You have a very interesting question. I've wondered about this myself. It's an even more interesting question with a lot of probability and statistics issues if the coins were an assortment of pennies, nickels, dimes and quarters. It involves how well coins -- shaped like flat little cylinders -- pack into a volume. And if it was a question of weight, rather than volume, then it includes a rather odd random variable, in what decade was it filled? Years back the materials used in coins changed to a copper-nickel sandwich (making them cheaper to manufacture), and the stamping impressions have changed, so old coins don't weigh the same as newer coins. But let's first assume these are all modern coins and not the ones that are mostly in the hands of coin collectors.

Your first question asks what your bottle would be worth. If you take a similar bottle and fill it with water using a measuring cup, you can determine how much volume it has. We'll need that number to proceed if we want to know the total value of coins in your bottle. Then we would need to estimate a packing density of coins, which will be a number less than 100% (probably closer to 60% since they are round and even though they tend to lay in flat layers, they can never fill in all the gaps). But for now, let's address your "for instance" case of an arbitrary volume full of quarters versus dimes.

We'll trust the mint for coin specifications and use their values;

www.usmint.gov

A dime has a diameter of 0.7 inches (1.8 cm) diameter, and is about 0.06 inches (0.13 cm) thick. Dimes have a volume of 0.33 cm

Quarter are about 0.95 inches (2.4 cm) diameter and are 0.055 inches (0.14 cm) thick. They weigh about 5.67 grams and have a volume of 0.63 cm

But a quarter's face value is 25 cents versus a dime's value at 10 cents, so they are 2.5 times more valuable per coin.

The volumes are 0.63 cm

So from this, we can get 2.5 times more face value with quarters but a coin volume difference of less than 2.5, making quarters more valuable per unit volume. In other words, since it would take two and a half dimes to make the same monetary value as a quarter, and two and a half dimes are 0.52 the size per dime times 2.5, they are equal to 1.3 times the volume. Therefore, dimes are a 30% increase in volume.

A possible flaw I see is if this analysis is if the bottle is small enough to prevent quarters and dimes to lay in the same packing ratio. If the width of the bottle is small relative to the diameter of the coin, then it's a different problem entirely. With coins the size of dimes or quarters, 5 cubic inches would not be 'huge' enough, but 5 cubic feet certainly would be.

Your first question asks what your bottle would be worth. If you take a similar bottle and fill it with water using a measuring cup, you can determine how much volume it has. We'll need that number to proceed if we want to know the total value of coins in your bottle. Then we would need to estimate a packing density of coins, which will be a number less than 100% (probably closer to 60% since they are round and even though they tend to lay in flat layers, they can never fill in all the gaps). But for now, let's address your "for instance" case of an arbitrary volume full of quarters versus dimes.

We'll trust the mint for coin specifications and use their values;

www.usmint.gov

A dime has a diameter of 0.7 inches (1.8 cm) diameter, and is about 0.06 inches (0.13 cm) thick. Dimes have a volume of 0.33 cm

^{3}and weigh 2.27 grams, giving a single dime's volume a density of 2.27 grams per 0.33 cm^{3}or 6.85 grams per cm^{3}Quarter are about 0.95 inches (2.4 cm) diameter and are 0.055 inches (0.14 cm) thick. They weigh about 5.67 grams and have a volume of 0.63 cm

^{3}so this is a density of 8.96 grams per cm^{3}But a quarter's face value is 25 cents versus a dime's value at 10 cents, so they are 2.5 times more valuable per coin.

The volumes are 0.63 cm

^{3 }per quarter and 0.33 cm^{3}per dime which is a volumetric ratio of 1.91 dimes per quarter, or if we invert it, 0.52 quarters per dime.So from this, we can get 2.5 times more face value with quarters but a coin volume difference of less than 2.5, making quarters more valuable per unit volume. In other words, since it would take two and a half dimes to make the same monetary value as a quarter, and two and a half dimes are 0.52 the size per dime times 2.5, they are equal to 1.3 times the volume. Therefore, dimes are a 30% increase in volume.

A possible flaw I see is if this analysis is if the bottle is small enough to prevent quarters and dimes to lay in the same packing ratio. If the width of the bottle is small relative to the diameter of the coin, then it's a different problem entirely. With coins the size of dimes or quarters, 5 cubic inches would not be 'huge' enough, but 5 cubic feet certainly would be.