A yeast culture weighing 2 grams is removed from a refrigerator unit and is expected to grow at the rate of $W^{\prime}(t)=0.4 e^{0.2 t}$ grams per hour at a higher controlled temperature. How much will the weight of the culture increase during the first 10 hours of growth? How much will the weight of the culture increase from the end of the 10 th hour to the end of the 20 th hour of growth? The weight increase during the first 10 hours is approximately $\square$ grams. (Type an integer or decimal rounded to three decimal places as needed.) example Get more help . Clear all Check answer

Step 1 ：We are given the rate of growth of the yeast culture as $W^{\prime}(t)=0.4 e^{0.2 t}$ grams per hour. We need to find the increase in weight during the first 10 hours and from the 10th to the 20th hour.

Step 2 ：To find the increase in weight, we need to integrate the rate of growth function over the given time intervals.

Step 3 ：First, we integrate the function $W^{\prime}(t)=0.4 e^{0.2 t}$ from 0 to 10. This will give us the increase in weight during the first 10 hours.

Step 4 ：After performing the integration, we get the result which represents the increase in weight of the yeast culture during the first 10 hours of growth.

Step 5 ：Final Answer: The weight increase during the first 10 hours is approximately \(\boxed{result}\) grams.