MHB How Much Should the Radius Increase to Enlarge the Circle's Area by b Units?

[mathdad]

The radius of a circle is r units. By how many units should the radius be increased so that the area increases by b units?

The information in this problem tells me that using the area of a circle formula is needed.

A = pi•r^2

I think b should be added to r and squared.

A = pi(r + b)^2

Can someone provide a one or two hints?

 

[Opalg]

The radius of a circle is r units. By how many units should the radius be increased so that the area increases by b units?

The information in this problem tells me that using the area of a circle formula is needed.

A = pi•r^2

I think b should be added to r and squared.

A = pi(r + b)^2

Can someone provide a one or two hints?

You certainly should not add $b$ to $r$, because $b$ is an area and $r$ is a distance. Adding a two-dimensional quantity to a one-dimensional quantity does not make sense.

What you want to know is how much to increase $r$ so that $\pi r^2$ becomes $\pi r^2 + b$. Suppose that $x$ is the amount that has to be added to $r$. Then the equation is $\pi(r+x)^2 = \pi r^2 + b$. So you need to solve that equation for $x$.

 

[mathdad]

You certainly should not add $b$ to $r$, because $b$ is an area and $r$ is a distance. Adding a two-dimensional quantity to a one-dimensional quantity does not make sense.

What you want to know is how much to increase $r$ so that $\pi r^2$ becomes $\pi r^2 + b$. Suppose that $x$ is the amount that has to be added to $r$. Then the equation is $\pi(r+x)^2 = \pi r^2 + b$. So you need to solve that equation for $x$.

I understand what you are saying.

π(r + x)^2 = A + b

π(r + x)^2 = πr^2 + b

(r + x)^2 = (πr^2 + b)/π

sqrt{(r + x)^2} = sqrt{(πr^2 + b)/π}

r + x = sqrt{(πr^2 + b)/π}

x = sqrt{(πr^2 + b)/π} - r

Correct?

 

[Opalg]

π(r + x)^2 = A + b

π(r + x)^2 = πr^2 + b

(r + x)^2 = (πr^2 + b)/π

sqrt{(r + x)^2} = sqrt{(πr^2 + b)π}

r + x = sqrt{(πr^2 + b)π}

x = sqrt{(πr^2 + b)π} - r

Correct?

The method is correct, but in the last three lines you should have $(\pi r^2 + b)/\pi$ instead of $(\pi r^2 + b)\pi$.

 

[mathdad]

The method is correct, but in the last three lines you should have $(\pi r^2 + b)/\pi$ instead of $(\pi r^2 + b)\pi$.

I just forgot to include the slash symbol in the last 3 lines. It has now been edited.