, ldots) is at least 52. 1. **Identify the first term and common difference:** - First term ((a_1)): (22) - Common difference ((d)): (-3) 2. **Sum of the first (n) terms of an arithmetic series:** [ S_n = frac{n}{2} left(2a_1 + (n-1)dright) ] 3. **Substitute (a_1 = 22) and (d = -3) into the formula:** [ S_n = frac{n}{2} left(2 cdot 22 + (n-1) cdot (-3)right) ] [ S_n = frac{n}{2} left(44 - 3(n-1)right) ] 4. **Simplify the expression:** [ S_n = frac{n}{2} left(44 - 3n + 3right) ] [ S_n = frac{n}{2} left(47 - 3nright) ] [ S_n = frac{47n - 3n^2}{2} ] 5. **Set up the inequality (S_n geq 52):** [ frac{47n - 3n^2}{2} geq 52 ] 6. **Multiply both sides by 2 to clear the fraction:** [ 47n - 3n^2 geq 104 ] 7. **Rearrange the inequality to form a standard quadratic inequality:** [ 3n^2 - 47n + 104 leq 0 ] 8. **Solve the quadratic inequality using the quadratic formula, (n = frac{-b pm sqrt{b^2 - 4ac}}{2a}), where (a = 3), (b = -47), and (c = 104):** [ n = frac{47 pm sqrt{47^2 - 4 cdot 3 cdot 104}}{2 cdot 3} ] [ n = frac{47 pm sqrt{2209 - 1248}}{6} ] [ n = frac{47 pm sqrt{961}}{6} ] [ n = frac{47 pm 31}{6} ] 9. **Calculate the two possible values for (n):** [ n = frac{47 + 31}{6} quad text{and} quad n = frac{47 - 31}{6} ] [ n = frac{78}{6} quad text{and} quad n = frac{16}{6} ] [ n = 13 quad text{and} quad n = frac{8}{3} ] 10. **Identify the interval for the natural numbers:** Since (n) must be a natural number, [ n in left[3, 13right] ] 11. **Conclusion:** The values of (n) satisfying the given condition are: [ boxed{3, 4, 5, ldots, 13} ]

answer:First, let's evaluate each trigonometric function in the expression: 1. sin 30^{circ} = dfrac{1}{2} 2. tan 45^{circ} = 1 3. tan 60^{circ} = sqrt{3} Now, substitute these values back into the original expression: begin{align} 2sin 30^{circ}-tan 45^{circ}-sqrt{(1-tan 60^{circ})^{2}} &= 2 cdot dfrac{1}{2} - 1 - sqrt{(1 - sqrt{3})^{2}} &= 1 - 1 - |1 - sqrt{3}| &= -|1 - sqrt{3}| &= -(1 - sqrt{3}) quad text{(since } 1 - sqrt{3} < 0text{)} &= boxed{sqrt{3} - 1} end{align}