Two circles are tangent to each other...Find the radius of each circle if their combined area is 29pi square yards. PLS HELP

Question

Two circles are tangent to each other. The distance between the centers is 7 yards. Find the radius of each circle if their combined area is 29pi square yards.

Answer 1

Jim,

Here's one approach... given that the two circles are tangent and the distance between their respective centers is 7 yards, then: r₁ + r₂ = 7. Then, given that their combined areas equal 29π, (r₁)²π + (r₂)²π = 29π.

 

So let's solve for r₁. Since, (r₁)²π + (r₂)²π = 29π, let's first get rid of π by dividing it out across the entire equation to give: (r₁)² + (r₂)² = 29. Next, since r₁ + r₂ = 7, r₂ = 7 - r₁ so let's substitute 7 - r₁ for r₂.

 

Now, we have:

 

(r₁)² + (7 - r₁)² = 29                    FOIL the (7 - r₁)²     

(r₁)² + 49 - 14r₁ + (r₁)² = 29      Rearrange and consolidate like terms: (r₁)² + (r₁)²

2r₁² - 14r₁ + 49 = 29                   Subtract 29 from both sides to try and factor

2r₁² - 14r₁ + 20 = 0                     (2r₁ - 10)(r₁ - 2) = 0

 

So, r₁ = 5 or r₁ = 2                       Both are viable radii lengths, so try substituting both into r₁ + r₂ = 7

 

5 + r₂ = 7, therefore r₂ = 2, or  2 + r₂ = 7, therefore r₂ = 5

 

Notice that they are interchangeable, so one radius is length 2 yards and the other radius is length 5 yards.

 

Hope this helps!