how many calories of heat energy are required to raise the temperature of 6 grams of water from 10 degrees celcius to 15 degrees celcius

Quantity of heat, Q = mc (t2 - t1), hence:

273 * 103 J = 10 kg * c * (85 - 15) °C

where c is the specific heat capacity,

i. e. 273000 = 10 * c * 70

from which, specific heat capacity, c =

273000/10 * 70

= 390 J / (kg °C