answer:Let's denote the sides of the triangle as x-2, x, and x+2, where x is the middle term of the arithmetic sequence. We start by noting that the problem refers to an arithmetic progression (AP) for the sides of the triangle, which implies there are three sides in sequence: a smaller side (x-2), a middle side x, and a larger side (x+2). Knowing that one of the internal angles is 120^circ, and using the law of cosines, which states that for any triangle ABC with sides a, b, and c opposite angles A, B, and C respectively, we have: c^2 = a^2 + b^2 - 2abcos(C), we can relate the sides of the triangle to the cosine of the 120^circ angle. Substituting the given angle and expressing cos120^circ with the respective sides, we have: cos(120^circ) = frac{(x)^2 + (x - 2)^2 - (x + 2)^2}{2cdot xcdot (x - 2)} = -frac{1}{2}. Simplifying the equation: x^2 + (x - 2)^2 - (x + 2)^2 = -(xcdot (x - 2)), x^2 + x^2 - 4x + 4 - (x^2 + 4x + 4) = -x^2 + 2x, x^2 - 8x = -x^2 + 2x, 2x^2 - 10x = 0. Dividing through by 2, we get: x^2 - 5x = 0, x(x - 5) = 0. This yields two solutions, x = 0 and x = 5. However, x = 0 is not possible as it would lead to non-positive side lengths. Therefore, the middle side of the triangle is x = 5. Now we can determine the side lengths of the triangle: 3, 5, and 7. To compute the area of the triangle, we use the formula: text{Area} = frac{1}{2} times text{base} times text{height}. However, since we are not given the height, we will use the formula involving the sine of the enclosed angle between the base and one of the other sides: text{Area} = frac{1}{2} times (x-2) times x times sin(120^circ). Substituting the obtained values and knowing that sin(120^circ) = sin(180^circ - 60^circ) = sin(60^circ) = frac{sqrt{3}}{2}, the area S is: S = frac{1}{2} times 3 times 5 times frac{sqrt{3}}{2} = frac{15sqrt{3}}{4}. Hence, the area of the triangle ABC is boxed{frac{15sqrt{3}}{4}}.

answer:1. **Calculate the number of sides**: - To find the number of sides ( n ), use the formula relating the perimeter ( P ) to the side length ( s ): [ n = frac{P}{s} = frac{180}{15} = 12 ] The polygon has 12 sides. 2. **Calculate the area of the polygon**: - The formula for the area ( A ) of a regular polygon using the number of sides ( n ), side length ( s ), and apothem ( a ) is: [ A = frac{1}{2} times P times a = frac{1}{2} times 180 times 12 = 1080 text{ cm}^2 ] Therefore, the area of the polygon is ( boxed{1080 text{ cm}^2} ). Conclusion: The polygon has ( boxed{12} ) sides, and its area is ( boxed{1080 text{ cm}^2} ).

