The radius of the bull's-eye of the dartboard is 8 inches. The radius of each concentric circle is 8 inches more than the radius of the circle inside it. If a dart lands at random on the dartboard, what is the probability that the dart will hit in area c?
Answers
Answer:
5/16
(2k-1)/n²
Step-by-step explanation:
The radius of the bull's-eye of the dartboard is 8 inches. The radius of each concentric circle is 8 inches more than the radius of the circle inside it. If a dart lands at random on the dartboard, what is the probability that the dart will hit in area c?
Area of inner most circle = π 8² = (1 * 8 * 8) π inch²
Area of 2nd circle = π (8+8)² - π 8² = π (24 * 8) = π (3 * 8 * 8) inch²
Area of 3rd circle = π (16+8)² - π 16² = π (50 * 8) = π (5 * 8 * 8) inch²
Area will keep increasing for each circle this way
Area for nth circle = π ((2n-1) * 8 * 8) inch²
Total Area for n circle = (1 * 8 * 8) π + π (3 * 8 * 8) + π (5 * 8 * 8) + .........+ π ((2n-1) * 8 * 8)
= ((8 * 8) π) (1 + 3 + 5 + +(2n-1))
= (8 * 8) π) n²
or directly area of nth circle = π(8n)²
Probability of hitting for any kth circle = π ((2k-1) * 8 * 8) / (8 * 8) π) n²
= (2k-1)/n² ( k is the number of circle for which probability is being calculated and n is total number of circles)
As per picture attached for C
k = 3
n = 4
putting this
probability = (2*3 - 1)/4² = 5 /16
