If the radius of a circle is decreased by 6, the old area is 9 times the new area. What was the original radius?

Question

Write equations and factoe

Answer 1

The formula for the Area of a circle is ∏r².

 

Let's call the Area of the original circle A.

 

If ∏r² is the Area of the original circle, then you need to show the new radius for the smaller circle as r - 6, becuase it decreased by 6.

 

The formula for the area of the new circle will be ∏(r-6)² which will be 9 times smaller than ∏r² or

∏r² = 9 * ∏(r-6)²

 

Next divide both sides of the equation by pi to cancel it out, ad you are left with r² = 9 * (r-6)²

 

Next take the square root of both sides of the equation:

√(r² ) = √[9 (r-6)²]

 

and you get √(r*r ) = √[3*3(r-6)(r-6)]

 

which is r = 3(r-6)

 

Next divide both sides of the eqauation by 3 to get
r / 3 = r - 6, so you know that one-third of r is the same as when you take 6 away from r.

Your job then is to figure out what r is that when you divide it by 3 you get the same answer as when you subtract 6 from it, and then plug the answer back into the 2 original equations to make sure that ∏r² truely is 9 times greater than ∏(r-6)².

 

 

Answer 2

 

let r be the original radius, so:

 

9∏(r - 6)² = ∏r²

 

divide both sides by ∏

 

9(r - 6)² = r²

 

9(r² - 12r + 36) = r²

 

9r² - 108r + 324 = r²

 

subtract r² from both sides:

 

8r² - 108 + 324 = 0

 

Factor out the Greatest Common Factor (GCF), '4'.
4(81 + -27r + 2r²)

Factor a trinomial.
4[(9 + -2r)(9 + -1r)]

Final result:
4(9 + -2r)(9 + -1r)

 

so radius r = 9

 

You're right Byron. Thanks.

 

Answer 3

The formula for the area of a circle is A = πr², where r is the radius of the circle. The equation suggested by the sentence you're given is:

(old area) = 9 * (new area with 6 smaller radius)

 

We can substitute the area formula in so we get something we can solve.

 

πr² = 9 π(r-6)²

 

Conveniently, you can divide both sides by π so that you're left only with friendly integers.

 

r² = 9(r-6)²

 

To solve, you'll need to multiply out the right side first. Then, subtract r² from both sides so that you get the quadratic = 0. Finally, factor the quadratic and solve as you've done before.

 

If you have more questions, please comment and ask!