First, we need to determine the ratio of popcorn kernels to butter based on the original recipe.
The original recipe uses:
- \(0.5\) cup of butter (which is equivalent to \(8\) tablespoons, since \(1\) cup = \(16\) tablespoons),
- \(3\) tablespoons of kernels.
Now, we find the ratio of kernels to butter: \[ \text{Butter per tablespoon of kernels} = \frac{8 \text{ tablespoons of butter}}{3 \text{ tablespoons of kernels}} = \frac{8}{3} \text{ tablespoons of butter per tablespoon of kernels}. \]
Next, Jennie uses \(10\) tablespoons of kernels. To find the amount of butter she needs, we can multiply the number of tablespoons of kernels by the ratio we found: \[ \text{Butter needed} = 10 \text{ tablespoons of kernels} \times \frac{8}{3} \text{ tablespoons of butter per tablespoon of kernels}. \]
Calculating that: \[ \text{Butter needed} = 10 \times \frac{8}{3} = \frac{80}{3} \text{ tablespoons of butter} \approx 26.67 \text{ tablespoons of butter}. \]
Now, converting tablespoons back to cups (since \(1\) cup = \(16\) tablespoons): \[ \text{Butter in cups} = \frac{26.67 \text{ tablespoons}}{16} \approx 1.667 \text{ cups of butter}. \]
So the final answer, rounded to the nearest hundredth, is: \[ \text{Butter needed} \approx 1.67 \text{ cups}. \]