When bodies at two different temperatures interact, one of them loses heat, while the other gains some heat. In this process, the body with the higher temperature comes down to a lower temperature, while the one with the lower temperature will gain heat and hence be at a higher temperature.
When water 10 degrees hotter is added to the tub, the water that is already in the tub (if any) will gain heat and the supplied water will lose heat. However, to gain 10 degrees, all the water originally present in the tub will have to be replaced by the hot water from the solar heater.
Thus, 300 gallons of hot water is needed. Since the rate of hot water flow is 1 gallon per minute, it will take 300 min for the temperature of the tub's water to rise by 10 degrees F.
Hope this helps.
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