answer:```markdown 1. Start by rewriting the original inequality: [ x^{log_{13} x} + 7(sqrt[3]{x})^{log_{13} x} leqslant 7 + (sqrt[3]{13})^{log_{ sqrt{13}}^2 x} ] 2. Express each term with a common base of 13. Notice that ( x^{log_{13} x} = 13^{log_{13}^2 x} ) and ( (sqrt[3]{x})^{log_{13} x} = 13^{frac{1}{3}log_{13}^2 x} ). Thus, the inequality becomes: [ 13^{log_{13}^2 x} + 7 cdot 13^{frac{1}{3} log_{13}^2 x} leqslant 7 + 13^{frac{4}{3} log_{13}^2 x} ] 3. Rearrange the terms to bring all expressions to one side of the inequality: [ 13^{log_{13}^2 x} + 7 cdot 13^{frac{1}{3} log_{13}^2 x} - 7 - 13^{frac{4}{3} log_{13}^2 x} leqslant 0 ] 4. Notice that the left-hand side can be factored by grouping the terms. Group the first and fourth terms together, and the second and third terms together: [ (13^{log_{13}^2 x} - 13^{frac{4}{3} log_{13}^2 x}) + (7 cdot 13^{frac{1}{3} log_{13}^2 x} - 7) leqslant 0 ] 5. Factor each group separately: [ 13^{log_{13}^2 x}(1 - 13^{frac{1}{3} log_{13}^2 x}) + 7(13^{frac{1}{3} log_{13}^2 x} - 1) leqslant 0 ] 6. To simplify further, consider factorization of the whole expression: [ (13^{log_{13}^2 x} - 7)(1 - 13^{frac{1}{3} log_{13}^2 x}) leqslant 0 ] 7. Now, solve for the values of ( x ) that satisfy this inequality by considering two cases: - **Case a)** [ begin{cases} 13^{log_{13}^{2} x} - 7 geq 0 1 - 13^{frac{1}{3} log_{13}^{2} x} leq 0 end{cases} ] **Step 1:** Solve ( 13^{log_{13}^{2} x} geq 7 ) [ log_{13}^{2} x geq log_{13} 7 ] [ log_{13} x geq sqrt{log_{13} 7} ] **Step 2:** Solve ( 1 leq 13^{frac{1}{3} log_{13}^{2} x} ) [ 0 leq frac{1}{3} log_{13} x implies log_{13} x geq 0 ] Thus, combining these conditions, we get: [ x geq 13^{sqrt{ log_{13} 7}} ] - **Case b)** [ begin{cases} 13^{log_{13}^{2} x} - 7 leq 0 1 - 13^{frac{1}{3} log_{13}^{2} x} geq 0 end{cases} ] **Step 1:** Solve ( 13^{log_{13}^{2} x} leq 7 ) [ log_{13}^{2} x leq log_{13} 7 ] [ log_{13} x leq sqrt{log_{13} 7} ] **Step 2:** Solve ( 1 geq 13^{frac{1}{3} log_{13}^{2} x} ) [ 0 geq frac{1}{3} log_{13} x implies log_{13} x leq 0 ] Rewrite combining these conditions: [ -sqrt{ log_{13} 7} leq log_{13} x leq 0 ] [ 13^{-sqrt{ log_{13} 7}} leq x leq 1 ] Combining the solutions from both cases, the complete solution set is: [ x in left(0; 13^{- sqrt{log_{13} 7}} right] cup {1} cup left[ 13^{sqrt{log_{13} 7}}, +infty right) ] Therefore, the complete solution to the inequality is: [ boxed{x in left(0; 13^{- sqrt{log_{13} 7}} right] cup {1} cup left[ 13^{sqrt{log_{13} 7}}, +infty right)} ] ```

answer:First, let's calculate how much Mrs. Smith would need for each type of clothing item if she needed two-fifths more money than she had budgeted. For dresses: Original budget: 300 Two-fifths more: 2/5 * 300 = 120 Total needed for dresses: 300 + 120 = 420 For shoes: Original budget: 150 Two-fifths more: 2/5 * 150 = 60 Total needed for shoes: 150 + 60 = 210 For accessories: Original budget: 50 Two-fifths more: 2/5 * 50 = 20 Total needed for accessories: 50 + 20 = 70 Now, let's add up the total amount she would need without the discount: Total needed without discount: 420 (dresses) + 210 (shoes) + 70 (accessories) = 700 Next, let's calculate the discount she would receive on the 700: Discount: 15% of 700 = 0.15 * 700 = 105 Now, let's subtract the discount from the total amount needed without the discount to find out how much she would need to pay after the discount: Total needed with discount: 700 - 105 = 595 Finally, let's subtract the amount of money she has (500) from the total amount she would need to pay after the discount to find out how much more money she will still need: Additional money needed: 595 - 500 = 95 Mrs. Smith will still need an additional boxed{95} .