A dartboard consists of five concentric circles. The radius of the smallest circle is about 1 inch. The radius of the second circle is about 3 inches longer. The radius of the third circle is about 1 inch longer than the previous circle. The radius of the fourth circle is about 2 inches longer than the previous circle. The radius of the largest circle is about 0.75 inches greater than the previous circle. What is the difference between the circumferences of the largest and smallest circle? Round your answer to the nearest tenth of an inch.
42.2
Well, let's start by figuring out the radius of each circle.
The radius of the smallest circle is 1 inch.
The radius of the second circle is 1 inch + 3 inches = 4 inches.
The radius of the third circle is 4 inches + 1 inch = 5 inches.
The radius of the fourth circle is 5 inches + 2 inches = 7 inches.
Finally, the radius of the largest circle is 7 inches + 0.75 inches = 7.75 inches.
To find the circumference of a circle, we use the formula C = 2πr, where r is the radius.
So, the circumference of the smallest circle is approximately 2π(1) = 2π inches.
And the circumference of the largest circle is approximately 2π(7.75) = 15.4π inches.
To find the difference between the two circumferences, we subtract them:
15.4π - 2π = 13.4π inches.
Since we need to round our answer to the nearest tenth of an inch, the difference is approximately 13.4(3.14) = 42 inches.
Therefore, the difference between the circumferences of the largest and smallest circles is approximately 42 inches.
To find the difference between the circumferences of the largest and smallest circle, we first need to calculate the circumference of each circle.
The formula for finding the circumference of a circle is:
Circumference = 2 * π * radius
Let's calculate the circumference of each circle step-by-step:
1. Smallest Circle:
Radius = 1 inch
Circumference = 2 * π * 1 = 2π inches
2. Second Circle:
Radius = 1 + 3 = 4 inches (3 inches longer than the smallest circle)
Circumference = 2 * π * 4 = 8π inches
3. Third Circle:
Radius = 4 + 1 = 5 inches (1 inch longer than the second circle)
Circumference = 2 * π * 5 = 10π inches
4. Fourth Circle:
Radius = 5 + 2 = 7 inches (2 inches longer than the third circle)
Circumference = 2 * π * 7 = 14π inches
5. Largest Circle:
Radius = 7 + 0.75 = 7.75 inches (0.75 inches greater than the fourth circle)
Circumference = 2 * π * 7.75 = 15.5π inches
Now, let's calculate the difference between the circumferences of the largest and smallest circles:
Difference = Circumference of Largest Circle - Circumference of Smallest Circle
Difference = 15.5π - 2π
To round the answer to the nearest tenth of an inch, we can use the value of π as approximately 3.14.
Difference = (15.5 - 2) * 3.14
Difference ≈ 13.5 * 3.14
Difference ≈ 42.39 inches
Therefore, the difference between the circumferences of the largest and smallest circles is approximately 42.39 inches when rounded to the nearest tenth of an inch.
To find the difference between the circumferences of the largest and smallest circle, we need to calculate each circumference first.
The formula for the circumference of a circle is given by:
C = 2πr
Let's calculate the radii of each circle:
- Smallest Circle Radius = 1 inch
- Second Circle Radius = 1 + 3 = 4 inches (3 inches longer)
- Third Circle Radius = 4 + 1 = 5 inches (1 inch longer)
- Fourth Circle Radius = 5 + 2 = 7 inches (2 inches longer)
- Largest Circle Radius = 7 + 0.75 = 7.75 inches (0.75 inches greater)
Now, let's calculate the circumferences using the formula mentioned earlier:
- Smallest Circle Circumference: C₁ = 2π(1) = 2π inches
- Largest Circle Circumference: C₅ = 2π(7.75) = 15.41π inches
To find the difference, we subtract the smallest circle circumference from the largest:
Difference = C₅ - C₁ = 15.41π - 2π ≈ 13.41π
Using the approximation of π as 3.14, we can calculate the numerical value of the difference:
Difference ≈ 13.41 * 3.14 ≈ 42.05 inches
Therefore, the difference between the circumferences of the largest and smallest circle is approximately 42.05 inches (rounded to the nearest tenth).
radius 1 = 1
radius 2 = 1+3 = 4
radius 3 = 4+1 = 5
radius 4 = 5+2 = 7
radius 5 = 7+.75 = 7.75
circumference 1 = 2π(1) = 2π
circumference 5 = 2π(7.75) = 15.5π
difference in circumference = 15.5π - 2π= 13.5π
= appr. 48.7 inches