When Erica and Ken explored a cave, they each found a gold nugget. Erica’s nugget is similar to Ken’s nugget. They measured the length of two matching parts of the nuggets and found that Erica’s nugget is five times longer than Ken’s. When they took their nuggets to the metallurgist to be analyzed, they learned that it would cost $\$30$ to have the surface area and weight of the smaller nugget calculated, and $\$150$ to have the same analysis done on the larger nugget.  

“I won’t have that kind of money until I sell my nugget, and then I won’t need it analyzed!” Erica says.

“Wait, Erica. Don’t worry. I’m pretty sure we can get all the information we need for only $\$30$.”

1. Explain how they can get all the information they need for $\$30$.

   Answer (a):

   Since the two nuggets are similar, if they spend $\$30$ to get Ken's nugget analyzed, they can calculate the surface area and weight of Erica's nuggets using the ratios of similarity.

2. If Ken’s nugget has a surface area of $110 \; \text{cm}^2$, what is the surface area of Erica’s nugget?

   Hint (b):

   We know the linear scale factor is $5$, so the ratio of the areas is $5^2$, $25$.

3. If Ken’s nugget weighs $5.6$ g (about $1.8$ oz), what is the weight of Erica’s nugget?

   Hint (c):

   Since the nuggets are solid gold and have the same density throughout, calculate the weight using the ratio of the volumes.

   Answer (c):

   $7000 \; \text{g} \approx 14 \; \text{pounds}$