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question:Determine the value of frac{tan 22.5^{circ}}{1 tan^2 22.5^{circ}}. The options are: A: frac{1}{2} B: frac{1}{2} C: 1 D: 1
answer:We start with the given expression: frac{tan 22.5^{circ}}{1 - tan^2 22.5^{circ}} Using the double-angle formula for tangent, which is: tan 2theta = frac{2tan theta}{1 - tan^2 theta} We can rewrite our expression as: frac{1}{2} tan (2 cdot 22.5^{circ}) Simplify the angle inside the tangent function: frac{1}{2} tan 45^{circ} Since tan 45^{circ} = 1, we can substitute this value into the expression: frac{1}{2} cdot 1 = boxed{frac{1}{2}}
question:the quadrilateral ( 11 square ABCD ), points ( A, B, ) and ( C ) correspond to the complex numbers ( 2+i, 4+3i, ) and ( 3+5i ) respectively. What is the corresponding complex number for point ( D )? A. ( 3i ) B. ( 1+3i ) C. ( 5+7i ) D. ( 9+9i )
answer:Given the vertices ( A, B, C, D ) of a quadrilateral ( 11ABCD ), we need to determine the complex number corresponding to point ( D ). The known complex numbers corresponding to the points are: - ( z_A = 2 + i ) - ( z_B = 4 + 3i ) - ( z_C = 3 + 5i ) We can determine the complex number for point ( D ) using the relation: [ z_D = z_A - z_B + z_C ] Let's compute this step-by-step. 1. **Calculate ( z_A - z_B )**: [ z_A - z_B = (2 + i) - (4 + 3i) = 2 + i - 4 - 3i = (2 - 4) + (i - 3i) = -2 - 2i ] 2. **Add ( z_C ) to the result from step 1**: [ z_D = (-2 - 2i) + (3 + 5i) = -2 + 3 + (-2i + 5i) = 1 + 3i ] Hence, the complex number corresponding to point ( D ) is: [ z_D = 1 + 3i ] Conclusion: [ boxed{B} ]
question:A marine biologist is studying the population of trout in a river. On April 1, she captures and tags 80 trout, then releases them back into the river. On August 1, she captures another sample of 100 trout, noticing that 4 of these are tagged. She estimates that 30% of the original tagged trout have left the river system (due to predation or migration), and that 50% of the trout in the August 1 sample were not in the river on April 1 (newly born or migrated into the river). What is the biologist's estimate for the number of trout in the river on April 1?
answer:From the problem: - On August 1, 50% of the trout are new, so 50% of the 100 trout sample were present on April 1, which amounts to 50 trout. - Assuming the proportion of tagged trout in August reflects the proportion of tagged trout in April, the equation relating the two is frac{4}{50} = frac{80}{x}. Calculation: [ frac{4}{50} = frac{80}{x} Rightarrow 4x = 4000 Rightarrow x = 1000. ] Thus, the biologist estimates that there were boxed{1000} trout in the river on April 1.
question:A grocery store priced a gallon of milk at 25% above the wholesale price of 4. If a customer comes in with a coupon for 5% off milk, how much does that customer pay?
answer:First, we need to calculate the retail price of the milk, which is 25% above the wholesale price. 25% of the wholesale price (4) is: 0.25 * 4 = 1 So, the retail price of the milk is: 4 + 1 = 5 Now, the customer has a coupon for 5% off the retail price. 5% of the retail price (5) is: 0.05 * 5 = 0.25 So, the discount amount with the coupon is 0.25. Finally, we subtract the discount from the retail price to find out how much the customer pays: 5 - 0.25 = 4.75 The customer pays boxed{4.75} for the gallon of milk with the coupon.