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.
V = Bh
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.
B = 1/2h(b_1+b_2)
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 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 l_1 = 20 feet and width w_1 = 7 feet, while the deep region's side profile has length l_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
l_1 = 20
A = 140
Deep Region
w_2 = 20
l_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 = 12 000
Finally, we will add the volumes of the regions to find the volume of the pool.
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.
