To find out how many cups can be filled from the cooler, we first need to determine the volume of one paper cup, which is shaped like a cone.
The volume \( V \) of a cone can be calculated using the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where:
- \( r \) is the radius of the base of the cone,
- \( h \) is the height of the cone, and
- \( \pi \) is approximately 3.14.
Given that the height \( h \) of the cup is 10 cm and the radius \( r \) is 6 cm, we can substitute these values into the formula:
\[ V = \frac{1}{3} \times 3.14 \times (6)^2 \times 10 \]
Calculating the radius squared:
\[ (6)^2 = 36 \]
Now substitute that into the volume equation:
\[ V = \frac{1}{3} \times 3.14 \times 36 \times 10 \]
Calculating \( 36 \times 10 \):
\[ 36 \times 10 = 360 \]
Now, substituting back into the equation for volume:
\[ V = \frac{1}{3} \times 3.14 \times 360 \]
Calculating \( 3.14 \times 360 \):
\[ 3.14 \times 360 = 1130.4 \]
Now, dividing by 3:
\[ V = \frac{1130.4}{3} = 376.8 \text{ cm}^3 \]
Now we know the volume of one cup is 376.8 cm³.
Next, we need to find out how many cups can be filled from the cooler which has a total volume of \( 16,956^3 \) cm³.
Calculating \( 16,956^3 \):
\[ 16,956^3 = 16,956 \times 16,956 \times 16,956 \]
Calculating \( 16,956^2 \):
\[ 16,956 \times 16,956 = 287,458,436 \]
Then multiplying by \( 16,956 \):
\[ 287,458,436 \times 16,956 = 4,861,550,940,856 \]
Thus, the cooler has approximately 4,861,550,940,856 cm³ of water.
To find the number of cups that can be filled, divide the total volume of water in the cooler by the volume of one cup:
\[ \text{Number of cups} = \frac{4,861,550,940,856}{376.8} \]
Calculating that gives us:
\[ \text{Number of cups} \approx 12,892,748,433.5 \]
Since we can only fill a whole number of cups, we round down:
Therefore, the number of cups that can be filled from the cooler is approximately \( 12,892,748,433 \) cups.
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