Question

Mr. Adams has a circular flower bed with a

diameter of 2 feet. He wishes to increase the size of this bed so that it will have nine times as much planting area. What must be the diameter of the new bed?
(F) 6 feet
(G) 8 feet
(H) 12 feet
(J) 16 feet
(K) none of these

Please Help!!!

Answer 1

Answer: (F) 6 feet

Explanation:

The area of a circle is proportional to the square of its radius. So if we increase the diameter of the flower bed by a factor of x, its area will increase by a factor of x^2. We want to find the diameter of the new bed that will have nine times the planting area of the old bed.

Let d be the diameter of the new bed. Then its radius is r = d/2, and its area is A = πr^2. We want:

9A = π(D/2)^2

where D is the diameter of the old bed. Substituting r = D/2, we get:

9A = π(D/2)^2

= πr^2

So we can solve for d by equating the right-hand sides and taking the square root:

9πr^2 = πd^2

d^2 = 9r^2

d = 3r

Therefore, the diameter of the new bed must be three times the diameter of the old bed. The diameter of the old bed is 2 feet, so the diameter of the new bed is 3 times that, or 6 feet.

Answer 2

The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius. The diameter of the original flower bed is 2 feet, which means the radius is 1 foot. The area of the original flower bed is:

A = πr^2
A = π(1^2)
A = π

To increase the planting area by a factor of 9, we need a new area of:

9A = 9π

To find the radius of the new flower bed, we can use the formula for the area of a circle again:

9π = πr^2
9 = r^2
r = 3

So the radius of the new flower bed is 3 feet, which means the diameter is:

D = 2r = 2(3) = 6 feet

Therefore, the diameter of the new bed is 6 feet, which corresponds to answer choice (F).