Question
A circle with a radius of 4 inches and a square with a side length of 8 inches are both inscribed in a rectangle with a width of 19 inches and a length of 21 inches. The circle and square inscribed in the rectangle are beside each other. Determine the probability that a random point placed in the figure is not inside the circle or inside the square. Round your answer to the nearest hundredth.(1 point) Responses 0.29 0.29 0.87 0.87 0.84 0.84 0.71
Answers
GPT-4o mini
To solve the problem, we need to calculate the areas involved and then find out the probability that a random point placed in the rectangle is not inside the circle or inside the square.
### Step 1: Calculate the area of the rectangle
The area of the rectangle is given by the formula:
\[
\text{Area of Rectangle} = \text{length} \times \text{width} = 21 \, \text{inches} \times 19 \, \text{inches} = 399 \, \text{square inches}
\]
### Step 2: Calculate the area of the circle
The area of the circle can be calculated using the formula:
\[
\text{Area of Circle} = \pi r^2
\]
Where \( r = 4 \, \text{inches} \):
\[
\text{Area of Circle} = \pi (4)^2 = 16\pi \approx 50.27 \, \text{square inches} \quad (\text{using } \pi \approx 3.14)
\]
### Step 3: Calculate the area of the square
The area of the square is given by:
\[
\text{Area of Square} = \text{side}^2 = 8^2 = 64 \, \text{square inches}
\]
### Step 4: Calculate the total area of the circle and square
Next, we add the areas of the circle and the square:
\[
\text{Total Area of Circle and Square} = \text{Area of Circle} + \text{Area of Square} = 16\pi + 64 \approx 50.27 + 64 \approx 114.27 \, \text{square inches}
\]
### Step 5: Calculate the area that is not inside the circle or square
Now we need to find the area of the rectangle that is not inside the circle or square:
\[
\text{Area outside both} = \text{Area of Rectangle} - \text{Total Area of Circle and Square}
\]
\[
\text{Area outside both} = 399 - 114.27 \approx 284.73 \, \text{square inches}
\]
### Step 6: Calculate the probability
The probability that a random point placed in the rectangle is not inside the circle or the square is given by:
\[
\text{Probability} = \frac{\text{Area outside both}}{\text{Area of Rectangle}} = \frac{284.73}{399} \approx 0.712
\]
### Step 7: Round to the nearest hundredth
Rounding \( 0.712 \) to the nearest hundredth gives \( 0.71 \).
### Final Answer
Thus, the probability that a random point placed in the figure is not inside the circle or inside the square is **0.71**.
### Step 1: Calculate the area of the rectangle
The area of the rectangle is given by the formula:
\[
\text{Area of Rectangle} = \text{length} \times \text{width} = 21 \, \text{inches} \times 19 \, \text{inches} = 399 \, \text{square inches}
\]
### Step 2: Calculate the area of the circle
The area of the circle can be calculated using the formula:
\[
\text{Area of Circle} = \pi r^2
\]
Where \( r = 4 \, \text{inches} \):
\[
\text{Area of Circle} = \pi (4)^2 = 16\pi \approx 50.27 \, \text{square inches} \quad (\text{using } \pi \approx 3.14)
\]
### Step 3: Calculate the area of the square
The area of the square is given by:
\[
\text{Area of Square} = \text{side}^2 = 8^2 = 64 \, \text{square inches}
\]
### Step 4: Calculate the total area of the circle and square
Next, we add the areas of the circle and the square:
\[
\text{Total Area of Circle and Square} = \text{Area of Circle} + \text{Area of Square} = 16\pi + 64 \approx 50.27 + 64 \approx 114.27 \, \text{square inches}
\]
### Step 5: Calculate the area that is not inside the circle or square
Now we need to find the area of the rectangle that is not inside the circle or square:
\[
\text{Area outside both} = \text{Area of Rectangle} - \text{Total Area of Circle and Square}
\]
\[
\text{Area outside both} = 399 - 114.27 \approx 284.73 \, \text{square inches}
\]
### Step 6: Calculate the probability
The probability that a random point placed in the rectangle is not inside the circle or the square is given by:
\[
\text{Probability} = \frac{\text{Area outside both}}{\text{Area of Rectangle}} = \frac{284.73}{399} \approx 0.712
\]
### Step 7: Round to the nearest hundredth
Rounding \( 0.712 \) to the nearest hundredth gives \( 0.71 \).
### Final Answer
Thus, the probability that a random point placed in the figure is not inside the circle or inside the square is **0.71**.