Question

What is an expression for the radius of the circle? What is the least possible integer value of x for the circle to exist? Explain

Answer 1

450

Explanation:

calculate it by solving the radius as you get the radius you will be able to do th question

Answer 2

The area of a circle is given by the next formula:

`A=\pi *r^2`

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For the given expression of area:

`\pi *r^2=4\pi x^2+12\pi x+9\pi`

Use the equation above to solve r (radius):

1. Divide both sides of the equation into π

`\begin{gathered} (\pi *r^2)/(\pi)=(4\pi x^2+12\pi x+9\pi)/(\pi) \\ \\ r^2=4x^2+12x+9 \end{gathered}`

2. Factor the expression on the right using the notable product perfect square binomial:

`\begin{gathered} (a+b)\placeholder{⬚}^2=a^2+2ab+b^2 \\ \\ 4x^2+12x+9=(2x)\placeholder{⬚}^2+2(2x)(3)+3^2 \\ 4x^2+12x+9=(2x+3)\placeholder{⬚}^2 \\ \\ \\ r^2=(2x+3)\placeholder{⬚}^2 \end{gathered}`

3. Find square root of both sides of the equation:

`\begin{gathered} √(r^2)=\sqrt{(2x+3)\placeholder{⬚}^2} \\ \\ r=2x+3 \end{gathered}`

Then, the radius is given by the expression 2x+3.

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To find the least possible integer value of x: As r is the radius of a circle, it cannot be a negative amount or 0, then r needs to be greather than 0:

`\begin{gathered} 2x+3>0 \\ \end{gathered}`

Solve the ineqaulity above:

`\begin{gathered} 2x+3-3>0-3 \\ 2x>-3 \\ \\ (2x)/(2)>-(3)/(2) \\ \\ x>-(3)/(2) \end{gathered}`

x needs to be greater than -3/2 (-1.5).

Then, the least possible integer value of x is -1