The circumference of a circle is 4 times the circumference of a circle with a radius 1 unit less. What is the radius of each circle?
radius of larger circle --- r
radius of smaller circle --- r-1
circumf of large ---- 2πr
circumf of smaller = 2π(r-10
but 2πr = 4(2π(r-1))
2πr = 8πr - 8π
divide each term by π and rearrange equation
-6r = -8
r = 8/6 = 4/3
larger has radius 4/3
smaller has radius 1/3
I will leave the checking up to you,
HATS OFF ! Amazing is a less adjective for you. Thank you.
To find the radius of each circle, let's break down the problem into smaller steps:
Step 1: Understand the problem.
The given information states that the circumference of a circle is 4 times the circumference of another circle with a radius 1 unit less. We need to find the radius of each circle.
Step 2: Define the variables.
Let's assign variables to the radii of the circles for easier calculation. Let's call the radius of the first circle R and the radius of the second circle r.
Step 3: Relate the circumferences of the circles.
The circumference of a circle is given by the formula: C = 2πr, where π is a constant value approximately equal to 3.14.
According to the problem statement, the first circle's circumference (C1) is four times the second circle's circumference (C2), so we can express this as an equation:
C1 = 4 * C2
Using the circumference formula, we can rewrite this equation in terms of the radii:
2πR = 4 * 2π(r - 1)
Step 4: Solve the equation.
To find the radius, we can simplify and solve the equation:
2πR = 8π(r - 1)
Divide both sides of the equation by 2π to isolate the variables:
R = 4(r - 1)
Expand the brackets, distribute 4, and simplify:
R = 4r - 4
Rearrange the equation to solve for r:
4r = R + 4
r = (R + 4) / 4
Step 5: Substitute values to calculate the radius.
To find the specific values of the radii, we need more information. If we are given a specific value for R, we can substitute it into the equation to solve for r.
For example, let's assume R = 5 units. Substituting this value into the equation:
r = (5 + 4) / 4
r = 9 / 4
r = 2.25 units
Therefore, with a radius of 5 units for the first circle, the second circle has a radius of 2.25 units.