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question:Find the length of the first side of a triangle if the other two sides measure 10 centimeters and 15 centimeters, and the perimeter of the triangle is 32 centimeters.
answer:To find the length of the first side of the triangle, we can use the perimeter formula for a triangle, which is the sum of the lengths of all three sides. Let's denote the length of the first side as x. According to the problem, the other two sides measure 10 cm and 15 cm, and the perimeter is 32 cm. So we can set up the following equation: x + 10 + 15 = 32 Now, we can solve for x: x + 25 = 32 x = 32 - 25 x = 7 Therefore, the length of the first side of the triangle is boxed{7} centimeters.
question:Ana, Bob, and CAO bike at constant rates of 8.6 meters per second, 6.2 meters per second, and 5 meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point D on the south edge of the field. Cao arrives at point D at the same time that Ana and Bob arrive at D for the first time. The ratio of the field's length to the field's width to the distance from point D to the southeast corner of the field can be represented as p : q : r, where p, q, and r are positive integers with p and q relatively prime. Find p+q+r.
answer:1. **Assign Variables and Set Up Equations**: Let a be the length of the field (west direction), b be the width of the field (south direction), and c be the distance from point D to the southeast corner of the field. Assume without loss of generality (WLOG) that the time taken for all to meet at D is 1 second. 2. **Write Equations for Ana, Bob, and Cao's Travel**: - Ana's distance: 2a + b + c = 8.6 meters (since she travels the entire length twice and then some more). - Bob's distance: b + c = 6.2 meters. - Cao's distance (diagonal): sqrt{b^2 + c^2} = 5 meters. 3. **Solve for a**: - Subtract Bob's equation from Ana's: (2a + b + c) - (b + c) = 8.6 - 6.2 Rightarrow 2a = 2.4 Rightarrow a = 1.2 meters. 4. **Solve for b and c Using Cao's and Bob's Equations**: - From Bob's equation: b + c = 6.2. - From Cao's equation: b^2 + c^2 = 25. - Square the sum equation: (b + c)^2 = 6.2^2 = 38.44. - Expand and rearrange: b^2 + 2bc + c^2 = 38.44 Rightarrow 2bc = 38.44 - 25 = 13.44. - Solve for (b - c)^2: b^2 - 2bc + c^2 = 25 - 13.44 = 11.56 Rightarrow b - c = sqrt{11.56} = 3.4. 5. **Solve the System for b and c**: - b + c = 6.2 and b - c = 3.4. - Adding these equations: 2b = 9.6 Rightarrow b = 4.8. - Subtracting the second from the first: 2c = 2.8 Rightarrow c = 1.4. 6. **Determine the Ratios**: - The ratios of the field's length to width to the distance from D to the southeast corner are a : b : c = 1.2 : 4.8 : 1.4. - Simplify the ratios: frac{1.2}{1.2} : frac{4.8}{1.2} : frac{1.4}{1.2} = 1 : 4 : 1.1667. - Convert to integers: 6 : 24 : 7 (by multiplying each term by 6). 7. **Calculate p + q + r**: - p = 6, q = 24, r = 7. - Therefore, p + q + r = 6 + 24 + 7 = boxed{37}.
question:Each interior angle of a regular polygon measures 120^circ. How many sides does this polygon have?
answer:Let n be the number of sides of the polygon. The sum of all interior angles in an n-sided polygon is given by 180(n-2) degrees. Given that each interior angle measures 120^circ, the sum of the interior angles is 120n. Equating the two expressions gives: [ 180(n-2) = 120n ] Expanding and simplifying, we have: [ 180n - 360 = 120n ] [ 60n = 360 ] [ n = frac{360}{60} = 6 ] Thus, the polygon is a hexagon. boxed{6} Additionally, we can verify this using the exterior angles. Each exterior angle is 180^circ - 120^circ = 60^circ. Since the sum of all exterior angles of any polygon is 360^circ, we have: [ frac{360^circ}{60^circ} = 6 ] This confirms that the polygon indeed has 6 sides.
question:What is the domain of the function f(x) = sqrt{x5} + sqrt[3]{2x6}~? Express your answer in interval notation.
answer:To find the domain of the function: 1. Analyze the square root term, sqrt{x-5}, which requires x-5 geq 0. Thus x geq 5. 2. Analyze the cubic root term, sqrt[3]{2x-6}. Since the cubic root is defined for all real numbers, there are no restrictions from this term. By combining these conditions, we observe that: - sqrt{x-5} restricts x to x geq 5. - sqrt[3]{2x-6} allows all x values. The stricter condition is x geq 5. Thus, the domain of the function f(x) is boxed{[5, infty)}.