I just don't understand how to do this...
A kidney-shaped swimming pool is located in a rectangular back yard. The sides of the pool consist of four connecting semicircles. A parachutist is hired to land in the back yard during a party. Assuming the parachutist lands in the back yard randomly, what is the probability that he will land on dray land ( not the pool)?
The back yard dimmensions: 30ft by 50 ft
large semicircle: 16ft
Two smaller semicircles: 8 ft ( both of them are )
Please help! I don't understand how! thank you!
If the parachute lands randomly inside the yard, the probability od landing in the pool equals the pool area divided by the total yard area (including pool)
The total yard area is 1500 sq. feet.
The total pool water area is 16 pi + 64 pi + (16x8). Draw the pool with connected semicircles with the large and small ones opposite each other in pairs) and you will see why.
I assumed that the given semicircle dimensions were diameters. You didn't say. If they were radii, the pool is four times larger.
The probability of hitting dry land is 1 - (Apool/Atotal)
To find the probability that the parachutist will land on dry land, we need to compare the area of the dry land to the total area of the backyard.
Step 1: Calculate the area of the entire backyard.
The backyard is rectangular, with dimensions 30 ft by 50 ft. The area of a rectangle is calculated by multiplying the length by the width. Therefore, the area of the backyard is 30 ft * 50 ft = 1500 square ft.
Step 2: Calculate the area of the pool.
The pool consists of four connecting semicircles. The larger semicircle has a diameter of 16 ft, so its radius is 8 ft. The smaller semicircles also have a diameter of 8 ft, so their radii are 4 ft. To calculate the area of a semicircle, we use the formula (π * r^2) / 2, where π is a mathematical constant approximately equal to 3.14159.
Let's calculate the area of the large semicircle:
Area of large semicircle = (π * (8 ft)^2) / 2 = (3.14159 * 64 ft^2) / 2 ≈ 100.53141 square ft.
Now, calculate the area of the two smaller semicircles:
Area of each small semicircle = (π * (4 ft)^2) / 2 = (3.14159 * 16 ft^2) / 2 ≈ 25.13274 square ft.
Total area of two small semicircles = 2 * 25.13274 square ft ≈ 50.26548 square ft.
The total area of the pool is the sum of the areas of the large and small semicircles:
Total area of the pool = 100.53141 square ft + 50.26548 square ft ≈ 150.79689 square ft.
Step 3: Calculate the area of the dry land.
The dry land is the remaining area after subtracting the pool area from the total backyard area:
Area of dry land = Total area of the backyard - Total area of the pool = 1500 square ft - 150.79689 square ft ≈ 1349.20311 square ft.
Step 4: Calculate the probability.
Probability is defined as the ratio of the favorable outcomes to the total possible outcomes. In this case, the favorable outcome is the parachutist landing on the dry land, and the total possible outcome is the entire backyard.
Probability = Area of dry land / Total area of the backyard = 1349.20311 square ft / 1500 square ft ≈ 0.89947.
Therefore, the probability that the parachutist will land on dry land (not the pool) is approximately 0.89947 or 89.95%.