Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. a(t) = 2 i + 6t j + 12t2 k, v(0) = i, r(0) = 7 j − 4 k

v(t) = (2t + 1)i + 3t²j + 4t³k

r(t) = (t² + t)i + (t³ + 7)j + (t⁴ - 4)k

Explanation:

a(t) = 2i + 6tj + 12t²k

v(t) = ∫a(t)dt

     = ∫(2i + 6tj + 12t²k)dt

     = 2ti + (6t²/2)j + (12t³/3)k + c

     = 2ti + 3t²j + 4t³k + c

v(0) = i

    i = 0i + 0j + 0k + c

    c = i

∴ v(t) = 2ti + 3t²j + 4t³k + i

v(t) = (2t + 1)i + 3t²j + 4t³k

r(t) = ∫ v(t)dt

    = i ∫ (2t + 1)dt + 3j ∫ t²dt + 4k ∫ t³dt

    = i (2t²/2 + t) + 3j(t³/3) + 4k(t⁴/4) + d

    = i (t² + t) + jt³ + t⁴k + d

r(0) = 7j - 4k

0i + 0j + 0k + d = 7j - 4k

d = 7j - 4k

∴ r(t) = (t² + t)i + t³j + t⁴k + 7j - 4k

r(t) = (t² + t)i + (t³ + 7)j + (t⁴ - 4)k

ans.

heterothermic organisms include those organisms that can switch between homeothermic (having body temperature higher than environment) and poikilothermic (having variation in internal body temperature) strategies.

some migratory birds show heterothermy as they decrease their metabolic rate during cold nights to keep their body temperature low. drop in metabolic rate allows increase in production efficiency of birds that reduces the internal body temperature to make it equal to external cold temperature during cold nights.

thus, the correct answer is 'birds allow their temperature drop in order to increase their production efficiency.'

a manufacturer of printed circuit boards uses exponential smoothing with trend to forecast monthly demand of its product. at the end of december, the company wishes to forecast sales for january. the estimate of trend through november has been 200 additional boards sold per month. average sales have been around 1000 units per month. the demand for december was 1100 units. the company uses α = 0.20 and β = 0.10. make a forecast including trend for the month of january.