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question:In a side-angle-side triangle, the lengths of two sides are 10. units and 8. units, and the included angle measures frac{41 pi }{60} radians. Find the coordinates of the triangle's vertices and the measures of its three interior angles.
answer:The vertices of the triangle are located at: {{0,0}, {15.8475,0}, {6.78791,4.23371}} The measures of the three interior angles, in radians, are: {0.557674, 0.437163, 2.14675} These values can be confirmed using trigonometric functions and the law of cosines to calculate the missing side, followed by the law of sines to determine the angles.
question:What is the general form of functions whose derivatives are greater than or equal to the function itself, and how can we derive it?
answer:Functions that have derivatives greater than or equal to the function itself have the general form: f(x) = ke^{int g(x)dx} where k is a constant and g(x) is a function such that g(x) ge 1 for all x. This form can be derived by solving the differential equation: frac{f'(x)}{f(x)} = g(x) which arises from the condition that the derivative of the function is greater than or equal to the function itself.
question:Martinez Company's relevant range of production is 7,500 units to 12,500 units. If the company plans to produce and sell 11,000 units, what is the total expected variable cost?
answer:137,500 Explanation: Variable cost per unit = 12.5 (as calculated in the original answer) Total variable cost for 11,000 units = 12.5 x 11,000 = 137,500
question:An air traffic controller observes two aircraft on his radar screen. The first is at an altitude of 950 m, a horizontal distance of 19.9 km, and 23.5° south of west. The second aircraft is at an altitude of 1200 m, a horizontal distance of 17.4 km, and 22.0° south of west. What is the distance between the two aircraft? (Place the x-axis west, the y-axis south, and the z-axis vertical.)
answer:To find the distance between the two aircraft, we can use the Pythagorean theorem in three dimensions. First, we need to find the horizontal distance between the two aircraft and then use that to find the total distance. The horizontal distance between the two aircraft is: ``` d_horizontal = sqrt((d1x - d2x)^2 + (d1y - d2y)^2) ``` where d1x and d1y are the horizontal components of the distance to the first aircraft, and d2x and d2y are the horizontal components of the distance to the second aircraft. We can find these components using trigonometry: ``` d1x = d1 * cos(theta1) d1y = d1 * sin(theta1) d2x = d2 * cos(theta2) d2y = d2 * sin(theta2) ``` where d1 and d2 are the horizontal distances to the first and second aircraft, and theta1 and theta2 are the angles south of west. Plugging in the given values, we get: ``` d1x = 19.9 km * cos(23.5°) = 18.2495 km d1y = 19.9 km * sin(23.5°) = 7.9351 km d2x = 17.4 km * cos(22.0°) = 16.1329 km d2y = 17.4 km * sin(22.0°) = 6.5182 km ``` Now we can find the horizontal distance: ``` d_horizontal = sqrt((18.2495 km - 16.1329 km)^2 + (7.9351 km - 6.5182 km)^2) d_horizontal = 2.5593 km ``` Finally, we can use the Pythagorean theorem to find the total distance: ``` d_total = sqrt(d_horizontal^2 + (a2 - a1)^2) ``` where a1 and a2 are the altitudes of the first and second aircraft. Plugging in the given values, we get: ``` d_total = sqrt(2.5593 km^2 + (1200 m - 950 m)^2) d_total = 2.5593 km ``` Therefore, the distance between the two aircraft is 2.5593 km.