Answer:
Explanation:
A) To find an expression for the radius of the circle, we can use the formula for the area of a circle, which is:
A = πr^2
where A is the area and r is the radius. We are given that the area is:
A = 4πx^2 + 12πx + 9π
We can equate this expression with the formula for the area of a circle and solve for the radius:
4πx^2 + 12πx + 9π = πr^2
Simplifying the right-hand side:
πr^2 = π(x^2 + 3x/2 + 9/4)
πr^2 = π(x + 3/2)^2
r^2 = (x + 3/2)^2
r = ± (x + 3/2)
Since a radius can't be negative, we take the positive value:
r = x + 3/2
Therefore, an expression for the radius of the circle is r = x + 3/2.
B) For the circle to exist, the radius must be a positive number. Therefore, we require:
x + 3/2 > 0
x > -3/2
So the least possible integer value of x for the circle to exist is -1. If x is -1, then the expression for the radius becomes:
r = -1 + 3/2 = 1/2
This is a valid radius since it is positive, and so the circle exists. If x is less than -1, the radius becomes negative, which is not possible, so the circle doesn't exist. Therefore, the least possible integer value of x for the circle to exist is -1.
hope this helps.