We are given that conical paper cups have a slant height of 4.5 inches and a diameter of 3 inches. A water cooler holds 2000 cubic inches of water. We are asked to find how many cups can be filled from the water cooler.

1. Problem Description

We are given that conical paper cups have a slant height of 4.5 inches and a diameter of 3 inches. A water cooler holds 2000 cubic inches of water. We are asked to find how many cups can be filled from the water cooler.

2. Solution Steps

First, we need to find the radius of the cone.

$radius = diameter / 2$

$r = 3 / 2 = 1.5$ inches.

Next, we need to find the height of the cone. We are given the slant height, $l = 4.5$ inches. We have a right triangle with the height $h$, radius $r$, and slant height $l$.

$h^2 + r^2 = l^2$

$h^2 = l^2 - r^2$

$h = \sqrt{l^2 - r^2}$

$h = \sqrt{(4.5)^2 - (1.5)^2}$

$h = \sqrt{20.25 - 2.25}$

$h = \sqrt{18}$

$h = 3\sqrt{2} \approx 4.24$ inches

Now, we can find the volume of the cone.

$V = (1/3) \pi r^2 h$

$V = (1/3) \pi (1.5)^2 (3\sqrt{2})$

$V = \pi (2.25) \sqrt{2}$

$V \approx \pi (2.25)(1.414)$

$V \approx 10.01$ cubic inches.

The water cooler holds 2000 cubic inches. The number of cups that can be filled is the total volume of water divided by the volume of each cup.

$Number\ of\ cups = 2000 / V$

$Number\ of\ cups = 2000 / 10.01$

$Number\ of\ cups \approx 199.8$

Since we can only fill a whole number of cups, we round down to

1

9

9.

3. Final Answer

199 cups can be filled.