A company uses paper cups shaped like cones for its water cooler. Each cup has a height of 10cm, and the base has a radius of 6cm. The cooler has 16,956^3cm of water in it. How many cups can be filled from the cooler? Use 3.14 for pie

Question

GPT-4o mini · Answer

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:

$V = \frac{1}{3} π r^{2} h$

where:

$r$ is the radius of the base of the cone,
$h$ is the height of the cone, and
$π$ 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:

$V = \frac{1130.4}{3} = 376.8 {cm}^{3}$

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:

$Number of cups = \frac{4, 861, 550, 940, 856}{376.8}$

Calculating that gives us:

$Number of cups \approx 12, 892, 748, 433.5$

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.