. A circle has a circumference of C inches and an area of A square inches. If C = 2/5 * A what is the diameter of the circle? A. 0.4 inches B. 0.8 inches C. 2.5 inches D. 10 inches
r = C / 2pi
C=(2/5) * pi r^2
2 pi r = (2/5) pi r^2
r = 5
so D = 2 r = 10
To find the diameter of the circle, we can use the formulas for circumference and area.
Let's start by using the formula for circumference:
C = 2πr
where C is the circumference and r is the radius of the circle. We can rewrite this equation as:
2πr = (2/5)A
Now, let's use the formula for area:
A = πr^2
We are given that C = (2/5)A, so we can substitute (2/5)A for C in the circumference formula:
2πr = (2/5)A
Now we can substitute πr^2 for A in the above equation:
2πr = (2/5)(πr^2)
We can cancel out π from both sides of the equation:
2r = (2/5)r^2
Now, to solve for the radius, we can divide both sides of the equation by r:
2 = (2/5)r
Now, we can solve for r:
r = (2/5)(5/2)
r = 1
The radius of the circle is 1 inch.
Finally, the diameter of the circle is twice the radius, so the diameter is:
d = 2r = 2(1) = 2 inches
Therefore, the correct answer is C. 2.5 inches.
To find the diameter of a circle given its circumference and area, we can use the formulas for circumference and area of a circle.
The circumference of a circle (C) can be calculated using the formula: C = 2πr, where r is the radius of the circle.
The area of a circle (A) can be calculated using the formula: A = πr^2.
According to the given information, C = (2/5)A. We can substitute the value of C from the circumference formula into this equation.
So, we have (2/5)A = 2πr.
To find the diameter, we need to find the value of r, and then double it.
To solve for r, we can divide both sides of the equation by 2π:
(2/5)A ÷ 2π = r.
Now that we have the value of r, we can calculate the diameter by doubling it:
Diameter = 2r.
Let's plug in the option values to check which one satisfies the equation:
A. If the diameter is 0.4 inches, then the radius is 0.2 inches, and the area is πr^2 = π(0.2)^2 = 0.04π square inches. The circumference would be C = 2πr = 2π(0.2) = 0.4π inches. But C ≠ (2/5)A, so option A is not correct.
B. If the diameter is 0.8 inches, then the radius is 0.4 inches, and the area is πr^2 = π(0.4)^2 = 0.16π square inches. The circumference would be C = 2πr = 2π(0.4) = 0.8π inches. But C ≠ (2/5)A, so option B is not correct.
C. If the diameter is 2.5 inches, then the radius is 1.25 inches, and the area is πr^2 = π(1.25)^2 = 1.5625π square inches. The circumference would be C = 2πr = 2π(1.25) = 2.5π inches. Let's check if C = (2/5)A: (2/5)A = (2/5)(1.5625π) = 0.625π square inches. Since 2.5π ≠ 0.625π, option C is not correct.
D. If the diameter is 10 inches, then the radius is 5 inches, and the area is πr^2 = π(5)^2 = 25π square inches. The circumference would be C = 2πr = 2π(5) = 10π inches. Let's check if C = (2/5)A: (2/5)A = (2/5)(25π) = 10π square inches. C = (2/5)A, so option D is correct.
Therefore, the diameter of the circle is 10 inches (option D).