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question:Calculate the dot product of the following two vectors: left( begin{array}{c} 1.78 -0.83 end{array} right) cdot left( begin{array}{c} 0.73 -4.43 end{array} right)
answer:The dot product is calculated as the sum of the products of corresponding components: (1.78 cdot 0.73) + (-0.83 cdot -4.43) = 1.2994 + 3.6549 = 4.9543 Hence, the dot product of the two vectors is 4.9543.
question:Classify the conic represented by the equation -10x^2 + 4x + y^2 - 3y + 10 = 0, and rewrite it in standard form. Then, state its relevant properties, including the center, foci, eccentricity, and asymptotes.
answer:Classification: Hyperbola Rewritten Equation: left(y - frac{3}{2}right)^2 - 10left(x - frac{1}{5}right)^2 = -frac{163}{20} Standard Form: frac{left(y - frac{3}{2}right)^2}{frac{163}{20}} - frac{left(x - frac{1}{5}right)^2}{frac{163}{200}} = 1 Properties: - Center: left{frac{1}{2} left(frac{1}{20} left(4-sqrt{3586}right) + frac{1}{20} left(4+sqrt{3586}right)right), frac{3}{2}right} - Foci: left( begin{array}{cc} frac{1}{20} left(4-sqrt{3586}right) & frac{3}{2} frac{1}{20} left(4+sqrt{3586}right) & frac{3}{2} end{array} right) - Eccentricity: e = sqrt{11} - Asymptotes: y = sqrt{10}x + frac{1}{10} left(15 - 2sqrt{10}right) and y = frac{1}{10} left(15 + 2sqrt{10}right) - sqrt{10}x
question:What are the key drivers of growth in the Indian biotechnology sector?
answer:Key drivers of growth in the Indian biotechnology sector include: * Projected compound annual growth rate (CAGR) of 30.46% by 2025. * India's position as a hub for the biotechnology industry. * Ranking among the top 12 biotechnology destinations globally and third in the Asia Pacific region. * Thriving job market in sectors such as pharmaceuticals, healthcare, and food processing. * Advancements in clinical testing and diagnostic tools.
question:A 60-turn coil with a radius of 4.8 cm rotates in a uniform magnetic field of magnitude 0.52 T. If the coil carries a current of 30 mA, what is the maximum torque exerted on the coil?
answer:The torque on a current loop in a magnetic field is given by: {eq}tau = NIABsintheta {/eq} where: * {eq}N{/eq} is the number of turns in the coil * {eq}I{/eq} is the current in the coil * {eq}A{/eq} is the area of the coil * {eq}B{/eq} is the magnitude of the magnetic field * {eq}theta{/eq} is the angle between the magnetic field and the normal to the coil In this problem, we are given: * {eq}N = 60{/eq} * {eq}I = 30 text{ mA} = 0.03 text{ A}{/eq} * {eq}r = 4.8 text{ cm} = 0.048 text{ m}{/eq} * {eq}B = 0.52 text{ T}{/eq} The area of the coil is: {eq}A = pi r^2 = pi (0.048)^2 = 7.23 times 10^{-3} text{ m}^2 {/eq} The maximum torque occurs when {eq}theta = 90^circ{/eq}, so {eq}sintheta = 1{/eq}. Therefore, the maximum torque is: {eq}tau_{max} = NIAB = (60)(0.03)(7.23 times 10^{-3})(0.52) = 6.77 times 10^{-3} text{ N} cdot text{m} {/eq} The maximum torque exerted on the coil is: {eq}tau_{max} = 6.77 times 10^{-3} text{ N} cdot text{m} {/eq}