Exercise 33 Page 936

We are given a side profile of a pool in a polar bear exhibit at a local zoo. We know that the pool is $20$ feet wide, and a bear is equally likely to swim anywhere in the pool. We want to find the probability that the bear will be in the incline region.

To find the probability we can use geometric probability. The probability that the bear is in the incline region is the ratio of the volume of the incline region to the volume of the entire pool.

Finding the Volume of the Incline Region

First, we will find the volume of the incline region of the pool. Let's take a closer look at the side profile of this region.

As we can see, this region of the pool is a right prism. Its base is a trapezoid and its height is $20$ feet. Remember the formula for the volume $V$ of a prism. In this formula $B$ is the area of the base of the prism and $h$ is its height. The base of our prism is a trapezoid with base lengths $b_1=7$ feet and $b_2=20$ feet, and height is $h=25$ feet. Let's recall the formula for the area $B$ of a trapezoid. By substituting $25$ for $h,$ $7$ for $b_1,$ and $20$ for $b_2$ in this formula, we can find the area of the base of the incline region. The base of the incline region has an area of $337.5$ square feet. Now we can calculate the volume of the incline region of the pool. The volume of the incline region is $6750$ cubic feet.

Finding the Volume of the Pool

Now let's find the volume of the entire pool. To do so, note that each region of the pool is a right prism. The base of each prism is the side profile of the region and their height is $20$ feet. To calculate the volume of the pool, we will find the volume of each region. We already know that the volume of the incline region is $V_I=6750$ cubic feet. Let's look at the two other regions.

Both of the regions have side profiles which are rectangles. The shallow region's side profile has length ${\ell }_1=20$ feet and width $w_1=7$ feet, while the deep region's side profile has length ${\ell }_2=30$ feet and width $w_2=20$ feet. Let's calculate the areas of the side profiles of these regions.

	Width	Length	Area
Shallow Region	$w_1=7$	${\ell }_1=20$	$A=140$
Deep Region	$w_2=20$	${\ell }_2=30$	$A=600$

Now, let's calculate the volumes of these areas.

	Area of Side Profile	Height	Volume
Shallow Region	$A=140$	$h=20$	$V_S=2800$
Deep Region	$A=600$	$h=20$	$V_D=12000$

Finally, we will add the volumes of the regions to find the volume of the pool. The volume of the pool is $V=21550$ cubic feet.

Calculating the Probability

We will calculate the probability that the bear will be in the incline region using geometric probability. The probability is the ratio of the volume $V_i$ of the incline region to the volume $V$ of the pool. The probability that the bear is in the incline region of the pool is approximately $0.31,$ or approximately $31\%.$