What is the probability that a randomly thrown dart hits a shaded region, given that the dart lands within the rectangle? All of the circles are congruent, and the diameter of each circle is 6 cm.
0.785 trust me i'm from the future from 2025
better explain some more about the diagram.
To determine the probability that a randomly thrown dart hits a shaded region, given that the dart lands within the rectangle, we need to calculate the area of the shaded region and divide it by the area of the rectangle.
First, let's calculate the area of one circle. The radius of each circle is half the diameter, so the radius is 6 cm / 2 = 3 cm. The area of one circle can be found using the formula A = πr^2, where r is the radius:
A = π(3 cm)^2
A = π(9 cm^2)
A ≈ 28.27 cm^2
Since all the circles are congruent, the shaded region consists of 6 circles. Hence, the total area of the shaded region is 6 times the area of one circle:
Total shaded area = 6 × 28.27 cm^2
Total shaded area ≈ 169.62 cm^2
Now, let's consider the rectangle. Since the diameter of each circle is 6 cm, the length of the rectangle is equal to the sum of the diameter of the circle and the width of the rectangle. Hence, the length of the rectangle is 6 cm + 6 cm + 6 cm = 18 cm. The width of the rectangle is equal to the diameter of the circle, which is 6 cm.
Hence, the area of the rectangle can be calculated as follows:
Area of rectangle = length × width
Area of rectangle = 18 cm × 6 cm
Area of rectangle = 108 cm^2
Finally, to find the probability, we divide the area of the shaded region by the area of the rectangle:
Probability = (Total shaded area) / (Area of rectangle)
Probability ≈ 169.62 cm^2 / 108 cm^2
Probability ≈ 1.57
Therefore, the probability that a randomly thrown dart hits a shaded region, given that the dart lands within the rectangle, is approximately 1.57.
To find the probability that a randomly thrown dart hits a shaded region within a rectangle, we need to compare the area of the shaded region to the total area of the rectangle.
Let's break down the problem into steps:
Step 1: Calculate the area of one circle.
Given that the diameter of each circle is 6 cm, we can find its radius by dividing the diameter by 2. Therefore, the radius is 6/2 = 3 cm. The area of a circle is calculated using the formula A = πr^2, where A represents the area and r represents the radius. Substituting the values, we have A = π(3)^2 = 9π cm^2.
Step 2: Determine the total area of the shaded region.
Since all of the circles are congruent, the shaded region consists of 4 circles. Therefore, the total area of the shaded region is 4 times the area of one circle, which gives us 4 * 9π = 36π cm^2.
Step 3: Calculate the area of the rectangle.
To calculate the area of the rectangle, we need its dimensions. Unfortunately, the dimensions of the rectangle are not mentioned in the given information. Please provide the dimensions of the rectangle so that we can calculate its area.
Once we have the area of the rectangle, we can proceed to the final step.
Step 4: Calculate the probability of hitting the shaded region.
The probability is determined by dividing the area of the shaded region by the area of the rectangle. The formula for probability is P = A_shaded / A_total, where P represents the probability, A_shaded represents the area of the shaded region, and A_total represents the total area of the rectangle.
With the area of the shaded region and the area of the rectangle determined, the equation for probability becomes P = 36π / A_total. Substitute the value of A_total and evaluate to find the probability of hitting the shaded region.